Abstract

The properties of injection-locking chaos synchronization and communication in closed-loop external-cavity semiconductor lasers (ECSL) subject to phase-conjugate feedback (PCF) are investigated systematically. We theoretically analyze the general conditions for the injection-locking, and numerically investigate the properties of injection-locking chaos synchronization in the phase and intensity domains, the influences of frequency detuning and intrinsic parameter mismatch on the injection-locking chaos synchronization, as well as the performance of injection-locking chaos synchronization-based communication in closed-loop PCF-ECSL systems. The numerical results demonstrate that with respect to the conventional optical feedback (COF) scenario, the injection-locking chaos synchronization in a PCF-ECSLs configuration shows a significantly wider high-quality synchronization region and excellent feasibility, and the performance of chaos communication can also be enhanced.

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

1. Introduction

Chaos communication has attracted great attention for its advantage of physical-layer security enhancement [13]. Due to the external optical feedback-induced rich dynamic behaviors, external cavity semiconductor lasers (ECSLs) can easily generate wideband noise-like chaotic signal with excellent complexity [46], as such ECSLs are extensively applied as chaotic carrier sources for all-optical chaos communication. The experimental demonstrations of ECSLs-based chaos communication in the commercial networks in Athens (Greece) and recent literatures have confirmed the feasibility of this technology [710].

In a chaos communication system, message is encoded onto (or into) the noise-like chaotic carrier and decoded in virtue of chaos synchronization. Therefore, the achievement of chaos synchronization between two ECSLs is the basis of chaos communication. Up to present, most of the investigations on ECSLs-based chaos synchronization and communication are concentrated on the unidirectional coupling and mutual coupling chaotic systems composed of ECSLs subject to conventional optical feedback (COF). Here the so-called COF means the feedback light is a linear replica of the output light with a time delay that is determined by the length of feedback loop [7,1113]. In the unidirectional coupling COF-ECSL systems, two types of chaos synchronization, namely the complete chaos synchronization and the generalized chaos synchronization, which are achieved respectively based on the mechanisms of symmetric operation and injection-locking effect, can be achieved [12,13]. In the mutual coupling COF-ECSL systems, the achievement of chaos synchronization is mainly based on the symmetric operation mechanism, and two types of synchronization referred to as the isochronal chaos synchronization and the leader-laggard chaos synchronization are achievable [1416]. On the other hand, there is another type of optical feedback called as phase-conjugate feedback (PCF) by using a phase-conjugate mirror as the external cavity reflector [1720]. Similar to the COF-ECSLs, ECSLs subject to PCF also show rich dynamic behaviors, such as the symmetry-breaking and restoring bifurcations, super-harmonic self-pulsation [17,19]. While with respect to the COF scenario, the PCF exhibits enhancements in the bandwidth and statistical complexity of chaos [20]. Regarding the chaos synchronization in PCF-ECSL systems, a brief demonstration of the complete chaos synchronization based on the symmetric operation mechanism is reported in [21]. However, with the best of our knowledge, the more practical injection-locking chaos synchronization in the PCF-ECSL system has never been studied, which motivates a thorough investigation on the theoretical conditions, the properties of mismatch robustness and detuning tolerance of the injection-locking chaos synchronization between PCF-ECSLs, as well as the performance of PCF-ECSL-based chaos communication.

In this paper, the achievement conditions and properties of the injection-locking chaos synchronization in PCF-ECSL system are systematically investigated. The paper is organized as follows. In Section 2, based on the well-known Lang-Kobayashi rate equation model, the theoretical conditions for the injection-locking chaos synchronization are derived. In Section 3, the phase locking characteristics, the influences of operation parameters on the quality of injection-locking chaos synchronization, as well as the mismatch robustness and the frequency detuning tolerance are studied thoroughly. Section 4 investigates the performance of chaos-based communication. Finally, a brief conclusion is given.

2. Architecture and theoretical analysis

Figure 1 shows the schematic diagram of the closed-loop PCF-ECSL system. It is composed of two ECSLs that are respectively referred to as the master semiconductor laser (MSL) and the slave semiconductor laser (SSL). Different from the COF-ECSL systems, here the MSL and SSL suffer external optical feedback from phase-conjugate mirrors (PCMs). With proper feedback strength, both MSL and SSL can easily work in chaotic regimes. Message is firstly encrypted onto the chaotic carrier outputted by MSL, and the modulated chaotic carrier (chaos + message) is then propagated to the receiver end through a fiber link unidirectionally and injected into SSL. Based on the injection-locking chaos synchronization between MSL and SSL, the message is decrypted by subtracting the output of SSL from the injection chaotic carrier.

 figure: Fig. 1.

Fig. 1. Schematic of the closed-loop PCF-ECSL system. MSL, Master semiconductor laser; SSL, slave semiconductor laser; PCM: phase-conjugate mirror, OC: optical coupler, OI: optical isolator, PS: phase shifter, M: modulator, PD: photodetector.

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For the numerical purpose, the modified Lang-Kobayashi rate equations that take the PCF terms into consideration are adopted to describe the dynamic behaviors of MSL and SSL. The rate equations for the MSL are written as [2022]:

$$\frac{{d{E_M}(t)}}{{dt}} = \frac{{1 + i\alpha }}{2}[{G_M}(t) - \frac{1}{{{\tau _p}}}]{E_M}(t) + {k_f}E^{\ast}_{M}(t - {\tau _f})\exp (i{\phi _{PCMM}}) + \sqrt {2\beta {N_M}(t)} {\chi _M}(t)$$
$$\frac{{d{N_M}(t)}}{{dt}} = \frac{I}{q} - \frac{{{N_M}(t)}}{{{\tau _e}}} - {G_M}(t){|{{E_M}(t)} |^2}$$
For the SSL, the rate equations are
$$\begin{aligned} \frac{{d{E_S}(t)}}{{dt}} = &\frac{{1 + i\alpha }}{2}[{G_S}(t) - \frac{1}{{{\tau _p}}}]{E_S}(t) + {k_f}E^{\ast}_{S}(t - {\tau _f})\exp (i{\phi _{PCMS}})\\ &+ {k_i}{E_M}(t - {\tau _i})\exp ( - i{\omega _M}{\tau _i})\exp (i\Delta \omega t) + \sqrt {2\beta {N_S}(t)} {\chi _S}(t) \end{aligned}$$
$$\frac{{d{N_S}(t)}}{{dt}} = \frac{I}{q} - \frac{{{N_S}(t)}}{{{\tau _e}}} - {G_S}(t){|{{E_S}(t)} |^2}$$
In these equations, the subscripts M and S stand for MSL and SSL, respectively. E(t) is the slowly-varying complex electronic field, N(t) is the carrier number in laser cavity. The optical gain in laser cavity is defined as G(t)=g[N(t)-N0]/[1+ɛ|E(t)|2], where g denotes the differential gain, N0 is the transparent carrier number, and ɛ is the saturation factor. The other parameters are the laser bias current I, the linewidth enhancement factor α, the electron charge q, the photon lifetime τp, the carrier lifetime τe, the spontaneous emission factor β, the operation frequency ω, and the detuning frequency Δω=ωM-ωS. χ is a Gaussian white noise with zero mean and unity variance, which is introduced here to model the spontaneous emission noise. The second terms in the right hand of Eqs. (1) and (3) denote the phase-conjugate feedback of MSL and SSL, respectively. Therein, kf is the feedback strength, τf is the feedback delay and ϕPCM is the phase change of PCM. It is worth mentioning that the PCF term kfE*(t-τf)exp(iϕPCM) is different from the COF term kfE(t-τf)exp(-iωτf). Except the conjugation in the amplitude of feedback electronic field, the PCF term does not include the phase term exp(-iωτf) because of the properties of phase-conjugate field [20]. Moreover, the third term in the right hand of Eq. (3) is the injection term, wherein ki is the injection strength and τi is the corresponding flight time from MSL to SSL.

To analyze the injection-locking properties in both of intensity and phase domains, the rate equations are further developed as follows, by substituting E(t)=A(t)exp[jϕ(t)] into Eqs. (1)–(4).

$$\begin{aligned} \frac{{d{A_{M,S}}(t)}}{{dt}} = &\frac{1}{2}[{G_{M,S}}(t) - \frac{1}{{{\tau _p}}}]{A_{M,S}}(t) + {k_f}{A_{M,S}}(t){A_{M,S}}(t - {\tau _f})\cos [{\theta _{fM,fS}}(t)]\\ &+ {k_i}{A_S}(t){A_M}(t - {\tau _i})\cos [{\theta _i}(t)] \end{aligned}$$
$$\begin{aligned} \frac{{d{\phi _{M,S}}(t)}}{{dt}} = &\frac{\alpha }{2}[{G_{M,S}}(t) - \frac{1}{{{\tau _p}}}] - {k_f}\frac{{{A_{M,S}}(t - {\tau _f})}}{{{A_{M,S}}(t)}}\sin [{\theta _{fM,fS}}(t)]\\ &- {k_i}\frac{{{A_M}(t - {\tau _i})}}{{{A_S}(t)}}\sin [{\theta _i}(t)] \end{aligned}$$
$$\frac{{d{N_{M,S}}(t)}}{{dt}} = \frac{I}{q} - \frac{{{N_{M,S}}(t)}}{{{\tau _e}}} - {G_{M,S}}(t){|{{A_{M,S}}(t)} |^2}$$
$${\theta _{fM,fS}}(t) = {\phi _{M,S}}(t) + {\phi _{M,S}}(t - {\tau _f}) - {\phi _{PCMM,PCMS}}$$
$${\theta _i}(t) = {\omega _M}{\tau _i} + {\phi _S}(t) - {\phi _M}(t - {\tau _i}) - \Delta \omega t$$
In the equations, A(t) is the amplitude of slowly-varying complex electronic field, ϕ(t) is the corresponding phase, θf (t) and θi(t)) are the phase changes induced by feedback and injection, respectively. It is worth mentioning that when the output of SSL is locked to that of MSL, the effect of the spontaneous emission noise is neglectable, as such for the sake of simplicity, the spontaneous emission noise terms are not considered here.

In the master-slave lasers system, when the slave laser is locked to the master laser, the relationship between the phases of them can be described as [23,24]

$${\phi _S}(t) - {\phi _M}(t) = \Delta \omega t + {\phi _L},$$
where ϕL is the locking phase difference between the master laser and the slave laser, and it is determined by [24]
$${\phi _L} = {\sin ^{ - 1}}\{ - \frac{{\Delta \omega }}{{{k_i}\sqrt {1 + {\alpha ^2}} }}\} - {\tan ^{ - 1}}\alpha .$$
Moreover, the difference of feedback phase changes of MSL and SSL should be zero, namely
$${\phi _M}(t) + {\phi _M}(t - {\tau _f}) - {\phi _{PCMM}} = {\phi _S}(t) + {\phi _S}(t - {\tau _f}) - {\phi _{PCMS}}$$
which can be rewritten as
$${\phi _M}(t) - {\phi _S}(t) + {\phi _M}(t - {\tau _f}) - {\phi _S}(t - {\tau _f}) - {\phi _{PCMM}} + {\phi _{PCMS}} = 0$$
Substituting Eqs. (10) and (11) into Eq. (12), we get
$$\textrm{2}{\phi _L} + 2\Delta \omega t - \Delta \omega {\tau _f} - \Delta {\phi _{PCM}} = 0$$
where ΔϕPCM=ϕPCMS-ϕPCSM is the phase change difference of PCMs. This is the essential condition for the achievement of injection-locking chaos synchronization in the closed-loop PCF-ECSL system. Since the values of ϕL, τf, and ΔϕPCM are constant, this condition is difficult to be satisfied, unless Δω=0 and ΔϕPCM=2ϕL. Therefore, it is necessary to add an additional phase shifter (PS) in the feedback loop of SSL to modify the feedback phase, and the phase change should meet
$${\phi _{PS1}} = \textrm{2}{\phi _L} + 2\Delta \omega t - \Delta \omega {\tau _f} - \Delta {\phi _{PCM}}$$

On the other hand, the phase shift can also be added in the injection light. In this case, the injection term in Eq. (3) should be modified as kiEM(t-τi)exp(-Mτi)exp(iΔωt)exp(iϕPS2). Under such a scenario, the injection-locking phase relation in Eq. (10) should be modified as

$${\phi _S}(t) - {\phi _M}(t) = \Delta \omega t + {\phi _{PS2}} + {\phi _L}.$$
Subsequently, with the aforementioned analysis method, the phase change induced by the PS should meet
$${\phi _{PS2}} ={-} {\phi _L} - \Delta \omega t + \frac{1}{2}\Delta \omega {\tau _f} + \frac{1}{2}\Delta {\phi _{PCM}}$$
To obtain a phase change in Eq. (15), a phase modulator that is driven by a periodic sawtooth signal with a frequency of Δω/π can be added in the feedback loop of SSL to act as the PS1. Similarly, a phase modulator driven by a periodic sawtooth signal with a frequency of Δω/2π can be added in the injection link to act as the PS2 for the second scenario defined by Eq. (17). In practice, the frequency detuning can be eliminated by properly tuning the operation temperatures of MSL and SSL, and then only a fixed optical phase shifter that provides a constant phase shift of ϕPS1=2ϕLϕPCM or ϕPS2=-ϕLϕPCM/2, can guarantee the phase locking conditions in Eqs. (15) and (17).

To quantitatively measure the synchronization quality between MSL and SSL, we adopt the cross-correlation function, which is defined as [2527]

$$\rho (\tau )= \frac{{\left\langle {\left[ {{P_M}({t - \tau } )- \left\langle {{P_M}({t - \tau } )} \right\rangle } \right]\left[ {{P_S}(t )- \left\langle {{P_S}(t )} \right\rangle } \right]} \right\rangle }}{{\sqrt {\left\langle {{{\left[ {{P_M}({t - \tau } )- \left\langle {{P_M}({t - \tau } )} \right\rangle } \right]}^2}} \right\rangle \left\langle {{{\left[ {{P_S}(t )- \left\langle {{P_S}(t )} \right\rangle } \right]}^2}} \right\rangle } }}$$
where 〈·〉 means the time averaging, P=|A(t)|2 is the photon number in the laser cavity, and τ is the varying time shift.

In our simulations, the rate equations are solved by using the fourth-order Runge-Kutta algorithm. Unless otherwise stated, the frequency detuning between MSL and SSL is set as Δω=0, and the values of intrinsic parameters of MSL and SSL are identical, which are chosen to be those reported in [28]: the operation wavelength λ=1550 nm, the linewidth enhancement factor α=5, the gain saturation coefficient ɛ=5×10−7, the carrier lifetime τe=2 ns, the photon lifetime τp=2 ps, the differential gain coefficient g=1.5×10−8 ps−1, the transparency carrier density N0 = 1.5×108, the spontaneous emission factor β=1.5×10−6 ns−1, and the electron charge q=1.602×10−19 C. The bias currents of MSL and SSL are fixed at I=2Ith, where Ith=14.7 mA is the threshold current, the feedback strength is fixed at kf = 15 ns−1 and the feedback delay is set at τf= 3ns. For the sake of simplicity, the injection delay from MSL to SSL is set as τi=0 ns, the phase change difference of PCMs is set as 0 rad. In the following sections, we numerically investigate the properties of injection-locking chaos synchronization and communication in the closed-loop PCF-ECSL system.

3. Results and analysis

We first investigate the phase locking phenomenon induced by the injection from MSL to SSL. When the emission frequency of SSL is locked to that of MSL, the phase relationship between them is determined by Eq. (10). Here we calculate the average phase difference Δϕ=ϕS(t)-ϕM(t)-Δωt〉 and compare it with the theoretical locking phase difference defined in Eq. (11). Figure 2 shows the simulation and analytical results for three exemplary frequency detuning cases with Δfω/2π) = 1GHz, 0, and −1GHz, as a function of injection strength ki. It is obvious that the variations of numerical phase difference Δϕ in all the three cases well agree with those of the analytical locking phase difference ϕL. That is, stable phase locking for chaotic light injection is easily achieved. For the case of zero detuning, the locking phase difference keeps at 0.563π which precisely equals to the theoretical results calculated by Eq. (11). While for the nonzero frequency detuning cases, the locked phase asymptotically gets closer and closer to that of zero tuning case, as the increase of injection strength. This phenomenon qualitatively agrees with that occurring in the open-loop semiconductor laser system with COF [24]. On the other hand, the variations of the cross correlation between MSL and SSL versus the tuning phase shift ϕPS in the feedback loop and the injection link are presented in Fig. 2(b). It is indicated that for the case with PS in feedback loop, high-quality chaos synchronization only occurs when the value of ϕPS1=1.126π which is twice of the steady state locking phase ϕL. For the case with PS in the injection link, high-quality chaos synchronization occurs when the value of ϕPS2=0.437π (π-ϕL) and 1.437π (-ϕL). These phenomena are perfectly in line with the theoretical conditions in Eqs. (15) and (17). It is worth noting that due to the synchronization quality is quantified by using Eq. (18) with τ=0 to calculate the cross-correlation coefficient between the output intensities of MSL and SSL, the phase correlation between the outputs of MSL[EM(t)) and SSL (ES(t)] is ambiguous. In fact, for the case of ϕPS2=0.437π, there is a phase difference of π between EM(t) and ES(t), this also agrees with the analytical results in Eq. (16). That is, under such a scenario, the synchronization between MSL and SSL is inverse-phase synchronization. Since the results of intensity chaos synchronization for the cases of ϕPS2=0.437π and ϕPS2=1.437π are quite similar, for the sake of simplicity, ϕPS2 is fixed at 1.437π and ϕPS1 is fixed at 1.126π in the following investigations on the injection-locking intensity chaos synchronization and chaotic communication in PCF-ECSLs.

 figure: Fig. 2.

Fig. 2. (a) Phase-locking characteristic versus the variation of ki in the closed-loop PCF-ECSLs system, (b) variation of the synchronization quality as a function of tuning phase difference ϕPS (normalized in [0∼2π]) with zero frequency detuning in SSL. Lines in (a) denote the analytical results of the phase ϕL defined in Eq. (11), while the marks stand for the numerical simulation results for Δϕ.

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Figure 3 shows the cross correlation between MSL and SSL in the parameter space of the injection strength and the feedback strength. For the purpose of comparison, the result for the closed-loop COF-ECSL configuration with identical parameters is also presented here. For a fixed injection, the synchronization quality is gradually degraded as the increase of feedback strength. This is because a stronger feedback shows stronger affection on the output intensity of SSL, which would comparatively weaken the affection of injection and consequently increase the difficulty of the injection-locking in intensity. For this reason, in the stronger feedback cases, stronger injection is necessary to obtain high-quality chaos synchronization. This phenomenon can be found in all cases of PCF and COF. The results for both PS cases shown in Figs. 3(a) and 3(b) are similar, and the high-quality synchronization regions where the cross-correlation coefficients are larger than 0.95 in the PCF-ECSL configurations are significantly broader than that of the COF-ECSL scenario shown in Fig. 3(c). That is, in the closed-loop PCF-ECSL system, high-quality intensity chaos synchronization is easier to be achieved with respect to that in the COF-ECSL system.

 figure: Fig. 3.

Fig. 3. Distributions of injection-locking synchronization quality as a function of the injection strength and feedback strength, for the cases of PCF with (a) ϕPS1, (b) ϕPS2, and (c) COF.

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Next, we discuss the practical feasibility of the injection-locking chaos synchronization in the PCF-ECSL system. Figure 4(a) shows the influence of the frequency detuning between MSL and SSL on the quality of injection-locking chaos synchronization. Here the results for the PCF configurations without tuning PS (diamond) in the feedback loop and injection link, as well as that for the COF-ECSL configuration (circle) are simultaneously presented for the purpose of comparison. Obviously, without the consideration of phase shift (diamond), it is unable to obtain high quality chaos synchronization in the PCF-ECSL system, even when the frequency detuning is zero. This is because without the tuning phase shift, the phase locking cannot be achieved, let alone to obtain injection-locking chaos synchronization in intensity. For the PCF case with a PS in the feedback loop (square), high-quality chaos synchronization can be achieved and it can tolerate a frequency detuning of several tens of GHz. The detuning tolerance characteristic in such case is similar to that of COF configuration (circle). For the PCF case with a PS in the injection (triangle), due to the phase change in the injection signal, there is a phase difference between the injection and the feedback of SSL, which results in that the quality of the injection-locking chaos synchronization is asymptotically degraded as the increase of frequency detuning. Nevertheless, high-quality chaos synchronization with a cross-correlation coefficient larger than 0.95 can be preserved in the frequency detuning range from −10 GHz to 10 GHz.

 figure: Fig. 4.

Fig. 4. Synchronization quality between MSL and SSL as functions of the frequency detuning (a), and the intrinsic parameters mismatch (b) in PCF and COF closed-loop schemes. The injection strength is set as 80ns−1.

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On the other hand, the intrinsic parameter mismatch robustness property of the PCF-ECSL and COF-ECSL configurations are presented in Fig. 4(b). Here, the mismatch is introduced according to the method reported in [15,29]: the intrinsic parameters for MSL are fixed, while the SSL’s internal parameters α, g and τp are decreased, and accordingly the parameters N0, s and τe are increased by the same amount (ratio). It is indicated that the quality variation of chaos synchronization versus the mismatch in the PCF-ECSL configuration is similar to that of COF-ECSL configuration. Although the synchronization quality is degraded gradually as the increase of mismatch ratio, the injection-locking chaos synchronization in the closed-loop PCF-ECSL system is robust to a relatively large mismatch up to a few tens of percentages. The cross-correlation coefficient between the intensities of MSL and SSL can be maintained larger than 0.95 in the mismatch range from −20% to 30%. Overall, high-quality injection-locking chaos synchronization in closed-loop PCF-ECSL system is feasible, as long as the analytical conditions in Eqs. (15) and (17) are satisfied.

Finally, we turn to investigate the performance of chaos communication in the closed-loop PCF-ECSL system. The message is encrypted in the chaotic carrier by the method of chaos modulation [28], which can be mathematically described as Emod(t)=EM(t)[1+Am(t)], where Emod(t) denotes the modulated chaotic carrier, the message m(t) is a random binary sequence, and Am=0.07 is the modulation index, it is small enough to guarantee the message being efficiently hidden in the chaotic carrier. The message decryption is carried out by the way of direct subtraction decoding, which is described as m'(t)=LPF[|ELmod(t)|2-|ES(t)|2], where ELmod(t) denotes the modulated chaotic carrier after fiber link transmission from MSL to SSL, and the LPF operation uses a five-order Butterworth low-pass-filter with a cutoff frequency equal to the message bit rate. Figure 5 illustrates the message transmission from MSL to SSL, through a 60km dispersion-shifted fiber (DSF) link with parameters identical to that in [30]. Here we considered two transmission cases with different bit rates of 2.5Gbit/s and 5Gbit/s, respectively. It is demonstrated that the messages encrypted in the chaotic carrier of MSL can be successfully decrypted by SSL.

 figure: Fig. 5.

Fig. 5. Illustrations of chaos communication in the closed-loop PCF-ECSL system for the cases of PCF with ϕPS1 (first row) and ϕPS2 (second row). The dark curves denote the original messages, while the red and blue ones denote the decrypted messages. The bit rates for message transmissions shown in the first and second columns are 2.5 Gbit/s and 5 Gbit/s, respectively. The horizontal axes have been properly shifted.

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To further investigate the chaos communication performance of the PCF-ECSL system, Fig. 6(a) shows the influence of the message bit rate on the Q-factor of the decrypted message. Here the Q-factor is calculated by Q=(P1-P0)/(σ1-σ0), where P1 and P0 are the average powers of bits “1” and “0”, respectively, and σ1 and σ0 are the corresponding standard deviations [3032]. The results (square and triangle curves) indicate that acceptable communication performance with a Q-factor greater than 5 can be maintained in the PCF-ECSL system, when the bit rate is lower than 6.5Gbit/s, even though the Q-factor is gradually degraded as the bit rate increases. While for the COF configuration (circle curve), identical acceptable communication performance requires that the message bit rate should be lower than 5.5Gbit/s. Comparison indicates that, with respect to the COF-ECSL configuration, the PCF-ECSL system can provide a larger Q-factor in the case of identical bit rate transmission, and supports a higher transmission capacity for identical Q-factor requirement. Based on this, we can conclude that the communication performance of PCF-ECSL system is enhanced with respect to the COF-ECSL configuration. The improvement of communication performance is intuitively attributed to the effective bandwidth enhancement effect of PCF. Figure 6(b) shows the effective bandwidth of chaotic carriers generated by COF-ECSL and PCF-ECSL, as a function of the feedback strength. Here the definition of effective bandwidth of chaotic carrier is identical to those in [6,20]. It is apparently indicated that the effective bandwidth of chaotic carrier generated by PCF-ECSL is always lager than that of COF-ECSL. The results are well in line with those in [20]. It is well known that, the larger the bandwidth of carrier is, the lager the transmission capacity is. For this reason, as shown in Fig. 6(a), the transmission capacity of the chaotic communication in PCF-ECSL system is relatively improved with respect to that in COF-ECSL system. In addition, our repeating simulation results indicate that as long as the conditions Eqs. (15) and (17) are satisfied, high-quality injection chaos synchronization can be easily achieved, and the additional phase shifter do not influences the maximum transmission bit rate.

 figure: Fig. 6.

Fig. 6. (a) Comparison of communication performance (Q-factor) in the closed-loop PCF-ECSL and COF-ECSL configurations as a function of the bit rate of message, (b) effective bandwidth (GHz) of chaotic carrier generated by PCF-ECSL and COF-ECSL versus the strength of feedback.

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4. Conclusions

In summary, we analytically and numerically investigate the properties of the injection-locking chaos synchronization and communication in the closed-loop PCF-ECSL system. By analyzing the phase relationship between MSL and SSL, we have derived the essential conditions for the achievement of injection-locking synchronization, the injection-locking conditions indicate that an additional phase shift is necessary to be added in the feedback loop of SSL or in the injection from MSL to SSL. The investigations on the properties of the injection-locking chaos synchronization in PCF-ECSL system demonstrate that with respect to the COF-ECSL configuration, high-quality chaos synchronization can be more easily achieved in PCF-ECSL system, and it shows practical feasible frequency tolerance and mismatch robustness properties. In addition, comparing with that in the COF-ECSL system, the performance of injection-locking synchronization-based chaos communication in PCF-ECSL system is also improved to some extent.

Funding

National Natural Science Foundation of China (616171119); Fundamental Research Funds for the Central Universities (ZYGX2019J003).

Disclosures

The authors declare no conflicts of interest related to this article.

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14. S. Y. Xiang, A. J. Wen, and W. Pan, “Synchronization regime of star-type laser network with heterogeneous coupling delays,” IEEE Photonics Technol. Lett. 28(18), 1988–1991 (2016). [CrossRef]  

15. N. Jiang, W. Pan, B. Luo, L. Yan, S. Xiang, L. Yang, D. Zheng, and N. Li, “Properties of leader-laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81(6), 066217 (2010). [CrossRef]  

16. J. G. Wu, Z. M. Wu, G. Q. Xia, T. Deng, X. D. Lin, X. Tang, and G. Feng, “Isochronous synchronization between chaotic semiconductor lasers over 40-km fiber links,” IEEE Photonics Technol. Lett. 23(24), 1854–1856 (2011). [CrossRef]  

17. B. Krauskopf, G. R. Gray, and D. Lenstra, “Semiconductor laser with phase-conjugate feedback: Dynamics and bifurcations,” Phys. Rev. E 58(6), 7190–7197 (1998). [CrossRef]  

18. J. S. Lawrence and D. M. Kane, “Contrasting conventional optical and phase-conjugate feedback in laser diodes,” Phys. Rev. A 63(3), 033805 (2001). [CrossRef]  

19. É. Mercier, D. Wolfersberger, and M. Sciamanna, “High-frequency chaotic dynamics enabled by optical phase-conjugation,” Sci. Rep. 6(1), 18988 (2016). [CrossRef]  

20. D. Rontani, É. Mercier, D. Wolfersberger, and M. Sciamanna, “Enhanced complexity of optical chaos in a laser diode with phase-conjugate feedback,” Opt. Lett. 41(20), 4637–4640 (2016). [CrossRef]  

21. A. Locquet, F. Rogister, M. Sciamanna, and P. Megret, “Synchronization of chaotic semiconductor lasers with phase-conjugate feedback,” in Pacific Rim Conference on Lasers and Electro-Optics, II-388–II-389 (2001).

22. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]  

23. F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985). [CrossRef]  

24. A. Murakami, “Phase locking and chaos synchronization in injection-locked semiconductor lasers,” IEEE J. Quantum Electron. 39(3), 438–447 (2003). [CrossRef]  

25. N. Li, R. M. Nguimdo, A. Locquet, and D. S. Citrin, “Enhancing optical-feedback-induced chaotic dynamics in semiconductor ring lasers via optical injection,” Nonlinear Dyn. 92(2), 315–324 (2018). [CrossRef]  

26. M. F. Xu, W. Pan, S. H. Xiang, and L. Zhang, “Cluster synchronization in symmetric VCSELs networks with variable-polarization optical feedback,” Opt. Express 26(8), 10754–10761 (2018). [CrossRef]  

27. N. Jiang, A. K. Zhao, S. Q. Liu, C. P. 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]  

28. D. Kanakidis, A. Argyris, A. Bogris, and D. Syvridis, “Influence of the decoding process on the performance of chaos encrypted optical communication systems,” J. Lightwave Technol. 24(1), 335–341 (2006). [CrossRef]  

29. A. Bogris, P. Rizomiliotis, K. E. Chlouverakis, A. Argyris, and D. Syvridis, “Feedback phase in optically generated chaos: A sceret key for cryptographic applications,” IEEE J. Quantum Electron. 44(2), 119–124 (2008). [CrossRef]  

30. D. Kanakidis, A. Bogris, A. Argyris, and D. Syvridis, “Numerical investigation of fiber transmission of a chaotic encrypted message using dispersion compensation schemes,” J. Lightwave Technol. 22(10), 2256–2263 (2004). [CrossRef]  

31. N. Li, W. Pan, S. Xiang, B. Luo, L. Yan, and X. Zou, “Hybrid chaos-based communication system consisting of three chaotic semiconductor ring lasers,” Appl. Opt. 52(7), 1523–1530 (2013). [CrossRef]  

32. N. Jiang, A. K. Zhao, Y. J. Wang, S. Q. Liu, J. M. Tang, and K. Qiu, “Security-enhanced chaotic communications with optical temporal encryption based on phase modulation and phase-to-intensity conversion,” OSA Continuum 2(12), 3422–3437 (2019). [CrossRef]  

References

  • View by:

  1. C. R. Mirasso, P. Colet, and P. García-Fernández, “Synchronization of chaotic semiconductor lasers: Application to encoded communications,” IEEE Photonics Technol. Lett. 8(2), 299–301 (1996).
    [Crossref]
  2. G. D. Vanwiggeren and R. Roy, “Communication with chaotic lasers,” Science 279(5354), 1198–1200 (1998).
    [Crossref]
  3. N. Jiang, A. K. Zhao, C. P. Xue, J. M. Tang, and K. Qiu, “Physical secure optical communication based on private chaotic spectral phase encryption/decryption,” Opt. Lett. 44(7), 1536–1539 (2019).
    [Crossref]
  4. N. Li, W. Pan, A. Locquet, and D. S. Citrin, “Time-delay concealment and complexity enhancement of an external-cavity laser through optical injection,” Opt. Lett. 40(19), 4416–4419 (2015).
    [Crossref]
  5. D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–1891 (2009).
    [Crossref]
  6. N. Jiang, Y. Wang, A. Zhao, S. Liu, Y. Zhang, L. Chen, B. Li, and K. Qiu, “Simultaneous bandwidth-enhanced and time delay signature-suppressed chaos generation in semiconductor laser subject to feedback from parallel coupling ring resonators,” Opt. Express 28(2), 1999–2009 (2020).
    [Crossref]
  7. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 438(7066), 343–346 (2005).
    [Crossref]
  8. J. X. Ke, L. L. Yi, G. Q. Xia, and W. Hu, “Chaotic optical communications over 100-km fiber transmission at 30-Gb/s bit rate,” Opt. Lett. 43(6), 1323–1326 (2018).
    [Crossref]
  9. T. Deng, G. Q. Xia, L. P. Cao, J. G. Chen, X. D. Lin, and Z. M. Wu, “Bidirectional chaos synchronization and communication in semiconductor lasers with optoelectronic feedback,” Opt. Commun. 282(11), 2243–2249 (2009).
    [Crossref]
  10. C. Xue, N. Jiang, Y. Lv, C. Wang, G. Li, S. Lin, and K. Qiu, “Security-enhanced chaos communication with time-delay signature suppression and phase encryption,” Opt. Lett. 41(16), 3690–3693 (2016).
    [Crossref]
  11. J. Ohtsubo, “Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback,” IEEE J. Quantum Electron. 38(9), 1141–1154 (2002).
    [Crossref]
  12. X. F. Li, W. Pan, B. Luo, and D. Ma, “Mismatch robustness and security of chaotic optical communications based on injection-locking chaos synchronization,” IEEE J. Quantum Electron. 42(9), 953–960 (2006).
    [Crossref]
  13. Y. H. Hong, M. W. Lee, J. Paul, P. S. Spencer, and K. A. Shore, “Enhanced chaos synchronization in unidirectionally coupled vertical-cavity surface-emitting semiconductor lasers with polarization-preserved injection,” Opt. Lett. 33(6), 587–589 (2008).
    [Crossref]
  14. S. Y. Xiang, A. J. Wen, and W. Pan, “Synchronization regime of star-type laser network with heterogeneous coupling delays,” IEEE Photonics Technol. Lett. 28(18), 1988–1991 (2016).
    [Crossref]
  15. N. Jiang, W. Pan, B. Luo, L. Yan, S. Xiang, L. Yang, D. Zheng, and N. Li, “Properties of leader-laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81(6), 066217 (2010).
    [Crossref]
  16. J. G. Wu, Z. M. Wu, G. Q. Xia, T. Deng, X. D. Lin, X. Tang, and G. Feng, “Isochronous synchronization between chaotic semiconductor lasers over 40-km fiber links,” IEEE Photonics Technol. Lett. 23(24), 1854–1856 (2011).
    [Crossref]
  17. B. Krauskopf, G. R. Gray, and D. Lenstra, “Semiconductor laser with phase-conjugate feedback: Dynamics and bifurcations,” Phys. Rev. E 58(6), 7190–7197 (1998).
    [Crossref]
  18. J. S. Lawrence and D. M. Kane, “Contrasting conventional optical and phase-conjugate feedback in laser diodes,” Phys. Rev. A 63(3), 033805 (2001).
    [Crossref]
  19. É. Mercier, D. Wolfersberger, and M. Sciamanna, “High-frequency chaotic dynamics enabled by optical phase-conjugation,” Sci. Rep. 6(1), 18988 (2016).
    [Crossref]
  20. D. Rontani, É. Mercier, D. Wolfersberger, and M. Sciamanna, “Enhanced complexity of optical chaos in a laser diode with phase-conjugate feedback,” Opt. Lett. 41(20), 4637–4640 (2016).
    [Crossref]
  21. A. Locquet, F. Rogister, M. Sciamanna, and P. Megret, “Synchronization of chaotic semiconductor lasers with phase-conjugate feedback,” in Pacific Rim Conference on Lasers and Electro-Optics, II-388–II-389 (2001).
  22. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
    [Crossref]
  23. F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985).
    [Crossref]
  24. A. Murakami, “Phase locking and chaos synchronization in injection-locked semiconductor lasers,” IEEE J. Quantum Electron. 39(3), 438–447 (2003).
    [Crossref]
  25. N. Li, R. M. Nguimdo, A. Locquet, and D. S. Citrin, “Enhancing optical-feedback-induced chaotic dynamics in semiconductor ring lasers via optical injection,” Nonlinear Dyn. 92(2), 315–324 (2018).
    [Crossref]
  26. M. F. Xu, W. Pan, S. H. Xiang, and L. Zhang, “Cluster synchronization in symmetric VCSELs networks with variable-polarization optical feedback,” Opt. Express 26(8), 10754–10761 (2018).
    [Crossref]
  27. N. Jiang, A. K. Zhao, S. Q. Liu, C. P. 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]
  28. D. Kanakidis, A. Argyris, A. Bogris, and D. Syvridis, “Influence of the decoding process on the performance of chaos encrypted optical communication systems,” J. Lightwave Technol. 24(1), 335–341 (2006).
    [Crossref]
  29. A. Bogris, P. Rizomiliotis, K. E. Chlouverakis, A. Argyris, and D. Syvridis, “Feedback phase in optically generated chaos: A sceret key for cryptographic applications,” IEEE J. Quantum Electron. 44(2), 119–124 (2008).
    [Crossref]
  30. D. Kanakidis, A. Bogris, A. Argyris, and D. Syvridis, “Numerical investigation of fiber transmission of a chaotic encrypted message using dispersion compensation schemes,” J. Lightwave Technol. 22(10), 2256–2263 (2004).
    [Crossref]
  31. N. Li, W. Pan, S. Xiang, B. Luo, L. Yan, and X. Zou, “Hybrid chaos-based communication system consisting of three chaotic semiconductor ring lasers,” Appl. Opt. 52(7), 1523–1530 (2013).
    [Crossref]
  32. N. Jiang, A. K. Zhao, Y. J. Wang, S. Q. Liu, J. M. Tang, and K. Qiu, “Security-enhanced chaotic communications with optical temporal encryption based on phase modulation and phase-to-intensity conversion,” OSA Continuum 2(12), 3422–3437 (2019).
    [Crossref]

2020 (1)

2019 (2)

2018 (4)

2016 (4)

C. Xue, N. Jiang, Y. Lv, C. Wang, G. Li, S. Lin, and K. Qiu, “Security-enhanced chaos communication with time-delay signature suppression and phase encryption,” Opt. Lett. 41(16), 3690–3693 (2016).
[Crossref]

S. Y. Xiang, A. J. Wen, and W. Pan, “Synchronization regime of star-type laser network with heterogeneous coupling delays,” IEEE Photonics Technol. Lett. 28(18), 1988–1991 (2016).
[Crossref]

É. Mercier, D. Wolfersberger, and M. Sciamanna, “High-frequency chaotic dynamics enabled by optical phase-conjugation,” Sci. Rep. 6(1), 18988 (2016).
[Crossref]

D. Rontani, É. Mercier, D. Wolfersberger, and M. Sciamanna, “Enhanced complexity of optical chaos in a laser diode with phase-conjugate feedback,” Opt. Lett. 41(20), 4637–4640 (2016).
[Crossref]

2015 (1)

2013 (1)

2011 (1)

J. G. Wu, Z. M. Wu, G. Q. Xia, T. Deng, X. D. Lin, X. Tang, and G. Feng, “Isochronous synchronization between chaotic semiconductor lasers over 40-km fiber links,” IEEE Photonics Technol. Lett. 23(24), 1854–1856 (2011).
[Crossref]

2010 (1)

N. Jiang, W. Pan, B. Luo, L. Yan, S. Xiang, L. Yang, D. Zheng, and N. Li, “Properties of leader-laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81(6), 066217 (2010).
[Crossref]

2009 (2)

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–1891 (2009).
[Crossref]

T. Deng, G. Q. Xia, L. P. Cao, J. G. Chen, X. D. Lin, and Z. M. Wu, “Bidirectional chaos synchronization and communication in semiconductor lasers with optoelectronic feedback,” Opt. Commun. 282(11), 2243–2249 (2009).
[Crossref]

2008 (2)

Y. H. Hong, M. W. Lee, J. Paul, P. S. Spencer, and K. A. Shore, “Enhanced chaos synchronization in unidirectionally coupled vertical-cavity surface-emitting semiconductor lasers with polarization-preserved injection,” Opt. Lett. 33(6), 587–589 (2008).
[Crossref]

A. Bogris, P. Rizomiliotis, K. E. Chlouverakis, A. Argyris, and D. Syvridis, “Feedback phase in optically generated chaos: A sceret key for cryptographic applications,” IEEE J. Quantum Electron. 44(2), 119–124 (2008).
[Crossref]

2006 (2)

D. Kanakidis, A. Argyris, A. Bogris, and D. Syvridis, “Influence of the decoding process on the performance of chaos encrypted optical communication systems,” J. Lightwave Technol. 24(1), 335–341 (2006).
[Crossref]

X. F. Li, W. Pan, B. Luo, and D. Ma, “Mismatch robustness and security of chaotic optical communications based on injection-locking chaos synchronization,” IEEE J. Quantum Electron. 42(9), 953–960 (2006).
[Crossref]

2005 (1)

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

2004 (1)

2003 (1)

A. Murakami, “Phase locking and chaos synchronization in injection-locked semiconductor lasers,” IEEE J. Quantum Electron. 39(3), 438–447 (2003).
[Crossref]

2002 (1)

J. Ohtsubo, “Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback,” IEEE J. Quantum Electron. 38(9), 1141–1154 (2002).
[Crossref]

2001 (1)

J. S. Lawrence and D. M. Kane, “Contrasting conventional optical and phase-conjugate feedback in laser diodes,” Phys. Rev. A 63(3), 033805 (2001).
[Crossref]

1998 (2)

B. Krauskopf, G. R. Gray, and D. Lenstra, “Semiconductor laser with phase-conjugate feedback: Dynamics and bifurcations,” Phys. Rev. E 58(6), 7190–7197 (1998).
[Crossref]

G. D. Vanwiggeren and R. Roy, “Communication with chaotic lasers,” Science 279(5354), 1198–1200 (1998).
[Crossref]

1996 (1)

C. R. Mirasso, P. Colet, and P. García-Fernández, “Synchronization of chaotic semiconductor lasers: Application to encoded communications,” IEEE Photonics Technol. Lett. 8(2), 299–301 (1996).
[Crossref]

1985 (1)

F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985).
[Crossref]

1980 (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Annovazzi-Lodi, V.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Argyris, A.

A. Bogris, P. Rizomiliotis, K. E. Chlouverakis, A. Argyris, and D. Syvridis, “Feedback phase in optically generated chaos: A sceret key for cryptographic applications,” IEEE J. Quantum Electron. 44(2), 119–124 (2008).
[Crossref]

D. Kanakidis, A. Argyris, A. Bogris, and D. Syvridis, “Influence of the decoding process on the performance of chaos encrypted optical communication systems,” J. Lightwave Technol. 24(1), 335–341 (2006).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

D. Kanakidis, A. Bogris, A. Argyris, and D. Syvridis, “Numerical investigation of fiber transmission of a chaotic encrypted message using dispersion compensation schemes,” J. Lightwave Technol. 22(10), 2256–2263 (2004).
[Crossref]

Bogris, A.

Cao, L. P.

T. Deng, G. Q. Xia, L. P. Cao, J. G. Chen, X. D. Lin, and Z. M. Wu, “Bidirectional chaos synchronization and communication in semiconductor lasers with optoelectronic feedback,” Opt. Commun. 282(11), 2243–2249 (2009).
[Crossref]

Chen, J. G.

T. Deng, G. Q. Xia, L. P. Cao, J. G. Chen, X. D. Lin, and Z. M. Wu, “Bidirectional chaos synchronization and communication in semiconductor lasers with optoelectronic feedback,” Opt. Commun. 282(11), 2243–2249 (2009).
[Crossref]

Chen, L.

Chlouverakis, K. E.

A. Bogris, P. Rizomiliotis, K. E. Chlouverakis, A. Argyris, and D. Syvridis, “Feedback phase in optically generated chaos: A sceret key for cryptographic applications,” IEEE J. Quantum Electron. 44(2), 119–124 (2008).
[Crossref]

Citrin, D. S.

N. Li, R. M. Nguimdo, A. Locquet, and D. S. Citrin, “Enhancing optical-feedback-induced chaotic dynamics in semiconductor ring lasers via optical injection,” Nonlinear Dyn. 92(2), 315–324 (2018).
[Crossref]

N. Li, W. Pan, A. Locquet, and D. S. Citrin, “Time-delay concealment and complexity enhancement of an external-cavity laser through optical injection,” Opt. Lett. 40(19), 4416–4419 (2015).
[Crossref]

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–1891 (2009).
[Crossref]

Colet, P.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

C. R. Mirasso, P. Colet, and P. García-Fernández, “Synchronization of chaotic semiconductor lasers: Application to encoded communications,” IEEE Photonics Technol. Lett. 8(2), 299–301 (1996).
[Crossref]

Deng, T.

J. G. Wu, Z. M. Wu, G. Q. Xia, T. Deng, X. D. Lin, X. Tang, and G. Feng, “Isochronous synchronization between chaotic semiconductor lasers over 40-km fiber links,” IEEE Photonics Technol. Lett. 23(24), 1854–1856 (2011).
[Crossref]

T. Deng, G. Q. Xia, L. P. Cao, J. G. Chen, X. D. Lin, and Z. M. Wu, “Bidirectional chaos synchronization and communication in semiconductor lasers with optoelectronic feedback,” Opt. Commun. 282(11), 2243–2249 (2009).
[Crossref]

Feng, G.

J. G. Wu, Z. M. Wu, G. Q. Xia, T. Deng, X. D. Lin, X. Tang, and G. Feng, “Isochronous synchronization between chaotic semiconductor lasers over 40-km fiber links,” IEEE Photonics Technol. Lett. 23(24), 1854–1856 (2011).
[Crossref]

Fischer, I.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

García-Fernández, P.

C. R. Mirasso, P. Colet, and P. García-Fernández, “Synchronization of chaotic semiconductor lasers: Application to encoded communications,” IEEE Photonics Technol. Lett. 8(2), 299–301 (1996).
[Crossref]

Garcia-Ojalvo, J.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Gray, G. R.

B. Krauskopf, G. R. Gray, and D. Lenstra, “Semiconductor laser with phase-conjugate feedback: Dynamics and bifurcations,” Phys. Rev. E 58(6), 7190–7197 (1998).
[Crossref]

Hong, Y. H.

Hu, W.

Jacobsen, G.

F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985).
[Crossref]

Jiang, N.

Kanakidis, D.

Kane, D. M.

J. S. Lawrence and D. M. Kane, “Contrasting conventional optical and phase-conjugate feedback in laser diodes,” Phys. Rev. A 63(3), 033805 (2001).
[Crossref]

Ke, J. X.

Kobayashi, K.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Krauskopf, B.

B. Krauskopf, G. R. Gray, and D. Lenstra, “Semiconductor laser with phase-conjugate feedback: Dynamics and bifurcations,” Phys. Rev. E 58(6), 7190–7197 (1998).
[Crossref]

Lang, R.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Larger, L.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Lawrence, J. S.

J. S. Lawrence and D. M. Kane, “Contrasting conventional optical and phase-conjugate feedback in laser diodes,” Phys. Rev. A 63(3), 033805 (2001).
[Crossref]

Lee, M. W.

Lenstra, D.

B. Krauskopf, G. R. Gray, and D. Lenstra, “Semiconductor laser with phase-conjugate feedback: Dynamics and bifurcations,” Phys. Rev. E 58(6), 7190–7197 (1998).
[Crossref]

Li, B.

Li, G.

Li, N.

N. Li, R. M. Nguimdo, A. Locquet, and D. S. Citrin, “Enhancing optical-feedback-induced chaotic dynamics in semiconductor ring lasers via optical injection,” Nonlinear Dyn. 92(2), 315–324 (2018).
[Crossref]

N. Li, W. Pan, A. Locquet, and D. S. Citrin, “Time-delay concealment and complexity enhancement of an external-cavity laser through optical injection,” Opt. Lett. 40(19), 4416–4419 (2015).
[Crossref]

N. Li, W. Pan, S. Xiang, B. Luo, L. Yan, and X. Zou, “Hybrid chaos-based communication system consisting of three chaotic semiconductor ring lasers,” Appl. Opt. 52(7), 1523–1530 (2013).
[Crossref]

N. Jiang, W. Pan, B. Luo, L. Yan, S. Xiang, L. Yang, D. Zheng, and N. Li, “Properties of leader-laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81(6), 066217 (2010).
[Crossref]

Li, X. F.

X. F. Li, W. Pan, B. Luo, and D. Ma, “Mismatch robustness and security of chaotic optical communications based on injection-locking chaos synchronization,” IEEE J. Quantum Electron. 42(9), 953–960 (2006).
[Crossref]

Lin, S.

Lin, X. D.

J. G. Wu, Z. M. Wu, G. Q. Xia, T. Deng, X. D. Lin, X. Tang, and G. Feng, “Isochronous synchronization between chaotic semiconductor lasers over 40-km fiber links,” IEEE Photonics Technol. Lett. 23(24), 1854–1856 (2011).
[Crossref]

T. Deng, G. Q. Xia, L. P. Cao, J. G. Chen, X. D. Lin, and Z. M. Wu, “Bidirectional chaos synchronization and communication in semiconductor lasers with optoelectronic feedback,” Opt. Commun. 282(11), 2243–2249 (2009).
[Crossref]

Liu, S.

Liu, S. Q.

Locquet, A.

N. Li, R. M. Nguimdo, A. Locquet, and D. S. Citrin, “Enhancing optical-feedback-induced chaotic dynamics in semiconductor ring lasers via optical injection,” Nonlinear Dyn. 92(2), 315–324 (2018).
[Crossref]

N. Li, W. Pan, A. Locquet, and D. S. Citrin, “Time-delay concealment and complexity enhancement of an external-cavity laser through optical injection,” Opt. Lett. 40(19), 4416–4419 (2015).
[Crossref]

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–1891 (2009).
[Crossref]

A. Locquet, F. Rogister, M. Sciamanna, and P. Megret, “Synchronization of chaotic semiconductor lasers with phase-conjugate feedback,” in Pacific Rim Conference on Lasers and Electro-Optics, II-388–II-389 (2001).

Luo, B.

N. Li, W. Pan, S. Xiang, B. Luo, L. Yan, and X. Zou, “Hybrid chaos-based communication system consisting of three chaotic semiconductor ring lasers,” Appl. Opt. 52(7), 1523–1530 (2013).
[Crossref]

N. Jiang, W. Pan, B. Luo, L. Yan, S. Xiang, L. Yang, D. Zheng, and N. Li, “Properties of leader-laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81(6), 066217 (2010).
[Crossref]

X. F. Li, W. Pan, B. Luo, and D. Ma, “Mismatch robustness and security of chaotic optical communications based on injection-locking chaos synchronization,” IEEE J. Quantum Electron. 42(9), 953–960 (2006).
[Crossref]

Lv, Y.

Ma, D.

X. F. Li, W. Pan, B. Luo, and D. Ma, “Mismatch robustness and security of chaotic optical communications based on injection-locking chaos synchronization,” IEEE J. Quantum Electron. 42(9), 953–960 (2006).
[Crossref]

Megret, P.

A. Locquet, F. Rogister, M. Sciamanna, and P. Megret, “Synchronization of chaotic semiconductor lasers with phase-conjugate feedback,” in Pacific Rim Conference on Lasers and Electro-Optics, II-388–II-389 (2001).

Mercier, É.

É. Mercier, D. Wolfersberger, and M. Sciamanna, “High-frequency chaotic dynamics enabled by optical phase-conjugation,” Sci. Rep. 6(1), 18988 (2016).
[Crossref]

D. Rontani, É. Mercier, D. Wolfersberger, and M. Sciamanna, “Enhanced complexity of optical chaos in a laser diode with phase-conjugate feedback,” Opt. Lett. 41(20), 4637–4640 (2016).
[Crossref]

Mirasso, C. R.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

C. R. Mirasso, P. Colet, and P. García-Fernández, “Synchronization of chaotic semiconductor lasers: Application to encoded communications,” IEEE Photonics Technol. Lett. 8(2), 299–301 (1996).
[Crossref]

Mogensen, F.

F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985).
[Crossref]

Murakami, A.

A. Murakami, “Phase locking and chaos synchronization in injection-locked semiconductor lasers,” IEEE J. Quantum Electron. 39(3), 438–447 (2003).
[Crossref]

Nguimdo, R. M.

N. Li, R. M. Nguimdo, A. Locquet, and D. S. Citrin, “Enhancing optical-feedback-induced chaotic dynamics in semiconductor ring lasers via optical injection,” Nonlinear Dyn. 92(2), 315–324 (2018).
[Crossref]

Ohtsubo, J.

J. Ohtsubo, “Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback,” IEEE J. Quantum Electron. 38(9), 1141–1154 (2002).
[Crossref]

Olesen, H.

F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985).
[Crossref]

Ortin, S.

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–1891 (2009).
[Crossref]

Pan, W.

M. F. Xu, W. Pan, S. H. Xiang, and L. Zhang, “Cluster synchronization in symmetric VCSELs networks with variable-polarization optical feedback,” Opt. Express 26(8), 10754–10761 (2018).
[Crossref]

S. Y. Xiang, A. J. Wen, and W. Pan, “Synchronization regime of star-type laser network with heterogeneous coupling delays,” IEEE Photonics Technol. Lett. 28(18), 1988–1991 (2016).
[Crossref]

N. Li, W. Pan, A. Locquet, and D. S. Citrin, “Time-delay concealment and complexity enhancement of an external-cavity laser through optical injection,” Opt. Lett. 40(19), 4416–4419 (2015).
[Crossref]

N. Li, W. Pan, S. Xiang, B. Luo, L. Yan, and X. Zou, “Hybrid chaos-based communication system consisting of three chaotic semiconductor ring lasers,” Appl. Opt. 52(7), 1523–1530 (2013).
[Crossref]

N. Jiang, W. Pan, B. Luo, L. Yan, S. Xiang, L. Yang, D. Zheng, and N. Li, “Properties of leader-laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81(6), 066217 (2010).
[Crossref]

X. F. Li, W. Pan, B. Luo, and D. Ma, “Mismatch robustness and security of chaotic optical communications based on injection-locking chaos synchronization,” IEEE J. Quantum Electron. 42(9), 953–960 (2006).
[Crossref]

Paul, J.

Pesquera, L.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Qiu, K.

Rizomiliotis, P.

A. Bogris, P. Rizomiliotis, K. E. Chlouverakis, A. Argyris, and D. Syvridis, “Feedback phase in optically generated chaos: A sceret key for cryptographic applications,” IEEE J. Quantum Electron. 44(2), 119–124 (2008).
[Crossref]

Rogister, F.

A. Locquet, F. Rogister, M. Sciamanna, and P. Megret, “Synchronization of chaotic semiconductor lasers with phase-conjugate feedback,” in Pacific Rim Conference on Lasers and Electro-Optics, II-388–II-389 (2001).

Rontani, D.

D. Rontani, É. Mercier, D. Wolfersberger, and M. Sciamanna, “Enhanced complexity of optical chaos in a laser diode with phase-conjugate feedback,” Opt. Lett. 41(20), 4637–4640 (2016).
[Crossref]

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–1891 (2009).
[Crossref]

Roy, R.

G. D. Vanwiggeren and R. Roy, “Communication with chaotic lasers,” Science 279(5354), 1198–1200 (1998).
[Crossref]

Sciamanna, M.

D. Rontani, É. Mercier, D. Wolfersberger, and M. Sciamanna, “Enhanced complexity of optical chaos in a laser diode with phase-conjugate feedback,” Opt. Lett. 41(20), 4637–4640 (2016).
[Crossref]

É. Mercier, D. Wolfersberger, and M. Sciamanna, “High-frequency chaotic dynamics enabled by optical phase-conjugation,” Sci. Rep. 6(1), 18988 (2016).
[Crossref]

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–1891 (2009).
[Crossref]

A. Locquet, F. Rogister, M. Sciamanna, and P. Megret, “Synchronization of chaotic semiconductor lasers with phase-conjugate feedback,” in Pacific Rim Conference on Lasers and Electro-Optics, II-388–II-389 (2001).

Shore, K. A.

Y. H. Hong, M. W. Lee, J. Paul, P. S. Spencer, and K. A. Shore, “Enhanced chaos synchronization in unidirectionally coupled vertical-cavity surface-emitting semiconductor lasers with polarization-preserved injection,” Opt. Lett. 33(6), 587–589 (2008).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Spencer, P. S.

Syvridis, D.

A. Bogris, P. Rizomiliotis, K. E. Chlouverakis, A. Argyris, and D. Syvridis, “Feedback phase in optically generated chaos: A sceret key for cryptographic applications,” IEEE J. Quantum Electron. 44(2), 119–124 (2008).
[Crossref]

D. Kanakidis, A. Argyris, A. Bogris, and D. Syvridis, “Influence of the decoding process on the performance of chaos encrypted optical communication systems,” J. Lightwave Technol. 24(1), 335–341 (2006).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

D. Kanakidis, A. Bogris, A. Argyris, and D. Syvridis, “Numerical investigation of fiber transmission of a chaotic encrypted message using dispersion compensation schemes,” J. Lightwave Technol. 22(10), 2256–2263 (2004).
[Crossref]

Tang, J. M.

Tang, X.

J. G. Wu, Z. M. Wu, G. Q. Xia, T. Deng, X. D. Lin, X. Tang, and G. Feng, “Isochronous synchronization between chaotic semiconductor lasers over 40-km fiber links,” IEEE Photonics Technol. Lett. 23(24), 1854–1856 (2011).
[Crossref]

Vanwiggeren, G. D.

G. D. Vanwiggeren and R. Roy, “Communication with chaotic lasers,” Science 279(5354), 1198–1200 (1998).
[Crossref]

Wang, C.

Wang, Y.

Wang, Y. J.

Wen, A. J.

S. Y. Xiang, A. J. Wen, and W. Pan, “Synchronization regime of star-type laser network with heterogeneous coupling delays,” IEEE Photonics Technol. Lett. 28(18), 1988–1991 (2016).
[Crossref]

Wolfersberger, D.

D. Rontani, É. Mercier, D. Wolfersberger, and M. Sciamanna, “Enhanced complexity of optical chaos in a laser diode with phase-conjugate feedback,” Opt. Lett. 41(20), 4637–4640 (2016).
[Crossref]

É. Mercier, D. Wolfersberger, and M. Sciamanna, “High-frequency chaotic dynamics enabled by optical phase-conjugation,” Sci. Rep. 6(1), 18988 (2016).
[Crossref]

Wu, J. G.

J. G. Wu, Z. M. Wu, G. Q. Xia, T. Deng, X. D. Lin, X. Tang, and G. Feng, “Isochronous synchronization between chaotic semiconductor lasers over 40-km fiber links,” IEEE Photonics Technol. Lett. 23(24), 1854–1856 (2011).
[Crossref]

Wu, Z. M.

J. G. Wu, Z. M. Wu, G. Q. Xia, T. Deng, X. D. Lin, X. Tang, and G. Feng, “Isochronous synchronization between chaotic semiconductor lasers over 40-km fiber links,” IEEE Photonics Technol. Lett. 23(24), 1854–1856 (2011).
[Crossref]

T. Deng, G. Q. Xia, L. P. Cao, J. G. Chen, X. D. Lin, and Z. M. Wu, “Bidirectional chaos synchronization and communication in semiconductor lasers with optoelectronic feedback,” Opt. Commun. 282(11), 2243–2249 (2009).
[Crossref]

Xia, G. Q.

J. X. Ke, L. L. Yi, G. Q. Xia, and W. Hu, “Chaotic optical communications over 100-km fiber transmission at 30-Gb/s bit rate,” Opt. Lett. 43(6), 1323–1326 (2018).
[Crossref]

J. G. Wu, Z. M. Wu, G. Q. Xia, T. Deng, X. D. Lin, X. Tang, and G. Feng, “Isochronous synchronization between chaotic semiconductor lasers over 40-km fiber links,” IEEE Photonics Technol. Lett. 23(24), 1854–1856 (2011).
[Crossref]

T. Deng, G. Q. Xia, L. P. Cao, J. G. Chen, X. D. Lin, and Z. M. Wu, “Bidirectional chaos synchronization and communication in semiconductor lasers with optoelectronic feedback,” Opt. Commun. 282(11), 2243–2249 (2009).
[Crossref]

Xiang, S.

N. Li, W. Pan, S. Xiang, B. Luo, L. Yan, and X. Zou, “Hybrid chaos-based communication system consisting of three chaotic semiconductor ring lasers,” Appl. Opt. 52(7), 1523–1530 (2013).
[Crossref]

N. Jiang, W. Pan, B. Luo, L. Yan, S. Xiang, L. Yang, D. Zheng, and N. Li, “Properties of leader-laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81(6), 066217 (2010).
[Crossref]

Xiang, S. H.

Xiang, S. Y.

S. Y. Xiang, A. J. Wen, and W. Pan, “Synchronization regime of star-type laser network with heterogeneous coupling delays,” IEEE Photonics Technol. Lett. 28(18), 1988–1991 (2016).
[Crossref]

Xu, M. F.

Xue, C.

Xue, C. P.

Yan, L.

N. Li, W. Pan, S. Xiang, B. Luo, L. Yan, and X. Zou, “Hybrid chaos-based communication system consisting of three chaotic semiconductor ring lasers,” Appl. Opt. 52(7), 1523–1530 (2013).
[Crossref]

N. Jiang, W. Pan, B. Luo, L. Yan, S. Xiang, L. Yang, D. Zheng, and N. Li, “Properties of leader-laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81(6), 066217 (2010).
[Crossref]

Yang, L.

N. Jiang, W. Pan, B. Luo, L. Yan, S. Xiang, L. Yang, D. Zheng, and N. Li, “Properties of leader-laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81(6), 066217 (2010).
[Crossref]

Yi, L. L.

Zhang, L.

Zhang, Y.

Zhao, A.

Zhao, A. K.

Zheng, D.

N. Jiang, W. Pan, B. Luo, L. Yan, S. Xiang, L. Yang, D. Zheng, and N. Li, “Properties of leader-laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81(6), 066217 (2010).
[Crossref]

Zou, X.

Appl. Opt. (1)

IEEE J. Quantum Electron. (7)

A. Bogris, P. Rizomiliotis, K. E. Chlouverakis, A. Argyris, and D. Syvridis, “Feedback phase in optically generated chaos: A sceret key for cryptographic applications,” IEEE J. Quantum Electron. 44(2), 119–124 (2008).
[Crossref]

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985).
[Crossref]

A. Murakami, “Phase locking and chaos synchronization in injection-locked semiconductor lasers,” IEEE J. Quantum Electron. 39(3), 438–447 (2003).
[Crossref]

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–1891 (2009).
[Crossref]

J. Ohtsubo, “Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback,” IEEE J. Quantum Electron. 38(9), 1141–1154 (2002).
[Crossref]

X. F. Li, W. Pan, B. Luo, and D. Ma, “Mismatch robustness and security of chaotic optical communications based on injection-locking chaos synchronization,” IEEE J. Quantum Electron. 42(9), 953–960 (2006).
[Crossref]

IEEE Photonics Technol. Lett. (3)

S. Y. Xiang, A. J. Wen, and W. Pan, “Synchronization regime of star-type laser network with heterogeneous coupling delays,” IEEE Photonics Technol. Lett. 28(18), 1988–1991 (2016).
[Crossref]

J. G. Wu, Z. M. Wu, G. Q. Xia, T. Deng, X. D. Lin, X. Tang, and G. Feng, “Isochronous synchronization between chaotic semiconductor lasers over 40-km fiber links,” IEEE Photonics Technol. Lett. 23(24), 1854–1856 (2011).
[Crossref]

C. R. Mirasso, P. Colet, and P. García-Fernández, “Synchronization of chaotic semiconductor lasers: Application to encoded communications,” IEEE Photonics Technol. Lett. 8(2), 299–301 (1996).
[Crossref]

J. Lightwave Technol. (2)

Nature (1)

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Nonlinear Dyn. (1)

N. Li, R. M. Nguimdo, A. Locquet, and D. S. Citrin, “Enhancing optical-feedback-induced chaotic dynamics in semiconductor ring lasers via optical injection,” Nonlinear Dyn. 92(2), 315–324 (2018).
[Crossref]

Opt. Commun. (1)

T. Deng, G. Q. Xia, L. P. Cao, J. G. Chen, X. D. Lin, and Z. M. Wu, “Bidirectional chaos synchronization and communication in semiconductor lasers with optoelectronic feedback,” Opt. Commun. 282(11), 2243–2249 (2009).
[Crossref]

Opt. Express (3)

Opt. Lett. (6)

OSA Continuum (1)

Phys. Rev. A (1)

J. S. Lawrence and D. M. Kane, “Contrasting conventional optical and phase-conjugate feedback in laser diodes,” Phys. Rev. A 63(3), 033805 (2001).
[Crossref]

Phys. Rev. E (2)

B. Krauskopf, G. R. Gray, and D. Lenstra, “Semiconductor laser with phase-conjugate feedback: Dynamics and bifurcations,” Phys. Rev. E 58(6), 7190–7197 (1998).
[Crossref]

N. Jiang, W. Pan, B. Luo, L. Yan, S. Xiang, L. Yang, D. Zheng, and N. Li, “Properties of leader-laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81(6), 066217 (2010).
[Crossref]

Sci. Rep. (1)

É. Mercier, D. Wolfersberger, and M. Sciamanna, “High-frequency chaotic dynamics enabled by optical phase-conjugation,” Sci. Rep. 6(1), 18988 (2016).
[Crossref]

Science (1)

G. D. Vanwiggeren and R. Roy, “Communication with chaotic lasers,” Science 279(5354), 1198–1200 (1998).
[Crossref]

Other (1)

A. Locquet, F. Rogister, M. Sciamanna, and P. Megret, “Synchronization of chaotic semiconductor lasers with phase-conjugate feedback,” in Pacific Rim Conference on Lasers and Electro-Optics, II-388–II-389 (2001).

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Figures (6)

Fig. 1.
Fig. 1. Schematic of the closed-loop PCF-ECSL system. MSL, Master semiconductor laser; SSL, slave semiconductor laser; PCM: phase-conjugate mirror, OC: optical coupler, OI: optical isolator, PS: phase shifter, M: modulator, PD: photodetector.
Fig. 2.
Fig. 2. (a) Phase-locking characteristic versus the variation of ki in the closed-loop PCF-ECSLs system, (b) variation of the synchronization quality as a function of tuning phase difference ϕPS (normalized in [0∼2π]) with zero frequency detuning in SSL. Lines in (a) denote the analytical results of the phase ϕL defined in Eq. (11), while the marks stand for the numerical simulation results for Δϕ.
Fig. 3.
Fig. 3. Distributions of injection-locking synchronization quality as a function of the injection strength and feedback strength, for the cases of PCF with (a) ϕPS1, (b) ϕPS2, and (c) COF.
Fig. 4.
Fig. 4. Synchronization quality between MSL and SSL as functions of the frequency detuning (a), and the intrinsic parameters mismatch (b) in PCF and COF closed-loop schemes. The injection strength is set as 80ns−1.
Fig. 5.
Fig. 5. Illustrations of chaos communication in the closed-loop PCF-ECSL system for the cases of PCF with ϕPS1 (first row) and ϕPS2 (second row). The dark curves denote the original messages, while the red and blue ones denote the decrypted messages. The bit rates for message transmissions shown in the first and second columns are 2.5 Gbit/s and 5 Gbit/s, respectively. The horizontal axes have been properly shifted.
Fig. 6.
Fig. 6. (a) Comparison of communication performance (Q-factor) in the closed-loop PCF-ECSL and COF-ECSL configurations as a function of the bit rate of message, (b) effective bandwidth (GHz) of chaotic carrier generated by PCF-ECSL and COF-ECSL versus the strength of feedback.

Equations (18)

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d E M ( t ) d t = 1 + i α 2 [ G M ( t ) 1 τ p ] E M ( t ) + k f E M ( t τ f ) exp ( i ϕ P C M M ) + 2 β N M ( t ) χ M ( t )
d N M ( t ) d t = I q N M ( t ) τ e G M ( t ) | E M ( t ) | 2
d E S ( t ) d t = 1 + i α 2 [ G S ( t ) 1 τ p ] E S ( t ) + k f E S ( t τ f ) exp ( i ϕ P C M S ) + k i E M ( t τ i ) exp ( i ω M τ i ) exp ( i Δ ω t ) + 2 β N S ( t ) χ S ( t )
d N S ( t ) d t = I q N S ( t ) τ e G S ( t ) | E S ( t ) | 2
d A M , S ( t ) d t = 1 2 [ G M , S ( t ) 1 τ p ] A M , S ( t ) + k f A M , S ( t ) A M , S ( t τ f ) cos [ θ f M , f S ( t ) ] + k i A S ( t ) A M ( t τ i ) cos [ θ i ( t ) ]
d ϕ M , S ( t ) d t = α 2 [ G M , S ( t ) 1 τ p ] k f A M , S ( t τ f ) A M , S ( t ) sin [ θ f M , f S ( t ) ] k i A M ( t τ i ) A S ( t ) sin [ θ i ( t ) ]
d N M , S ( t ) d t = I q N M , S ( t ) τ e G M , S ( t ) | A M , S ( t ) | 2
θ f M , f S ( t ) = ϕ M , S ( t ) + ϕ M , S ( t τ f ) ϕ P C M M , P C M S
θ i ( t ) = ω M τ i + ϕ S ( t ) ϕ M ( t τ i ) Δ ω t
ϕ S ( t ) ϕ M ( t ) = Δ ω t + ϕ L ,
ϕ L = sin 1 { Δ ω k i 1 + α 2 } tan 1 α .
ϕ M ( t ) + ϕ M ( t τ f ) ϕ P C M M = ϕ S ( t ) + ϕ S ( t τ f ) ϕ P C M S
ϕ M ( t ) ϕ S ( t ) + ϕ M ( t τ f ) ϕ S ( t τ f ) ϕ P C M M + ϕ P C M S = 0
2 ϕ L + 2 Δ ω t Δ ω τ f Δ ϕ P C M = 0
ϕ P S 1 = 2 ϕ L + 2 Δ ω t Δ ω τ f Δ ϕ P C M
ϕ S ( t ) ϕ M ( t ) = Δ ω t + ϕ P S 2 + ϕ L .
ϕ P S 2 = ϕ L Δ ω t + 1 2 Δ ω τ f + 1 2 Δ ϕ P C M
ρ ( τ ) = [ P M ( t τ ) P M ( t τ ) ] [ P S ( t ) P S ( t ) ] [ P M ( t τ ) P M ( t τ ) ] 2 [ P S ( t ) P S ( t ) ] 2

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