Abstract

We propose a type of semiconductor laser (SL) network that supports flexible chaos synchronization and multipoint-to-multipoint communications by using one-way isolation (OWI). The properties of chaos synchronization, influences of coupling strength and time delay mismatches on the quality of chaos synchronization, and the performance as well as the security of the SL network-based chaotic communications are systematically discussed. The numerical results demonstrate that, with the introduction of OWI, flexible chaos synchronization can be easily achieved in arbitrary-size SL clusters over wide parameter spaces of coupling strength and current factor. Based on the high-quality flexible chaos synchronization, satisfactory performance for Gb/s chaotic communications can be achieved in arbitrary-size clusters in the SL networks. Moreover, it is also indicated that in the SL networks, the security of intra-cluster communications can be guaranteed in three aspects. Firstly, the eavesdroppers cannot intercept any useful information by using a typical illegal attack. Secondly, due to the OWI, the chaotic carriers are only transmitted in the corresponding clusters, not transmitted among clusters, as such the security can be further improved. Thirdly, the high sensitivity of cross-correlation coefficient to the injection delay mismatches indicates that the injection delays of idle SLs to communicating SLs can be regarded as the keys of the communication clusters. The proposed scheme offers an alternative solution to flexible secure network-type communications.

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

1. Introduction

Laser chaos has attracted more and more attention due to its unpredictability and broadband bandwidth [17]. In 1990, chaos synchronization was firstly demonstrated between two circuits [8]. Since then, many efforts have been devoted to carrying out chaos-based communication in unidirectional and bidirectional transmission systems [916]. However, most of the previously reported chaotic communication systems only focused on the point-to-point transmission, while the multipoint-to-multipoint chaotic communication is rarely discussed.

Until recent years, a few chaos synchronization and communication schemes in the semiconductor laser (SL) networks with complex topologies are proposed [1722]. Zhang et al. theoretically and numerically investigated the cluster chaos synchronization in symmetric SL network, by dividing the SLs into a set of clusters [19]. Xiang et al. numerically studied the synchronization and complexity properties of a hierarchical tree-type network composed of mutually coupled SLs [20]. Li et al. numerically demonstrated the point-to-multipoint communications in a star-type SL network, the adjacent communication in a ring SL network, and the wavelength-division-multiplexing (WDM) based chaos communication in the star-type SL network by modulating the bias currents with pseudorandom digital messages [21,22]. In these schemes, chaos synchronization can be only achieved among symmetric SLs of the networks, and consequently, chaos communication can be only realized between a few specific SLs that are structurally symmetric. However, it is difficult to flexibly construct communication systems among any SLs, for this reason, the flexibility of multi-user chaos communication systems is limited. Therefore, it is of great significance to enhance the flexibility of multipoint-to-multipoint chaos communication in SL networks.

In this work, a type of SL network is proposed by applying the one-way isolation (OWI) to a global SL network in which each SL is mutually coupled with the other SLs, and the properties of the flexible isochronal chaos synchronization and bidirectional chaotic communication in the proposed SL network are systematically investigated. To the best of our knowledge, this is the first time to enhance the flexibility of network-type chaos synchronization and communication. In contrast with previously reported works, the flexibility of our proposed SL network is significantly improved with the contribution of OWI because both of high-quality chaos synchronization and secure bidirectional communications can be realized among any SLs of the global networks, instead of several specific SLs, and moreover, chaotic communications are simultaneously achieved in two independent communication clusters. Additionally, due to the effective isolation to chaotic carriers from communication clusters to idle cluster, the security of communication clusters is ensured. The proposed SL networks are also against illegal interceptions over the public links even in the low-rate transmission cases, as such the system security can be further enhanced.

2. Theoretical model

The topology of mutually coupled global SL network is depicted in Fig. 1(a). In the global semiconductor lasers network, each SL is identically and mutually coupled with the other SLs. That is, all of the SLs play exactly the same structural role. In this work, for the sake of simplicity, we take a 6-SLs global network for instance. Under a traditional scenario, all 6 SLs are regarded as a 6-SLs cluster and only the communication among 6 SLs is allowed when all the SLs are synchronized, while the partial communications within 6 synchronous SLs could not be supported with security ensured, because the mutual couplings between communication SLs and idle SLs would threaten the safety of communication carriers. Nevertheless, in our proposed scheme, we introduce a novel method referred to as OWI to achieve flexible chaos synchronization and security-enhanced chaotic communication. We divide the SLs in the global network into two parts that respectively contain communication SLs and idle SLs. The OWI is performed by preserving the injections from the idle SLs to the communication SLs, but cutting off those in the opposite directions. Under such a case, due to the symmetric breaking the global network can be divided into two SLs clusters, namely a communication cluster and an idle cluster where the intra-cluster SLs are structurally symmetric while the inter-cluster SLs are asymmetrical to each other. Similarly, two or more independent communication SL clusters can be obtained by performing OWI twice or multiple times in the global SL network. In general, by introducing the OWI to a global SL network, the global SL network can be divided into a set of arbitrary-size clusters that support intra-cluster chaotic communication. For the exemplary global 6-SLs network shown in Fig. 1(a), it can be divided into 2-SLs, 3-SLs, 4-SLs, and 5-SLs communication clusters as shown in Figs. 1(b)–1(e). Furthermore, Fig. 1(f) displays another available case in which two independent 2-SLs clusters simultaneously exist in a SL network. Here, the network topologies for same-size clusters composed of different communication SLs are similar, and for this reason the exemplary communication SLs in Figs. 1(b)–1(f) are taken for instance. Therefore, with the OWI, flexible message exchange among arbitrary 2 SLs, 3 SLs, 4 SLs, 5 SLs, and 6 SLs (traditional case) in the SL networks can be achieved based on the intra-cluster chaos synchronization, and simultaneous communications in two independent communication clusters can also be supported. Meanwhile by using OWI, the security of chaos communications in clusters can be guaranteed since the chaotic carriers are only limited in the corresponding SL clusters.

 

Fig. 1. The topologies of (a) a global 6-SLs network, and SL networks under OWI containing (b) 2-SLs cluster, (c) 3-SLs cluster, (d) 4-SLs cluster, (e) 5-SLs cluster, (f) two co-existing independent 2-SLs clusters. Here, the SLs denoted by the red nodes are the exemplary SLs for communications, and the blue nodes denote the idle SLs that are not involved in communications.

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The Lang-Kobayashi rate equations are modified to describe the dynamics of the SLs in the network. Considering the couplings inside the SL networks, the mathematical model can be expressed as follows [2326]:

$$\frac{{dE_i^{(l)}{(}t{)}}}{{dt}} = \frac{{(1 + i\alpha )}}{2}(G_i^{(l)}\textrm{(}t\textrm{)} - \frac{1}{{{\tau _p}}})E_i^{(l)}{(}t{)} + {\sigma ^{(l)}}\sum\limits_{j = 1}^6 {A_{ij}^{(l)}E_j^{(l)}(t - \tau )\exp ( - i\omega \tau )} + \sqrt {2\beta N_i^{(l)}(t)} \chi _i^{(l)}(t),$$
$$\frac{{dN_i^{(l)}(t)}}{{dt}} = \frac{{\mu {I_{th}}}}{q} - \frac{{N_i^{(l)}(t)}}{{{\tau _e}}} - G_i^{(l)}(t)||{E_i^{(l)}} (t{ ) ||^2},$$
$$G_i^{(l)}(t) = \frac{{g(N_i^{(l)}(t) - {N_\textrm{0}})}}{{1 + s||{E_i^{(l)}} (t{{ ) ||}^2}}},$$
$${A^{(1)}} = \left[ {\begin{array}{@{}cccc@{}} 0&1&1&{\begin{array}{@{}ccc@{}} 1&1&1 \end{array}}\\ 1&0&1&{\begin{array}{@{}ccc@{}} 1&1&1 \end{array}}\\ 1&1&0&{\begin{array}{@{}ccc@{}} 1&1&1 \end{array}}\\ {\begin{array}{@{}c@{}} 1\\ 1\\ 1 \end{array}}&{\begin{array}{@{}c@{}} 1\\ 1\\ 1 \end{array}}&{\begin{array}{@{}c@{}} 1\\ 1\\ 1 \end{array}}&{\begin{array}{@{}ccc@{}} {\begin{array}{@{}c@{}} 0\\ 1\\ 1 \end{array}}&{\begin{array}{@{}c@{}} 1\\ 0\\ 1 \end{array}}&{\begin{array}{@{}c@{}} 1\\ 1\\ 0 \end{array}} \end{array}} \end{array}} \right],\quad {A^{(2)}} = \left[ {\begin{array}{@{}cccc@{}} 0&1&1&{\begin{array}{@{}ccc@{}} 1&1&1 \end{array}}\\ 1&0&1&{\begin{array}{@{}ccc@{}} 1&1&1 \end{array}}\\ 0&0&0&{\begin{array}{@{}ccc@{}} 1&1&1 \end{array}}\\ {\begin{array}{@{}c@{}} 0\\ 0\\ 0 \end{array}}&{\begin{array}{@{}c@{}} 0\\ 0\\ 0 \end{array}}&{\begin{array}{@{}c@{}} 1\\ 1\\ 1 \end{array}}&{\begin{array}{@{}ccc@{}} {\begin{array}{@{}c@{}} 0\\ 1\\ 1 \end{array}}&{\begin{array}{@{}c@{}} 1\\ 0\\ 1 \end{array}}&{\begin{array}{@{}c@{}} 1\\ 1\\ 0 \end{array}} \end{array}} \end{array}} \right],\quad {A^{(3)}} = \left[ {\begin{array}{@{}cccc@{}} 0&1&1&{\begin{array}{@{}ccc@{}} 1&1&1 \end{array}}\\ 1&0&1&{\begin{array}{@{}ccc@{}} 1&1&1 \end{array}}\\ 1&1&0&{\begin{array}{@{}ccc@{}} 1&1&1 \end{array}}\\ {\begin{array}{@{}c@{}} 0\\ 0\\ 0 \end{array}}&{\begin{array}{@{}c@{}} 0\\ 0\\ 0 \end{array}}&{\begin{array}{@{}c@{}} 0\\ 0\\ 0 \end{array}}&{\begin{array}{@{}ccc@{}} {\begin{array}{@{}c@{}} 0\\ 1\\ 1 \end{array}}&{\begin{array}{@{}c@{}} 1\\ 0\\ 1 \end{array}}&{\begin{array}{@{}c@{}} 1\\ 1\\ 0 \end{array}} \end{array}} \end{array}} \right], $$
$${A^{(4)}} = \left[ {\begin{array}{@{}cccc@{}} 0&1&1&{\begin{array}{@{}ccc@{}} 1&1&1 \end{array}}\\ 1&0&1&{\begin{array}{@{}ccc@{}} 1&1&1 \end{array}}\\ 1&1&0&{\begin{array}{@{}ccc@{}} 1&1&1 \end{array}}\\ {\begin{array}{@{}c@{}} 1\\ 0\\ 0 \end{array}}&{\begin{array}{@{}c@{}} 1\\ 0\\ 0 \end{array}}&{\begin{array}{@{}c@{}} 1\\ 0\\ 0 \end{array}}&{\begin{array}{@{}ccc@{}} {\begin{array}{@{}c@{}} 0\\ 0\\ 0 \end{array}}&{\begin{array}{@{}c@{}} 1\\ 0\\ 1 \end{array}}&{\begin{array}{@{}c@{}} 1\\ 1\\ 0 \end{array}} \end{array}} \end{array}} \right],\quad {A^{(5)}} = \left[ {\begin{array}{@{}cccc@{}} 0&1&1&{\begin{array}{@{}ccc@{}} 1&1&1 \end{array}}\\ 1&0&1&{\begin{array}{@{}ccc@{}} 1&1&1 \end{array}}\\ 1&1&0&{\begin{array}{@{}ccc@{}} 1&1&1 \end{array}}\\ {\begin{array}{@{}c@{}} 1\\ 1\\ 0 \end{array}}&{\begin{array}{@{}c@{}} 1\\ 1\\ 0 \end{array}}&{\begin{array}{@{}c@{}} 1\\ 1\\ 0 \end{array}}&{\begin{array}{@{}ccc@{}} {\begin{array}{@{}c@{}} 0\\ 1\\ 0 \end{array}}&{\begin{array}{@{}c@{}} 1\\ 0\\ 0 \end{array}}&{\begin{array}{@{}c@{}} 1\\ 1\\ 0 \end{array}} \end{array}} \end{array}} \right],\quad {A^{(6)}} = \left[ {\begin{array}{@{}cccc@{}} 0&1&1&{\begin{array}{@{}ccc@{}} 0&0&1 \end{array}}\\ 1&0&1&{\begin{array}{@{}ccc@{}} 0&0&1 \end{array}}\\ 0&0&0&{\begin{array}{@{}ccc@{}} 0&0&1 \end{array}}\\ {\begin{array}{@{}c@{}} 0\\ 0\\ 0 \end{array}}&{\begin{array}{@{}c@{}} 0\\ 0\\ 0 \end{array}}&{\begin{array}{@{}c@{}} 1\\ 1\\ 1 \end{array}}&{\begin{array}{@{}ccc@{}} {\begin{array}{@{}c@{}} 0\\ 1\\ 0 \end{array}}&{\begin{array}{@{}c@{}} 1\\ 0\\ 0 \end{array}}&{\begin{array}{@{}c@{}} 1\\ 1\\ 0 \end{array}} \end{array}} \end{array}} \right], $$
where the subscripts i=1, 2, …, 6, j=1, 2, …, 6 denote SLi and SLj, and l=1, 2, …, 6 represents the SL networks containing different-size communication clusters which are shown in Figs. 1(a)–1(f), respectively. E and N are the slowly varying complex electric field and carrier density, respectively. G(t) is the optical gain. A is the adjacent matrix of the SL networks shown in Fig. 1 which is expressed by Eqs. (4) and (5), wherein Aij=1 means there is an injection from SLj to SLi, Aij=0 otherwise. The second term in the right hand of Eq. (1) denotes the couplings between SLi and the residual SLs, and the third term stands for the spontaneous emission noise that is modeled by the Gaussian noise χ(t) with unity variance and zero mean. The internal laser parameters are assumed to be identical and set as the typical values reported in [11,27,28]: α=5 is the linewidth-enhancement factor, τp=2ps is the photon lifetime, Ith=14.7 mA is the threshold current, q=1.6×10−19C is the electric charge, τe=2 ns is the carrier lifetime, β = 1.5×10−6ns-1 is the spontaneous emission rate, g=1.5×104s-1 is the differential gain coefficient, N0=1.5×108 is the transparency carrier density, s=5×10−7 is the gain saturation coefficient, λ0=1550 nm is the wavelength of SLs, ω=1.216×1015rad/s is the angle frequency, τ=5 ns is the coupling delay. Unless otherwise stated, the current factor is µ=2, and the overall coupling strengths of these SL networks are σ(1)=25ns-1, σ(2)=30ns-1, σ(3)=40ns-1, σ(4)=50ns-1, σ(5)=35ns-1, σ(6)=70ns-1. The fourth-order Runge–Kutta method with a step of 1 ps is used to solve the Eqs. (1)-(5).

To mathematically quantify the synchronization performance of SL clusters, the time-averaged root-mean synchronization error (RMS) is adopted and defined as [29,30]:

$$RMS\textrm{ = }\frac{{\sum\limits_{m = 1}^{{N_c}} {\sqrt {{{({I_m}(t) - \mathop I\limits^ \wedge (t))}^2}/{L_I}} } }}{{{N_c}\mathop I\limits^ \wedge (t)}},$$
where the subscript m=1, …, Nc denotes the m-th SL in the cluster containing Nc SLs, the average output intensity of a SL cluster is defined as $\mathop I\limits^ \wedge (t) = \sum\nolimits_{m = 1}^{{N_c}} {{I_m}(t)/{N_c}}$. LI is the length of Im(t) and $\mathop I\limits^ \wedge (t)$. The outputs of intra-cluster SLs are considered as synchronized when RMS<0.01 [30].

To evaluate the quality of chaos synchronization in two SLs, the usually used cross-correlation coefficient (CC) is employed, and the CC of SLi and SLj is given by [22,26]:

$${C_{i,j}} = \frac{{\left\langle {({P_i} - \left\langle {{P_i}} \right\rangle ) \cdot ({P_j} - \left\langle {{P_j}} \right\rangle )} \right\rangle }}{{\sqrt {\left\langle {{{({P_i} - \left\langle {{P_i}} \right\rangle )}^2}} \right\rangle \cdot \left\langle {{{({P_j} - \left\langle {{P_j}} \right\rangle )}^2}} \right\rangle } }},$$
where the P=|E2| is the time series of chaos outputted by SLs, <•> denotes time average. Here, two SLs are regarded as synchronized only when the value of CC is larger than 0.95.

3. Properties of chaos synchronization

High-quality chaos synchronization is the basis of successful chaotic communication. For the exemplary global 6-SLs network (6-SLs cluster) shown in Fig. 1(a), Fig. 2 presents the temporal waveforms and corresponding spectra for different coupling strength cases. When σ(1)=6ns-1, six inconsistent waveforms are observed, which indicate no synchronization is achieved. Correspondingly, for the spectrum of the exemplary SL (SL1), the energy is concentrated nearby the relaxation oscillation frequency, resulting in an effective bandwidth of 17.5 GHz. As the increase of coupling strength, global isochronous synchronization is realized among all the 6 SLs with a relatively large σ(1). Moreover, the energy distribution in the spectrum is more uniform, and the power spectrum is flatter. Specifically, some low-frequency components appear when σ(1)=18ns-1. Here, the 6 SLs of the global SL network can be regarded as a cluster of 6 SLs. Therefore, it is demonstrated that the coupling strength influences the chaos synchronization of 6-SLs cluster, and high-quality chaos synchronization is achievable among all of the SLs in the global SL network, by properly selecting coupling strength. As for the communication clusters of 2 SLs, 3SLs, 4 SLs, 5 SLs, and even two simultaneous communication clusters, the waveforms of intra-cluster SLs keep consistent by choosing proper coupling strengths. The results are similar to that of the 6-SLs communication cluster and not shown here for the sake of simplicity.

 

Fig. 2. Temporal waveforms of chaotic signals (first row), and corresponding power spectra (second row) in the global SL network with overall coupling strength (a) σ(1)=6ns-1, (b) σ(1)=12ns-1, (c) σ(1)=18ns-1. The spectra of SL1 are selected as representative.

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To further investigate the effects of key parameters on the chaos synchronization errors and obtain the synchronous parameter regions of different-size communication clusters, Fig. 3 presents the influences of the coupling strength and current factor on the intra-cluster RMS, for the scenarios of 2-SLs, 3-SLs, 4-SLs, 5-SLs, 6-SLs communication clusters and two co-existing 2-SLs communication clusters shown in Fig. 1. Here, the white dotted lines denote the boundaries of the synchronous regions (RMS<0.01) and the desynchronization ones (RMS>0.01), and the deep blue parameter regions (denoted by CS) indicate the high-quality chaos synchronization. It is shown that chaos synchronization can be achieved over wide parameter spaces of coupling strength and current factor. For the different OWI-SL networks shown in Figs. 1(a)–1(f), the topologies are different from each other, as such the parameter regions for intra-cluster chaos synchronization are distinct from each other. It is worth mentioning that, in the two co-existing 2-SLs clusters case, the parameter regions for chaos synchronization is obviously smaller than that of one 2-SLs cluster scenario. This is because the corresponding total external injections from the idle SLs are weaker compared with the 2-SLs cluster case under same injection strength. Since the parameter regions for chaos synchronization in different SL clusters are different, proper adjustment to the coupling strength is required if the network topology is changed. Over all, by properly selecting the parameters of current factor and coupling strength, intra-cluster chaos synchronization can be flexibly achieved in arbitrary communication clusters of different sizes.

 

Fig. 3. Two dimensional maps of RMS in SL networks containing a communication cluster of (a) 2 SLs, (b) 3 SLs, (c) 4 SLs, (d) 5 SLs, and (e) 6 SLs (global SL network), and containing (f) two 2-SLs communication clusters, in the parameter spaces of coupling strength and current factor. In the SL network of two simultaneous independent 2-SLs clusters, the results of two clusters are very similar, thus, the RMS of SL1 and SL2 are shown for simplicity.

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In the above discussions, the intra-cluster communication SLs are assumed to be coupled with identical coupling strengths and homogeneous coupling time delays. It is valuable to discuss the influences of mismatched coupling strengths and time delays on the quality of chaos synchronization, and the results are presented in Fig. 4. For simplicity, the mismatches of coupling strength and time delay [22], which are mathematically expressed as: σ’=σ(1+uσ), τ’=τ(1+uτ), are introduced to the common injections from an idle SL to two exemplary intra-cluster SLs of different-size communication clusters, and the CC of the mismatched intra-cluster SLs in different clusters is calculated. In particular, the parameter mismatches of the 6-SLs communication cluster which is not injected by any idle SL, are introduced to the common injection from a third communication SL to a pair one. As seen in Fig. 4(a), high-quality chaos synchronization with CC larger than 0.95 can be achieved in all of the different-size clusters over a wide range of coupling-strength mismatch. The synchronization characteristics of different clusters are different due to the structural differences among them as seen in Fig. 1. Regarding to the mismatch of time delay, as shown in Fig. 4(b), it is obviously observed that the communication clusters are sensitive to the mismatch of injection time delays or the intra-cluster injection time delays. Especially in the two independent 2-SLs clusters, the mismatched common injections from an idle SL account for a half of the total idle-cluster injections, and the corresponding proportion of the time-delay mismatch part is much higher than those in other communication clusters. Thus, the CC of the two-communication-cluster scenario with mismatched time delays is apparently smaller than those of one-communication-cluster cases. Therefore, we can conclude that the chaos synchronization of intra-cluster SL pairs is robust to the coupling-strength mismatch of common injections to some extent, while well-matched injection time delays from the idle SLs and well-matched coupling delays within communication clusters are desired for the achievement of high-quality chaos synchronization, indicating that the distances between communication SLs should be adjusted to be homogeneous and as fixed as possible to achieve stable chaos synchronization. Moreover, for the eavesdroppers, it is difficult to adjust the injection delays to be close to the private injection delays in our SL network, even if they obtain the chaotic outputs of idle cluster. Accordingly, the injection time delays from the idle clusters to the communication clusters can be regarded as the keys, which can further enhance the security of the communication clusters.

 

Fig. 4. The CC of different-size clusters versus parameter mismatches of (a) coupling strength and (b) time delay. Here, due to the similarity of two co-existing 2-SLs clusters shown in Fig. 1(f), the 2-SLs cluster composed of SL1 and SL2 is taken to present the results for simplicity.

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4. Multipoint-to-multipoint communications

Next, we turn to implement intra-cluster bidirectional chaotic communications and discuss the communication performance for different-size SL clusters in the SL networks. For the sake of simplicity, we take the 2-SLs cluster scenario in Fig. 1(b) for instance to demonstrate the feasibility of the proposed SL networks. Figure 5 illustrates the communication process for the 2-SLs cluster scenario. Under such a scenario, with the OWI, the global 6-SLs network is divided into two SL clusters, namely a 2-SLs communication cluster and an idle cluster composed of 4 idle SLs that are not involved for communication. The privacy of chaotic carriers is effectively protected since the carriers are only transmitted within the corresponding communication clusters. Regarding to the transmission from SL1 to SL2, the message m1(t) is firstly modulated into the chaotic carrier outputted by SL1, which is mathematically expressed by E1m(t)=E1(t)[1 + 0.05m1(t)], where the modulation index is chosen as 0.05 which is small enough to guarantee that the message is successfully hidden in the chaotic carrier [31,32]. After that, the modulated chaotic carrier (chaos + message) is transmitted to the receiver end. On the receiver end, the message is decoded by filtering the subtraction of |E1m|2 and |E2|2. The decryption process is presented as: m1’(t)=LPF[|E1m|2-|E2|2], wherein LPF denotes a filtering process using a low-pass fourth-order Butterworth filter with a cut-off frequency equaling to the message bit rate R. For the transmission from SL2 to SL1, the processes are similar and not stated here for the sake of simplicity.

 

Fig. 5. Schematic diagram of intra-cluster communication in the 2-SLs communication cluster, cluster of SL1 and SL2 is taken for instance, wherein SL: semiconductor laser, FC: fiber coupler, IM: intensity modulator, m(t): original message, OI: optical isolator.

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The recovered massages m1’(t) and m2’(t) in the 2-SLs intra-cluster transmission cases of 1 Gb/s, 5 Gb/s, and 10 Gb/s are presented in Fig. 6. As shown in Figs. 6(a1) and 6(a2), the exchanged massages are correctly decoded in the 1 Gb/s case. With the increase of message bit rate R, bigger fluctuations are observed in the waveforms of the messages recovered by receivers compared with the original messages. This is because the high-frequency components of chaotic carriers are more susceptible to noise than the low-frequency components and the corresponding synchronization performance of high-frequency components is relatively poor, as such the communication performance of low-rate transmission is better than that of high-rate cases. Therefore, it can be concluded that the quality of recovered messages in 2-SLs intra-cluster is influenced by the message bit rate.

 

Fig. 6. The original messages (black dotted lines) and the recovered messages m1’(t) (red solid lines) and m2’(t) (blue solid lines) in the 2-SLs communication cluster with message bit rates of (a) 1 Gb/s, (b) 5 Gb/s, and (c) 10 Gb/s.

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The quality of the decrypted messages can also be evaluated by the eye diagrams. Figure 7 shows the eye diagrams of recovered messages m1’(t) and m2’(t) as well as the intercepted messages m1e(t) and m2e(t) in the 2-SLs intra-cluster communication. Here, we consider the typical attack scenario of directly filtering the signals transmitted over public links by using a low pass filter with a cut-off frequency equaling to R [33,34]. It is assumed that the eavesdroppers know the bit rate of message. As presented in Fig. 7(a), widely-open eyes are observed in the 1-Gb/s case for the legal receivers. As the bit rate of message increases, the eye diagrams have narrower openings, which means the performance of low-R transmission case is better than that of high-R case for two main reasons. On the one hand, due to the chaotic filtering effect, more low-frequency noise in low-speed transmissions are filtered compared with the high-speed transmission cases. On the other hand, the effective bandwidth of synchronized chaotic carrier is around 30 GHz which limits the transmission capacity. For the illegal attack scenarios, the eyes are completely closed and indistinguishable in the transmission cases of different rates as shown in Figs. 7(d)–7(f). Thus, it is concluded that the messages can be successfully recovered by the legal receivers in the 2-SLs intra-cluster communication, and the security of communication system is guaranteed since the eavesdroppers cannot obtain any useful information through the public links.

 

Fig. 7. The eye diagrams of the recovered messages m1’(t) (first column) and m2’(t) (second column) in the 2-SLs communication cluster with message bit rate of (a) 1 Gb/s, (b) 5 Gb/s, and (c) 10 Gb/s, and the eye diagrams of the intercepted messages m1e (third column) and m2e (fourth column) in the cases of (d) 1 Gb/s, (e) 5 Gb/s, and (f) 10 Gb/s.

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Then, the communication performance is systematically and intuitively investigated in the 2-SLs, 3-SLs, 4-SLs, 5-SLs, 6-SLs, and two co-existing 2-SLs communication clusters. Figure 8 depicts the dependence of Q-factors of legal received messages and illegal intercepted messages in the transmission clusters on the message bit rate R. The definition of Q-factor is identical to that in Ref. [11], and satisfactory communication performance is obtained when the Q-factor value is greater than 6. For the sake of simplicity, the messages sent by the same transmitter are assumed to be identical, namely, the message sent by SL1 is denoted by m1, and that sent by SL2 is denoted by m2, etc. For the 2-SLs intra-cluster communication, it is shown in Fig. 8(a) that the Q-factors of recovered messages decrease as the R increases. Even so, satisfactory performance (Q>6) of chaos communication is still achieved with bit rate of message over 8 Gb/s. For communications in larger clusters given in Figs. 8(b)–8(e), the larger the R, the smaller the Q-factors, and the development trends are similar to that of 2-SLs intra-cluster communication. Moreover, it is found that the communication performance in clusters of smaller sizes is better than that in larger clusters. This is because more messages are simultaneously exchanged between SLs in the larger clusters, and consequently, the quality of chaos synchronization and communication performance would be easier to be influenced. Considering a possible communication scenario: simultaneous intra-cluster communication in two communication clusters, as shown in Fig. 8(f), intra-cluster communications of good performance are simultaneously achieved in two 2-SLs clusters of the SL network. The result indicates that two independent communication clusters can be simultaneously supported in the proposed SL networks with OWI. Besides, it is also noticed that the communication performance of two independent 2-SLs clusters is very similar. This is because the two clusters are completely symmetric to each other as seen in Fig. 1(f). For the illegal attack in the intra-cluster communications, the values of Q-factors all maintain at a very low level (Q<2.2) over the whole range of transmission bit rate since the uniform energy distribution of chaotic carriers presented in Fig. 2(c2) ensures the message can be successfully hidden even in the low-R scenarios. Compared with the traditional point-to-point communication which is easy to be attacked (Q>6) in the low-rate range [11,35], the security of intra-cluster communications in our proposed SL networks is significantly enhanced. Thus, it is demonstrated that by using OWI, Gb/s communications are successfully realized in the arbitrary-size clusters of the SL networks, and the flexibility and security of communication clusters are greatly improved. Furthermore, in terms of the OWI-SL network with larger size, multipoint-to-multipoint secure communications among more users can also be achieved, and the results are not presented here for the sake of simplicity.

 

Fig. 8. The Q-factors of legal recovered messages and the illegal intercepted messages versus message bit rate R in the communication clusters of the (a) 2 SLs, (b) 3 SLs, (c) 4 SLs, (d) 5 SLs, and (e) 6 SLs, as well as (f) two independent 2-SLs clusters for simultaneous communications. Here, the messages intercepted by the eavesdroppers are denoted by m1e, m2e, m3e, m4e, m5e, and m6e.

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In the above discussions, we assume that messages transmissions are simultaneously performed in all SL pairs of a communication cluster, and the messages sent by the same transmitter (SL) are identical. To further validate the generality of our proposed scheme, we also discuss two types of intra-cluster communications, which are more common in practice. One type is that not all pairwise SLs in the clusters communicate simultaneously, and it is performed by assuming that messages are only exchanged between SL1 and SL2, SL2 and SL3, while SL1 and SL3 are not communicating. Here, false messages are embedded into the chaotic carriers transmitted between SL1 and SL3 to avoid exposing of synchronized carrier [18]. The other type is that the transmitted messages of different transmitter and receiver pairs are different, and the message transmitted from SLi to SLj is denoted by mij. The network-type communication of 3-SL cluster is taken for instance to demonstrate two types of common scenarios, and the corresponding Q-factors as the function of message bit rate are illustrated in Fig. 9. The results show that satisfactory transmission performance is realized beyond 6 Gb/s in two common scenarios, and the results are similar to those under two assumptions shown in Fig. 8(b). Thus, we can conclude that the proposed SL networks under OWI support partial intra-cluster communication and communication with different messages between different intra-cluster SL pairs. Moreover, under such scenarios, the low-value Q-factors mean that the eavesdroppers cannot obtain any useful information, and with the contribution of OWI, the privacy of the carriers is ensured because the chaotic carriers are only transmitted in the corresponding communication cluster, which further improve the security of flexible multipoint-to-multipoint communications.

 

Fig. 9. The Q-factors of legal recovered messages and the illegal intercepted messages in 3-SLs communication cluster versus message bit rate R, in the common cases of (a) non-simultaneous message exchange, and (b) different messages among different transmitter and receiver pairs. For the second case, mij stands for message transmitted from SLi to SLj, and the messages bidirectionally intercepted by the eavesdroppers are denoted as m12e, m21e, m13e, m31e, m23e, and m32e.

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5. Discussions

Up to present, relevant previously reported approaches mainly focused on the chaos synchronization in fixed SL networks, in which chaos synchronization can be achieved in fixed SL clusters [1820]. In our previous work [18] and Ref. [19], cluster chaos synchronization was investigated in different types of SL networks, including the small-world network, random network, the Nepal power grid network, etc., and the results showed that isochronous chaos synchronization can be achieved among intra-cluster SLs. In Ref. [20], hierarchical chaos synchronization was demonstrated in tree-type SL networks where SLs belonging to the same layer can be synchronized. In these reported works, only the intra-cluster symmetric SLs can be synchronized while the SLs of different clusters are asynchronous. Besides, it is difficult to change the network topologies to satisfy different demands, and consequently, the flexibility of chaos synchronization in SL networks is limited. Compared with the existing methods of chaos synchronization, our SL networks can support high-quality chaos synchronization between any SLs with the contribution of OWI, which indicate that the flexibility of the chaos synchronization in SL networks is greatly improved. Additionally, the intra-cluster chaos synchronization of all the different-size clusters is robust to the mismatch of coupling strength to some extent, and the sensitivity of intra-cluster chaos synchronization to the mismatch of external injection time delays means that the time delays of the injections from idle SLs to the communication SLs can be employed as the keys. For this reason, the privacy of chaos synchronization among any SLs can be ensured. On the basis of flexible chaos synchronization, secure key distributions and chaotic communications in arbitrary-size SL clusters can be available.

Regarding to the network-type chaotic communication, broadcast message transmissions from the center SL to the side SLs in a star-type SL network as well as the bidirectional chaotic communication between neighboring SLs in a ring SL network was numerically achieved with message bit rate of several hundreds of Mbit/s in Ref. [21]. Besides, in Ref. [22], WDM-based communication system was constructed in a star-type SL network, and Gb/s secure communication was realized between two equal parts composed of side SLs. A complex network composed of two-layer chaotic nodes is proposed and demonstrated in Ref. [36], where secure chaotic communication is achieved between pairs of matching nodes by using one-time-one-cipher encryption method. In our previous work, intra-cluster message exchange in the SL network composed of 12 SLs was investigated with bit rate over Gbit/s based on cluster chaos synchronization [18]. In most of these reported schemes, message can be only exchanged among specific users which are symmetric to each other, and the various transmission demands in real world could not be satisfied. For the purpose of realizing flexible chaotic communication, this paper systematically discusses the multipoint-to-multipoint transmissions in the communication clusters composed of different number of users, by using OWI. With respect to the existing multi-user communication schemes, our work proposes a novel OWI method to realize the flexible multipoint-to-multipoint communications, in which the communicating SLs are not limited to several specific SLs. Moreover, the security of the proposed communication system is demonstrated from three aspects. Firstly, by using a typical attack method, the eye diagrams of the intercepted messages are completely close and indistinguishable. Secondly, the chaotic carriers are only transmitted within the communication clusters since the injections from the communication clusters to the idle cluster are cut off with the introduction of OWI, and for this reason eavesdroppers are unable to simultaneously obtain the unmodulated carriers and the modulated carriers to recover the correct messages. Thirdly, as demonstrated in Fig. 4(b), the injection time delays from idle SLs to the communication SLs can be regarded as the keys of the communication system, and consequently the security can be obviously enhanced.

6. Conclusion

In summary, flexible chaos synchronization and multipoint-to-multipoint communications are systematically investigated in SL networks by using one-way isolation. In the proposed SL networks, it is demonstrated that chaos synchronization can be realized in the arbitrary-size clusters over a wide parameter range. Besides, the quality of chaos synchronization in different clusters is all robust to the mismatch of coupling strengths to some extent but sensitive to the mismatched time delays. Based on high-quality intra-cluster chaos synchronization, messages bidirectionally transmitted in arbitrary-size SL clusters are successfully recovered with bit rate over several Gb/s, and simultaneous communications in two independent 2-SL communication clusters of a SL network are also supported. Moreover, the security of flexible intra-cluster message transmissions is guaranteed because the communications in arbitrary-size SL clusters are all against interceptions over the public links during the whole range of message bit rate. Meanwhile, with the contribution of one-way isolation, on the one hand, the privacy of the chaotic carriers is protected since the idle SLs cannot obtain any chaotic injections from the communication SLs, on the other hand, the security of communication clusters is further improved because the injection time delays from the idle SLs to the communication SLs can be used as the keys. Compared with the previously reported communication schemes, the flexibility of multipoint-to-multipoint communications is obviously enhanced since message exchange among any SLs is achievable. Our work provides a novel multipoint-to-multipoint communication scheme to realize flexible secure multiple-user communications.

Funding

National Natural Science Foundation of China (61671119, 61805031); Fundamental Research Funds for the Central Universities (ZYGX2019J003).

Disclosures

The authors declare no conflicts of interest.

References

1. M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015). [CrossRef]  

2. M. C. Soriano, F. Ruiz-Oliveras, P. Colet, and C. R. Mirasso, “Synchronization properties of coupled semiconductor lasers subject to filtered optical feedback,” Phys. Rev. E 78(4), 046218 (2008). [CrossRef]  

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

4. C. C. Cui, X. L. Fu, and S. C. Chan, “Double-locked semiconductor laser for radio-over-fiber uplink transmission,” Opt. Lett. 34(24), 3821–3823 (2009). [CrossRef]  

5. 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]  

6. S. Y. Xiang, Z. Ren, Y. Zhang, Z. Song, and Y. Hao, “All-optical neuromorphic XOR operation with inhibitory dynamics of a single photonic spiking neuron based on VCSEL-SA,” Opt. Lett. 45(5), 1104–1107 (2020). [CrossRef]  

7. N. Jiang, Y. J. Wang, A. K. Zhao, S. Q. Liu, Y. Q. Zhang, L. Chen, B. C. 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]  

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

9. G. Q. Xia, Z. M. Wu, and J. G. Wu, “Theory and simulation of dual-channel optical chaotic communication system,” Opt. Express 13(9), 3445–3453 (2005). [CrossRef]  

10. J. G. Wu, Z. M. Wu, X. Tang, L. Fan, W. Deng, and G. Q. Xia, “Experimental demonstration of LD-based bidirectional fiber-optic chaos communication,” IEEE Photonics Technol. Lett. 25(6), 587–590 (2013). [CrossRef]  

11. 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]  

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

13. D. M. Wang, L. S. Wang, Y. Y. Guo, Y. C. Wang, and A. B. Wang, “Key space enhancement of optical chaos secure communication: chirped FBG feedback semiconductor laser,” Opt. Express 27(3), 3065–3073 (2019). [CrossRef]  

14. X. J. Gao, M. F. Cheng, L. Deng, M. M. Zhang, S. N. Fu, and D. M. Liu, “Robust chaotic-shift-keying scheme based on electro-optical hybrid feedback system,” Opt. Express 28(8), 10847–10858 (2020). [CrossRef]  

15. M. F. Cheng, C. K. Luo, X. X. Jiang, L. Deng, M. M. Zhang, C. J. Ke, S. N. Fu, M. Tang, P. Shum, and D. M. Liu, “An electrooptic chaotic system based on a hybrid feedback loop,” J. Lightwave Technol. 36(19), 4259–4266 (2018). [CrossRef]  

16. Y. H. Hong, M. W. Lee, and K. A. Shore, “Optimised message extraction in laser diode based optical chaos communications,” IEEE J. Quantum Electron. 46(2), 253–257 (2010). [CrossRef]  

17. N. Q. Li, W. Pan, L. S. Yan, B. Luo, and X. H. Zou, “Enhanced chaos synchronization and communication in cascade-coupled semiconductor ring lasers,” Commun. Nonlinear Sci. Numer. Simul. 19(6), 1874–1883 (2014). [CrossRef]  

18. S. Q. Liu, N. Jiang, A. K. Zhao, Y. Q. Zhang, and K. Qiu, “Secure optical communication based on cluster chaos synchronization in semiconductor lasers network,” IEEE Access 8, 11872–11879 (2020). [CrossRef]  

19. L. Y. Zhang, W. Pan, L. S. Yan, B. Luo, X. H. Zou, and M. F. Xu, “Cluster synchronization of coupled semiconductor lasers network with complex topology,” IEEE J. Sel. Top. Quantum Electron. 25(6), 1–7 (2019). [CrossRef]  

20. S. Y. Xiang, Y. N. Han, H. N. Wang, A. J. Wen, and Y. Hao, “Zero-lag chaos synchronization properties in a hierarchical tree-type network consisting of mutually coupled semiconductor lasers,” Nonlinear Dyn. 99(4), 2893–2906 (2020). [CrossRef]  

21. Q. L. Li, Q. Bao, D. W. Chen, S. N. Yang, M. Hu, R. Zeng, H. Chi, and Q. Li, “Point-to-multipoint and ring network communication based on chaotic semiconductor lasers with optical feedback,” Appl. Opt. 58(4), 1025–1032 (2019). [CrossRef]  

22. X. Bao, Q. Li, T. Wu, Y. Deng, M. Hu, and R. Zeng, “WDM-based bidirectional chaotic communication for semiconductor lasers system with time delay concealment,” Opt. Commun. 472, 125868 (2020). [CrossRef]  

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

24. N. Q. Li, H. Susanto, B. R. Cemlyn, I. D. Henning, and M. J. Adams, “Modulation properties of solitary and optically injected phased-array semiconductor lasers,” Photonics Res. 6(9), 908–917 (2018). [CrossRef]  

25. P. Li, Q. Cai, J. G. Zhang, B. J. Xu, Y. M. Liu, A. Bogris, K. A. Shore, and Y. C. Wang, “Observation of flat chaos generation using an optical feedback multi-mode laser with a band-pass filter,” Opt. Express 27(13), 17859–17867 (2019). [CrossRef]  

26. X. Z. Li and S. C. Chan, “Detection dependencies of statistical properties for semiconductor laser chaos,” IEEE J. Sel. Top. Quantum Electron. 25(6), 1–9 (2019). [CrossRef]  

27. S. Y. Xiang, J. K. Gong, H. Zhang, X. X. Guo, H. N. Wang, Y. H. Zhang, and A. J. Wen, “Zero-lag intensity correlation properties in small ring laser network with heterogeneous delays,” J. Opt. Soc. Am. B 35(2), 287–294 (2018). [CrossRef]  

28. Y. H. Hong and S. K. Ji, “Effect of digital acquisition on the complexity of chaos,” Opt. Lett. 42(13), 2507–2510 (2017). [CrossRef]  

29. L. Y. Zhang, W. Pan, L. S. Yan, B. Luo, X. Zou, and M. F. 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]  

30. M. F. Xu, W. Pan, and L. Y. Zhang, “Isolated desynchronization and intertwined synchronization in networks of semiconductor lasers,” Opt. Eng. 57(4), 046106 (2018). [CrossRef]  

31. N. Q. Li, H. Susanto, B. Cemlyn, I. D. Henning, and M. J. Adams, “Secure communication systems based on chaos in optically pumped spin-VCSELs,” Opt. Lett. 42(17), 3494–3497 (2017). [CrossRef]  

32. D. M. Wang, L. S. Wang, P. Li, T. Zhao, Z. W. Jia, Z. S. Gao, Y. Y. Guo, Y. C. Wang, and A. B. Wang, “Bias current of semiconductor laser: an unsafe key for secure chaos communication,” Photonics 6(2), 59 (2019). [CrossRef]  

33. A. Bogris, A. Argyris, and D. Syvridis, “Encryption efficiency analysis of chaotic communication systems based on photonic integrated chaotic circuits,” IEEE J. Quantum Electron. 46(10), 1421–1429 (2010). [CrossRef]  

34. 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]  

35. A. Jacobo, M. C. Soriano, C. R. Mirasso, and P. Colet, “Chaos-based optical communications: encryption versus nonlinear filtering,” IEEE J. Quantum Electron. 46(4), 499–505 (2010). [CrossRef]  

36. L. Zhou and F. Tan, “A chaotic secure communication scheme based on synchronization of double-layered and multiple complex networks,” Nonlinear Dyn. 96(2), 869–883 (2019). [CrossRef]  

References

  • View by:
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  • |
  • |

  1. M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015).
    [Crossref]
  2. M. C. Soriano, F. Ruiz-Oliveras, P. Colet, and C. R. Mirasso, “Synchronization properties of coupled semiconductor lasers subject to filtered optical feedback,” Phys. Rev. E 78(4), 046218 (2008).
    [Crossref]
  3. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
    [Crossref]
  4. C. C. Cui, X. L. Fu, and S. C. Chan, “Double-locked semiconductor laser for radio-over-fiber uplink transmission,” Opt. Lett. 34(24), 3821–3823 (2009).
    [Crossref]
  5. 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]
  6. S. Y. Xiang, Z. Ren, Y. Zhang, Z. Song, and Y. Hao, “All-optical neuromorphic XOR operation with inhibitory dynamics of a single photonic spiking neuron based on VCSEL-SA,” Opt. Lett. 45(5), 1104–1107 (2020).
    [Crossref]
  7. N. Jiang, Y. J. Wang, A. K. Zhao, S. Q. Liu, Y. Q. Zhang, L. Chen, B. C. 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]
  8. L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64(8), 821–824 (1990).
    [Crossref]
  9. G. Q. Xia, Z. M. Wu, and J. G. Wu, “Theory and simulation of dual-channel optical chaotic communication system,” Opt. Express 13(9), 3445–3453 (2005).
    [Crossref]
  10. J. G. Wu, Z. M. Wu, X. Tang, L. Fan, W. Deng, and G. Q. Xia, “Experimental demonstration of LD-based bidirectional fiber-optic chaos communication,” IEEE Photonics Technol. Lett. 25(6), 587–590 (2013).
    [Crossref]
  11. 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]
  12. J. X. Ke, L. L. Yi, G. Q. Xia, and W. S. Hu, “Chaotic optical communications over 100-km fiber transmission at 30-Gb/s bit rate,” Opt. Lett. 43(6), 1323–1326 (2018).
    [Crossref]
  13. D. M. Wang, L. S. Wang, Y. Y. Guo, Y. C. Wang, and A. B. Wang, “Key space enhancement of optical chaos secure communication: chirped FBG feedback semiconductor laser,” Opt. Express 27(3), 3065–3073 (2019).
    [Crossref]
  14. X. J. Gao, M. F. Cheng, L. Deng, M. M. Zhang, S. N. Fu, and D. M. Liu, “Robust chaotic-shift-keying scheme based on electro-optical hybrid feedback system,” Opt. Express 28(8), 10847–10858 (2020).
    [Crossref]
  15. M. F. Cheng, C. K. Luo, X. X. Jiang, L. Deng, M. M. Zhang, C. J. Ke, S. N. Fu, M. Tang, P. Shum, and D. M. Liu, “An electrooptic chaotic system based on a hybrid feedback loop,” J. Lightwave Technol. 36(19), 4259–4266 (2018).
    [Crossref]
  16. Y. H. Hong, M. W. Lee, and K. A. Shore, “Optimised message extraction in laser diode based optical chaos communications,” IEEE J. Quantum Electron. 46(2), 253–257 (2010).
    [Crossref]
  17. N. Q. Li, W. Pan, L. S. Yan, B. Luo, and X. H. Zou, “Enhanced chaos synchronization and communication in cascade-coupled semiconductor ring lasers,” Commun. Nonlinear Sci. Numer. Simul. 19(6), 1874–1883 (2014).
    [Crossref]
  18. S. Q. Liu, N. Jiang, A. K. Zhao, Y. Q. Zhang, and K. Qiu, “Secure optical communication based on cluster chaos synchronization in semiconductor lasers network,” IEEE Access 8, 11872–11879 (2020).
    [Crossref]
  19. L. Y. Zhang, W. Pan, L. S. Yan, B. Luo, X. H. Zou, and M. F. Xu, “Cluster synchronization of coupled semiconductor lasers network with complex topology,” IEEE J. Sel. Top. Quantum Electron. 25(6), 1–7 (2019).
    [Crossref]
  20. S. Y. Xiang, Y. N. Han, H. N. Wang, A. J. Wen, and Y. Hao, “Zero-lag chaos synchronization properties in a hierarchical tree-type network consisting of mutually coupled semiconductor lasers,” Nonlinear Dyn. 99(4), 2893–2906 (2020).
    [Crossref]
  21. Q. L. Li, Q. Bao, D. W. Chen, S. N. Yang, M. Hu, R. Zeng, H. Chi, and Q. Li, “Point-to-multipoint and ring network communication based on chaotic semiconductor lasers with optical feedback,” Appl. Opt. 58(4), 1025–1032 (2019).
    [Crossref]
  22. X. Bao, Q. Li, T. Wu, Y. Deng, M. Hu, and R. Zeng, “WDM-based bidirectional chaotic communication for semiconductor lasers system with time delay concealment,” Opt. Commun. 472, 125868 (2020).
    [Crossref]
  23. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
    [Crossref]
  24. N. Q. Li, H. Susanto, B. R. Cemlyn, I. D. Henning, and M. J. Adams, “Modulation properties of solitary and optically injected phased-array semiconductor lasers,” Photonics Res. 6(9), 908–917 (2018).
    [Crossref]
  25. P. Li, Q. Cai, J. G. Zhang, B. J. Xu, Y. M. Liu, A. Bogris, K. A. Shore, and Y. C. Wang, “Observation of flat chaos generation using an optical feedback multi-mode laser with a band-pass filter,” Opt. Express 27(13), 17859–17867 (2019).
    [Crossref]
  26. X. Z. Li and S. C. Chan, “Detection dependencies of statistical properties for semiconductor laser chaos,” IEEE J. Sel. Top. Quantum Electron. 25(6), 1–9 (2019).
    [Crossref]
  27. S. Y. Xiang, J. K. Gong, H. Zhang, X. X. Guo, H. N. Wang, Y. H. Zhang, and A. J. Wen, “Zero-lag intensity correlation properties in small ring laser network with heterogeneous delays,” J. Opt. Soc. Am. B 35(2), 287–294 (2018).
    [Crossref]
  28. Y. H. Hong and S. K. Ji, “Effect of digital acquisition on the complexity of chaos,” Opt. Lett. 42(13), 2507–2510 (2017).
    [Crossref]
  29. L. Y. Zhang, W. Pan, L. S. Yan, B. Luo, X. Zou, and M. F. 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]
  30. M. F. Xu, W. Pan, and L. Y. Zhang, “Isolated desynchronization and intertwined synchronization in networks of semiconductor lasers,” Opt. Eng. 57(4), 046106 (2018).
    [Crossref]
  31. N. Q. Li, H. Susanto, B. Cemlyn, I. D. Henning, and M. J. Adams, “Secure communication systems based on chaos in optically pumped spin-VCSELs,” Opt. Lett. 42(17), 3494–3497 (2017).
    [Crossref]
  32. D. M. Wang, L. S. Wang, P. Li, T. Zhao, Z. W. Jia, Z. S. Gao, Y. Y. Guo, Y. C. Wang, and A. B. Wang, “Bias current of semiconductor laser: an unsafe key for secure chaos communication,” Photonics 6(2), 59 (2019).
    [Crossref]
  33. A. Bogris, A. Argyris, and D. Syvridis, “Encryption efficiency analysis of chaotic communication systems based on photonic integrated chaotic circuits,” IEEE J. Quantum Electron. 46(10), 1421–1429 (2010).
    [Crossref]
  34. 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]
  35. A. Jacobo, M. C. Soriano, C. R. Mirasso, and P. Colet, “Chaos-based optical communications: encryption versus nonlinear filtering,” IEEE J. Quantum Electron. 46(4), 499–505 (2010).
    [Crossref]
  36. L. Zhou and F. Tan, “A chaotic secure communication scheme based on synchronization of double-layered and multiple complex networks,” Nonlinear Dyn. 96(2), 869–883 (2019).
    [Crossref]

2020 (6)

S. Y. Xiang, Z. Ren, Y. Zhang, Z. Song, and Y. Hao, “All-optical neuromorphic XOR operation with inhibitory dynamics of a single photonic spiking neuron based on VCSEL-SA,” Opt. Lett. 45(5), 1104–1107 (2020).
[Crossref]

N. Jiang, Y. J. Wang, A. K. Zhao, S. Q. Liu, Y. Q. Zhang, L. Chen, B. C. 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]

X. J. Gao, M. F. Cheng, L. Deng, M. M. Zhang, S. N. Fu, and D. M. Liu, “Robust chaotic-shift-keying scheme based on electro-optical hybrid feedback system,” Opt. Express 28(8), 10847–10858 (2020).
[Crossref]

S. Q. Liu, N. Jiang, A. K. Zhao, Y. Q. Zhang, and K. Qiu, “Secure optical communication based on cluster chaos synchronization in semiconductor lasers network,” IEEE Access 8, 11872–11879 (2020).
[Crossref]

S. Y. Xiang, Y. N. Han, H. N. Wang, A. J. Wen, and Y. Hao, “Zero-lag chaos synchronization properties in a hierarchical tree-type network consisting of mutually coupled semiconductor lasers,” Nonlinear Dyn. 99(4), 2893–2906 (2020).
[Crossref]

X. Bao, Q. Li, T. Wu, Y. Deng, M. Hu, and R. Zeng, “WDM-based bidirectional chaotic communication for semiconductor lasers system with time delay concealment,” Opt. Commun. 472, 125868 (2020).
[Crossref]

2019 (9)

Q. L. Li, Q. Bao, D. W. Chen, S. N. Yang, M. Hu, R. Zeng, H. Chi, and Q. Li, “Point-to-multipoint and ring network communication based on chaotic semiconductor lasers with optical feedback,” Appl. Opt. 58(4), 1025–1032 (2019).
[Crossref]

P. Li, Q. Cai, J. G. Zhang, B. J. Xu, Y. M. Liu, A. Bogris, K. A. Shore, and Y. C. Wang, “Observation of flat chaos generation using an optical feedback multi-mode laser with a band-pass filter,” Opt. Express 27(13), 17859–17867 (2019).
[Crossref]

X. Z. Li and S. C. Chan, “Detection dependencies of statistical properties for semiconductor laser chaos,” IEEE J. Sel. Top. Quantum Electron. 25(6), 1–9 (2019).
[Crossref]

L. Y. Zhang, W. Pan, L. S. Yan, B. Luo, X. Zou, and M. F. 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]

D. M. Wang, L. S. Wang, P. Li, T. Zhao, Z. W. Jia, Z. S. Gao, Y. Y. Guo, Y. C. Wang, and A. B. Wang, “Bias current of semiconductor laser: an unsafe key for secure chaos communication,” Photonics 6(2), 59 (2019).
[Crossref]

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]

L. Zhou and F. Tan, “A chaotic secure communication scheme based on synchronization of double-layered and multiple complex networks,” Nonlinear Dyn. 96(2), 869–883 (2019).
[Crossref]

L. Y. Zhang, W. Pan, L. S. Yan, B. Luo, X. H. Zou, and M. F. Xu, “Cluster synchronization of coupled semiconductor lasers network with complex topology,” IEEE J. Sel. Top. Quantum Electron. 25(6), 1–7 (2019).
[Crossref]

D. M. Wang, L. S. Wang, Y. Y. Guo, Y. C. Wang, and A. B. Wang, “Key space enhancement of optical chaos secure communication: chirped FBG feedback semiconductor laser,” Opt. Express 27(3), 3065–3073 (2019).
[Crossref]

2018 (6)

2017 (2)

2016 (1)

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]

2015 (1)

M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015).
[Crossref]

2014 (1)

N. Q. Li, W. Pan, L. S. Yan, B. Luo, and X. H. Zou, “Enhanced chaos synchronization and communication in cascade-coupled semiconductor ring lasers,” Commun. Nonlinear Sci. Numer. Simul. 19(6), 1874–1883 (2014).
[Crossref]

2013 (1)

J. G. Wu, Z. M. Wu, X. Tang, L. Fan, W. Deng, and G. Q. Xia, “Experimental demonstration of LD-based bidirectional fiber-optic chaos communication,” IEEE Photonics Technol. Lett. 25(6), 587–590 (2013).
[Crossref]

2010 (3)

Y. H. Hong, M. W. Lee, and K. A. Shore, “Optimised message extraction in laser diode based optical chaos communications,” IEEE J. Quantum Electron. 46(2), 253–257 (2010).
[Crossref]

A. Bogris, A. Argyris, and D. Syvridis, “Encryption efficiency analysis of chaotic communication systems based on photonic integrated chaotic circuits,” IEEE J. Quantum Electron. 46(10), 1421–1429 (2010).
[Crossref]

A. Jacobo, M. C. Soriano, C. R. Mirasso, and P. Colet, “Chaos-based optical communications: encryption versus nonlinear filtering,” IEEE J. Quantum Electron. 46(4), 499–505 (2010).
[Crossref]

2009 (1)

2008 (1)

M. C. Soriano, F. Ruiz-Oliveras, P. Colet, and C. R. Mirasso, “Synchronization properties of coupled semiconductor lasers subject to filtered optical feedback,” Phys. Rev. E 78(4), 046218 (2008).
[Crossref]

2005 (2)

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

G. Q. Xia, Z. M. Wu, and J. G. Wu, “Theory and simulation of dual-channel optical chaotic communication system,” Opt. Express 13(9), 3445–3453 (2005).
[Crossref]

1990 (1)

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

Adams, M. J.

N. Q. Li, H. Susanto, B. R. Cemlyn, I. D. Henning, and M. J. Adams, “Modulation properties of solitary and optically injected phased-array semiconductor lasers,” Photonics Res. 6(9), 908–917 (2018).
[Crossref]

N. Q. Li, H. Susanto, B. Cemlyn, I. D. Henning, and M. J. Adams, “Secure communication systems based on chaos in optically pumped spin-VCSELs,” Opt. Lett. 42(17), 3494–3497 (2017).
[Crossref]

Annovazzi-Lodi, V.

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

Argyris, A.

A. Bogris, A. Argyris, and D. Syvridis, “Encryption efficiency analysis of chaotic communication systems based on photonic integrated chaotic circuits,” IEEE J. Quantum Electron. 46(10), 1421–1429 (2010).
[Crossref]

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

Bao, Q.

Bao, X.

X. Bao, Q. Li, T. Wu, Y. Deng, M. Hu, and R. Zeng, “WDM-based bidirectional chaotic communication for semiconductor lasers system with time delay concealment,” Opt. Commun. 472, 125868 (2020).
[Crossref]

Bogris, A.

P. Li, Q. Cai, J. G. Zhang, B. J. Xu, Y. M. Liu, A. Bogris, K. A. Shore, and Y. C. Wang, “Observation of flat chaos generation using an optical feedback multi-mode laser with a band-pass filter,” Opt. Express 27(13), 17859–17867 (2019).
[Crossref]

A. Bogris, A. Argyris, and D. Syvridis, “Encryption efficiency analysis of chaotic communication systems based on photonic integrated chaotic circuits,” IEEE J. Quantum Electron. 46(10), 1421–1429 (2010).
[Crossref]

Cai, Q.

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.

Cemlyn, B. R.

N. Q. Li, H. Susanto, B. R. Cemlyn, I. D. Henning, and M. J. Adams, “Modulation properties of solitary and optically injected phased-array semiconductor lasers,” Photonics Res. 6(9), 908–917 (2018).
[Crossref]

Chan, S. C.

X. Z. Li and S. C. Chan, “Detection dependencies of statistical properties for semiconductor laser chaos,” IEEE J. Sel. Top. Quantum Electron. 25(6), 1–9 (2019).
[Crossref]

C. C. Cui, X. L. Fu, and S. C. Chan, “Double-locked semiconductor laser for radio-over-fiber uplink transmission,” Opt. Lett. 34(24), 3821–3823 (2009).
[Crossref]

Chen, D. W.

Chen, L.

Cheng, M. F.

Chi, H.

Colet, P.

A. Jacobo, M. C. Soriano, C. R. Mirasso, and P. Colet, “Chaos-based optical communications: encryption versus nonlinear filtering,” IEEE J. Quantum Electron. 46(4), 499–505 (2010).
[Crossref]

M. C. Soriano, F. Ruiz-Oliveras, P. Colet, and C. R. Mirasso, “Synchronization properties of coupled semiconductor lasers subject to filtered optical feedback,” Phys. Rev. E 78(4), 046218 (2008).
[Crossref]

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

Cui, C. C.

Deng, L.

Deng, W.

J. G. Wu, Z. M. Wu, X. Tang, L. Fan, W. Deng, and G. Q. Xia, “Experimental demonstration of LD-based bidirectional fiber-optic chaos communication,” IEEE Photonics Technol. Lett. 25(6), 587–590 (2013).
[Crossref]

Deng, Y.

X. Bao, Q. Li, T. Wu, Y. Deng, M. Hu, and R. Zeng, “WDM-based bidirectional chaotic communication for semiconductor lasers system with time delay concealment,” Opt. Commun. 472, 125868 (2020).
[Crossref]

Fan, L.

J. G. Wu, Z. M. Wu, X. Tang, L. Fan, W. Deng, and G. Q. Xia, “Experimental demonstration of LD-based bidirectional fiber-optic chaos communication,” IEEE Photonics Technol. Lett. 25(6), 587–590 (2013).
[Crossref]

Fischer, I.

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

Fu, S. N.

Fu, X. L.

Gao, X. J.

Gao, Z. S.

D. M. Wang, L. S. Wang, P. Li, T. Zhao, Z. W. Jia, Z. S. Gao, Y. Y. Guo, Y. C. Wang, and A. B. Wang, “Bias current of semiconductor laser: an unsafe key for secure chaos communication,” Photonics 6(2), 59 (2019).
[Crossref]

García-Ojalvo, J.

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

Gong, J. K.

Guo, X. X.

Guo, Y. Y.

D. M. Wang, L. S. Wang, Y. Y. Guo, Y. C. Wang, and A. B. Wang, “Key space enhancement of optical chaos secure communication: chirped FBG feedback semiconductor laser,” Opt. Express 27(3), 3065–3073 (2019).
[Crossref]

D. M. Wang, L. S. Wang, P. Li, T. Zhao, Z. W. Jia, Z. S. Gao, Y. Y. Guo, Y. C. Wang, and A. B. Wang, “Bias current of semiconductor laser: an unsafe key for secure chaos communication,” Photonics 6(2), 59 (2019).
[Crossref]

Han, Y. N.

S. Y. Xiang, Y. N. Han, H. N. Wang, A. J. Wen, and Y. Hao, “Zero-lag chaos synchronization properties in a hierarchical tree-type network consisting of mutually coupled semiconductor lasers,” Nonlinear Dyn. 99(4), 2893–2906 (2020).
[Crossref]

Hao, Y.

S. Y. Xiang, Y. N. Han, H. N. Wang, A. J. Wen, and Y. Hao, “Zero-lag chaos synchronization properties in a hierarchical tree-type network consisting of mutually coupled semiconductor lasers,” Nonlinear Dyn. 99(4), 2893–2906 (2020).
[Crossref]

S. Y. Xiang, Z. Ren, Y. Zhang, Z. Song, and Y. Hao, “All-optical neuromorphic XOR operation with inhibitory dynamics of a single photonic spiking neuron based on VCSEL-SA,” Opt. Lett. 45(5), 1104–1107 (2020).
[Crossref]

Henning, I. D.

N. Q. Li, H. Susanto, B. R. Cemlyn, I. D. Henning, and M. J. Adams, “Modulation properties of solitary and optically injected phased-array semiconductor lasers,” Photonics Res. 6(9), 908–917 (2018).
[Crossref]

N. Q. Li, H. Susanto, B. Cemlyn, I. D. Henning, and M. J. Adams, “Secure communication systems based on chaos in optically pumped spin-VCSELs,” Opt. Lett. 42(17), 3494–3497 (2017).
[Crossref]

Hong, Y. H.

Y. H. Hong and S. K. Ji, “Effect of digital acquisition on the complexity of chaos,” Opt. Lett. 42(13), 2507–2510 (2017).
[Crossref]

Y. H. Hong, M. W. Lee, and K. A. Shore, “Optimised message extraction in laser diode based optical chaos communications,” IEEE J. Quantum Electron. 46(2), 253–257 (2010).
[Crossref]

Hu, M.

X. Bao, Q. Li, T. Wu, Y. Deng, M. Hu, and R. Zeng, “WDM-based bidirectional chaotic communication for semiconductor lasers system with time delay concealment,” Opt. Commun. 472, 125868 (2020).
[Crossref]

Q. L. Li, Q. Bao, D. W. Chen, S. N. Yang, M. Hu, R. Zeng, H. Chi, and Q. Li, “Point-to-multipoint and ring network communication based on chaotic semiconductor lasers with optical feedback,” Appl. Opt. 58(4), 1025–1032 (2019).
[Crossref]

Hu, W. S.

Jacobo, A.

A. Jacobo, M. C. Soriano, C. R. Mirasso, and P. Colet, “Chaos-based optical communications: encryption versus nonlinear filtering,” IEEE J. Quantum Electron. 46(4), 499–505 (2010).
[Crossref]

Ji, S. K.

Jia, Z. W.

D. M. Wang, L. S. Wang, P. Li, T. Zhao, Z. W. Jia, Z. S. Gao, Y. Y. Guo, Y. C. Wang, and A. B. Wang, “Bias current of semiconductor laser: an unsafe key for secure chaos communication,” Photonics 6(2), 59 (2019).
[Crossref]

Jiang, N.

Jiang, X. X.

Ke, C. J.

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]

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. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Lee, M. W.

Y. H. Hong, M. W. Lee, and K. A. Shore, “Optimised message extraction in laser diode based optical chaos communications,” IEEE J. Quantum Electron. 46(2), 253–257 (2010).
[Crossref]

Li, B. C.

Li, N. Q.

N. Q. Li, H. Susanto, B. R. Cemlyn, I. D. Henning, and M. J. Adams, “Modulation properties of solitary and optically injected phased-array semiconductor lasers,” Photonics Res. 6(9), 908–917 (2018).
[Crossref]

N. Q. Li, H. Susanto, B. Cemlyn, I. D. Henning, and M. J. Adams, “Secure communication systems based on chaos in optically pumped spin-VCSELs,” Opt. Lett. 42(17), 3494–3497 (2017).
[Crossref]

N. Q. Li, W. Pan, L. S. Yan, B. Luo, and X. H. Zou, “Enhanced chaos synchronization and communication in cascade-coupled semiconductor ring lasers,” Commun. Nonlinear Sci. Numer. Simul. 19(6), 1874–1883 (2014).
[Crossref]

Li, P.

P. Li, Q. Cai, J. G. Zhang, B. J. Xu, Y. M. Liu, A. Bogris, K. A. Shore, and Y. C. Wang, “Observation of flat chaos generation using an optical feedback multi-mode laser with a band-pass filter,” Opt. Express 27(13), 17859–17867 (2019).
[Crossref]

D. M. Wang, L. S. Wang, P. Li, T. Zhao, Z. W. Jia, Z. S. Gao, Y. Y. Guo, Y. C. Wang, and A. B. Wang, “Bias current of semiconductor laser: an unsafe key for secure chaos communication,” Photonics 6(2), 59 (2019).
[Crossref]

Li, Q.

X. Bao, Q. Li, T. Wu, Y. Deng, M. Hu, and R. Zeng, “WDM-based bidirectional chaotic communication for semiconductor lasers system with time delay concealment,” Opt. Commun. 472, 125868 (2020).
[Crossref]

Q. L. Li, Q. Bao, D. W. Chen, S. N. Yang, M. Hu, R. Zeng, H. Chi, and Q. Li, “Point-to-multipoint and ring network communication based on chaotic semiconductor lasers with optical feedback,” Appl. Opt. 58(4), 1025–1032 (2019).
[Crossref]

Li, Q. L.

Li, X. Z.

X. Z. Li and S. C. Chan, “Detection dependencies of statistical properties for semiconductor laser chaos,” IEEE J. Sel. Top. Quantum Electron. 25(6), 1–9 (2019).
[Crossref]

Liu, D. M.

Liu, S.

Liu, S. Q.

Liu, Y. M.

Luo, B.

L. Y. Zhang, W. Pan, L. S. Yan, B. Luo, X. Zou, and M. F. 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. Y. Zhang, W. Pan, L. S. Yan, B. Luo, X. H. Zou, and M. F. Xu, “Cluster synchronization of coupled semiconductor lasers network with complex topology,” IEEE J. Sel. Top. Quantum Electron. 25(6), 1–7 (2019).
[Crossref]

N. Q. Li, W. Pan, L. S. Yan, B. Luo, and X. H. Zou, “Enhanced chaos synchronization and communication in cascade-coupled semiconductor ring lasers,” Commun. Nonlinear Sci. Numer. Simul. 19(6), 1874–1883 (2014).
[Crossref]

Luo, C. K.

Mirasso, C. R.

A. Jacobo, M. C. Soriano, C. R. Mirasso, and P. Colet, “Chaos-based optical communications: encryption versus nonlinear filtering,” IEEE J. Quantum Electron. 46(4), 499–505 (2010).
[Crossref]

M. C. Soriano, F. Ruiz-Oliveras, P. Colet, and C. R. Mirasso, “Synchronization properties of coupled semiconductor lasers subject to filtered optical feedback,” Phys. Rev. E 78(4), 046218 (2008).
[Crossref]

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

Pan, W.

L. Y. Zhang, W. Pan, L. S. Yan, B. Luo, X. H. Zou, and M. F. Xu, “Cluster synchronization of coupled semiconductor lasers network with complex topology,” IEEE J. Sel. Top. Quantum Electron. 25(6), 1–7 (2019).
[Crossref]

L. Y. Zhang, W. Pan, L. S. Yan, B. Luo, X. Zou, and M. F. 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. F. Xu, W. Pan, and L. Y. Zhang, “Isolated desynchronization and intertwined synchronization in networks of semiconductor lasers,” Opt. Eng. 57(4), 046106 (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. Q. Li, W. Pan, L. S. Yan, B. Luo, and X. H. Zou, “Enhanced chaos synchronization and communication in cascade-coupled semiconductor ring lasers,” Commun. Nonlinear Sci. Numer. Simul. 19(6), 1874–1883 (2014).
[Crossref]

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.

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

Qiu, K.

Ren, Z.

Ruiz-Oliveras, F.

M. C. Soriano, F. Ruiz-Oliveras, P. Colet, and C. R. Mirasso, “Synchronization properties of coupled semiconductor lasers subject to filtered optical feedback,” Phys. Rev. E 78(4), 046218 (2008).
[Crossref]

Sciamanna, M.

M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015).
[Crossref]

Shore, K. A.

P. Li, Q. Cai, J. G. Zhang, B. J. Xu, Y. M. Liu, A. Bogris, K. A. Shore, and Y. C. Wang, “Observation of flat chaos generation using an optical feedback multi-mode laser with a band-pass filter,” Opt. Express 27(13), 17859–17867 (2019).
[Crossref]

M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015).
[Crossref]

Y. H. Hong, M. W. Lee, and K. A. Shore, “Optimised message extraction in laser diode based optical chaos communications,” IEEE J. Quantum Electron. 46(2), 253–257 (2010).
[Crossref]

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

Shum, P.

Song, Z.

Soriano, M. C.

A. Jacobo, M. C. Soriano, C. R. Mirasso, and P. Colet, “Chaos-based optical communications: encryption versus nonlinear filtering,” IEEE J. Quantum Electron. 46(4), 499–505 (2010).
[Crossref]

M. C. Soriano, F. Ruiz-Oliveras, P. Colet, and C. R. Mirasso, “Synchronization properties of coupled semiconductor lasers subject to filtered optical feedback,” Phys. Rev. E 78(4), 046218 (2008).
[Crossref]

Susanto, H.

N. Q. Li, H. Susanto, B. R. Cemlyn, I. D. Henning, and M. J. Adams, “Modulation properties of solitary and optically injected phased-array semiconductor lasers,” Photonics Res. 6(9), 908–917 (2018).
[Crossref]

N. Q. Li, H. Susanto, B. Cemlyn, I. D. Henning, and M. J. Adams, “Secure communication systems based on chaos in optically pumped spin-VCSELs,” Opt. Lett. 42(17), 3494–3497 (2017).
[Crossref]

Syvridis, D.

A. Bogris, A. Argyris, and D. Syvridis, “Encryption efficiency analysis of chaotic communication systems based on photonic integrated chaotic circuits,” IEEE J. Quantum Electron. 46(10), 1421–1429 (2010).
[Crossref]

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

Tan, F.

L. Zhou and F. Tan, “A chaotic secure communication scheme based on synchronization of double-layered and multiple complex networks,” Nonlinear Dyn. 96(2), 869–883 (2019).
[Crossref]

Tang, J. M.

Tang, M.

Tang, X.

J. G. Wu, Z. M. Wu, X. Tang, L. Fan, W. Deng, and G. Q. Xia, “Experimental demonstration of LD-based bidirectional fiber-optic chaos communication,” IEEE Photonics Technol. Lett. 25(6), 587–590 (2013).
[Crossref]

Wang, A. B.

D. M. Wang, L. S. Wang, Y. Y. Guo, Y. C. Wang, and A. B. Wang, “Key space enhancement of optical chaos secure communication: chirped FBG feedback semiconductor laser,” Opt. Express 27(3), 3065–3073 (2019).
[Crossref]

D. M. Wang, L. S. Wang, P. Li, T. Zhao, Z. W. Jia, Z. S. Gao, Y. Y. Guo, Y. C. Wang, and A. B. Wang, “Bias current of semiconductor laser: an unsafe key for secure chaos communication,” Photonics 6(2), 59 (2019).
[Crossref]

Wang, D. M.

D. M. Wang, L. S. Wang, P. Li, T. Zhao, Z. W. Jia, Z. S. Gao, Y. Y. Guo, Y. C. Wang, and A. B. Wang, “Bias current of semiconductor laser: an unsafe key for secure chaos communication,” Photonics 6(2), 59 (2019).
[Crossref]

D. M. Wang, L. S. Wang, Y. Y. Guo, Y. C. Wang, and A. B. Wang, “Key space enhancement of optical chaos secure communication: chirped FBG feedback semiconductor laser,” Opt. Express 27(3), 3065–3073 (2019).
[Crossref]

Wang, H. N.

S. Y. Xiang, Y. N. Han, H. N. Wang, A. J. Wen, and Y. Hao, “Zero-lag chaos synchronization properties in a hierarchical tree-type network consisting of mutually coupled semiconductor lasers,” Nonlinear Dyn. 99(4), 2893–2906 (2020).
[Crossref]

S. Y. Xiang, J. K. Gong, H. Zhang, X. X. Guo, H. N. Wang, Y. H. Zhang, and A. J. Wen, “Zero-lag intensity correlation properties in small ring laser network with heterogeneous delays,” J. Opt. Soc. Am. B 35(2), 287–294 (2018).
[Crossref]

Wang, L. S.

D. M. Wang, L. S. Wang, Y. Y. Guo, Y. C. Wang, and A. B. Wang, “Key space enhancement of optical chaos secure communication: chirped FBG feedback semiconductor laser,” Opt. Express 27(3), 3065–3073 (2019).
[Crossref]

D. M. Wang, L. S. Wang, P. Li, T. Zhao, Z. W. Jia, Z. S. Gao, Y. Y. Guo, Y. C. Wang, and A. B. Wang, “Bias current of semiconductor laser: an unsafe key for secure chaos communication,” Photonics 6(2), 59 (2019).
[Crossref]

Wang, Y. C.

Wang, Y. J.

Wen, A. J.

S. Y. Xiang, Y. N. Han, H. N. Wang, A. J. Wen, and Y. Hao, “Zero-lag chaos synchronization properties in a hierarchical tree-type network consisting of mutually coupled semiconductor lasers,” Nonlinear Dyn. 99(4), 2893–2906 (2020).
[Crossref]

S. Y. Xiang, J. K. Gong, H. Zhang, X. X. Guo, H. N. Wang, Y. H. Zhang, and A. J. Wen, “Zero-lag intensity correlation properties in small ring laser network with heterogeneous delays,” J. Opt. Soc. Am. B 35(2), 287–294 (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]

Wu, J. G.

J. G. Wu, Z. M. Wu, X. Tang, L. Fan, W. Deng, and G. Q. Xia, “Experimental demonstration of LD-based bidirectional fiber-optic chaos communication,” IEEE Photonics Technol. Lett. 25(6), 587–590 (2013).
[Crossref]

G. Q. Xia, Z. M. Wu, and J. G. Wu, “Theory and simulation of dual-channel optical chaotic communication system,” Opt. Express 13(9), 3445–3453 (2005).
[Crossref]

Wu, T.

X. Bao, Q. Li, T. Wu, Y. Deng, M. Hu, and R. Zeng, “WDM-based bidirectional chaotic communication for semiconductor lasers system with time delay concealment,” Opt. Commun. 472, 125868 (2020).
[Crossref]

Wu, Z. M.

J. G. Wu, Z. M. Wu, X. Tang, L. Fan, W. Deng, and G. Q. Xia, “Experimental demonstration of LD-based bidirectional fiber-optic chaos communication,” IEEE Photonics Technol. Lett. 25(6), 587–590 (2013).
[Crossref]

G. Q. Xia, Z. M. Wu, and J. G. Wu, “Theory and simulation of dual-channel optical chaotic communication system,” Opt. Express 13(9), 3445–3453 (2005).
[Crossref]

Xia, G. Q.

Xiang, S. Y.

S. Y. Xiang, Z. Ren, Y. Zhang, Z. Song, and Y. Hao, “All-optical neuromorphic XOR operation with inhibitory dynamics of a single photonic spiking neuron based on VCSEL-SA,” Opt. Lett. 45(5), 1104–1107 (2020).
[Crossref]

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L. Y. Zhang, W. Pan, L. S. Yan, B. Luo, X. H. Zou, and M. F. Xu, “Cluster synchronization of coupled semiconductor lasers network with complex topology,” IEEE J. Sel. Top. Quantum Electron. 25(6), 1–7 (2019).
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L. Y. Zhang, W. Pan, L. S. Yan, B. Luo, X. Zou, and M. F. 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).
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Zhang, Y.

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Zhao, A. K.

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N. Q. Li, W. Pan, L. S. Yan, B. Luo, and X. H. Zou, “Enhanced chaos synchronization and communication in cascade-coupled semiconductor ring lasers,” Commun. Nonlinear Sci. Numer. Simul. 19(6), 1874–1883 (2014).
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Appl. Opt. (1)

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Figures (9)

Fig. 1.
Fig. 1. The topologies of (a) a global 6-SLs network, and SL networks under OWI containing (b) 2-SLs cluster, (c) 3-SLs cluster, (d) 4-SLs cluster, (e) 5-SLs cluster, (f) two co-existing independent 2-SLs clusters. Here, the SLs denoted by the red nodes are the exemplary SLs for communications, and the blue nodes denote the idle SLs that are not involved in communications.
Fig. 2.
Fig. 2. Temporal waveforms of chaotic signals (first row), and corresponding power spectra (second row) in the global SL network with overall coupling strength (a) σ(1)=6ns-1, (b) σ(1)=12ns-1, (c) σ(1)=18ns-1. The spectra of SL1 are selected as representative.
Fig. 3.
Fig. 3. Two dimensional maps of RMS in SL networks containing a communication cluster of (a) 2 SLs, (b) 3 SLs, (c) 4 SLs, (d) 5 SLs, and (e) 6 SLs (global SL network), and containing (f) two 2-SLs communication clusters, in the parameter spaces of coupling strength and current factor. In the SL network of two simultaneous independent 2-SLs clusters, the results of two clusters are very similar, thus, the RMS of SL1 and SL2 are shown for simplicity.
Fig. 4.
Fig. 4. The CC of different-size clusters versus parameter mismatches of (a) coupling strength and (b) time delay. Here, due to the similarity of two co-existing 2-SLs clusters shown in Fig. 1(f), the 2-SLs cluster composed of SL1 and SL2 is taken to present the results for simplicity.
Fig. 5.
Fig. 5. Schematic diagram of intra-cluster communication in the 2-SLs communication cluster, cluster of SL1 and SL2 is taken for instance, wherein SL: semiconductor laser, FC: fiber coupler, IM: intensity modulator, m(t): original message, OI: optical isolator.
Fig. 6.
Fig. 6. The original messages (black dotted lines) and the recovered messages m1’(t) (red solid lines) and m2’(t) (blue solid lines) in the 2-SLs communication cluster with message bit rates of (a) 1 Gb/s, (b) 5 Gb/s, and (c) 10 Gb/s.
Fig. 7.
Fig. 7. The eye diagrams of the recovered messages m1’(t) (first column) and m2’(t) (second column) in the 2-SLs communication cluster with message bit rate of (a) 1 Gb/s, (b) 5 Gb/s, and (c) 10 Gb/s, and the eye diagrams of the intercepted messages m1e (third column) and m2e (fourth column) in the cases of (d) 1 Gb/s, (e) 5 Gb/s, and (f) 10 Gb/s.
Fig. 8.
Fig. 8. The Q-factors of legal recovered messages and the illegal intercepted messages versus message bit rate R in the communication clusters of the (a) 2 SLs, (b) 3 SLs, (c) 4 SLs, (d) 5 SLs, and (e) 6 SLs, as well as (f) two independent 2-SLs clusters for simultaneous communications. Here, the messages intercepted by the eavesdroppers are denoted by m1e, m2e, m3e, m4e, m5e, and m6e.
Fig. 9.
Fig. 9. The Q-factors of legal recovered messages and the illegal intercepted messages in 3-SLs communication cluster versus message bit rate R, in the common cases of (a) non-simultaneous message exchange, and (b) different messages among different transmitter and receiver pairs. For the second case, mij stands for message transmitted from SLi to SLj, and the messages bidirectionally intercepted by the eavesdroppers are denoted as m12e, m21e, m13e, m31e, m23e, and m32e.

Equations (7)

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d E i ( l ) ( t ) d t = ( 1 + i α ) 2 ( G i ( l ) ( t ) 1 τ p ) E i ( l ) ( t ) + σ ( l ) j = 1 6 A i j ( l ) E j ( l ) ( t τ ) exp ( i ω τ ) + 2 β N i ( l ) ( t ) χ i ( l ) ( t ) ,
d N i ( l ) ( t ) d t = μ I t h q N i ( l ) ( t ) τ e G i ( l ) ( t ) | | E i ( l ) ( t ) | | 2 ,
G i ( l ) ( t ) = g ( N i ( l ) ( t ) N 0 ) 1 + s | | E i ( l ) ( t ) | | 2 ,
A ( 1 ) = [ 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 0 ] , A ( 2 ) = [ 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1 0 ] , A ( 3 ) = [ 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 ] ,
A ( 4 ) = [ 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 1 1 1 0 ] , A ( 5 ) = [ 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 0 0 1 1 0 ] , A ( 6 ) = [ 0 1 1 0 0 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 ] ,
R M S  =  m = 1 N c ( I m ( t ) I ( t ) ) 2 / L I N c I ( t ) ,
C i , j = ( P i P i ) ( P j P j ) ( P i P i ) 2 ( P j P j ) 2 ,