Chaos synchronization and communication in closed-loop semiconductor lasers subject to common chaotic phase-modulated feedback

Ning Jiang; Anke Zhao; Shiqin Liu; Chenpeng Xue; Kun Qiu

doi:10.1364/OE.26.032404

1. Introduction

Chaos synchronization and communication in external-cavity semiconductor lasers (ECSLs) have attracted great attention in the last decade for its physical security enhancement property [1–3]. In chaos-based communication systems based on ECSLs, message is usually embedded into the chaotic carrier at the transmitter end by the ways of chaos modulation, chaos masking, or chaos shift keying, and at the receiver end the message is recovered on the basis of chaos synchronization [4–6]. It has been theoretically and experimentally confirmed that there are two types of chaos synchronization can be achieved in the unidirectional coupling ECSL systems. One is the complete chaos synchronization and the other one is the generalized chaos synchronization. The physical mechanisms for these two types of chaos synchronization are based on the symmetric operation and the injection-locking effect, respectively [7,8].

Up to present, in most of chaotic ECSL systems, the feedback light is a linear delayed replica of the output light, which results in that the bandwidth of chaotic carrier is limited nearby the relaxation oscillation frequency (f_RO) and the time delay signature (feedback delay time) is easily identified [9,10]. The limited bandwidth would restrict the transmission capacity of chaos communication system, and the TDS revealment would degrade the system security. In recent years, a lot of methods have been proposed to enhance the bandwidth and conceal the TDS of chaos in ECSLs, such as dual optical feedback [11], distributed feedback [12,13], phase-modulated feedback [5,14], delayed self-interference [15], electronic or optical heterodyning [16,17], coupling CW laser with chaotic laser [18], and propagating chaos generated by ECSL through optical time lens or optical fractional Fourier transformation module [19,20]. However, these studies are mainly focused on the stand-alone chaos sources, while the chaos synchronization and communication in the coupling chaos systems composed of these bandwidth-enhanced and TDS-concealed chaos sources are rarely investigated.

In this paper, we propose and demonstrate a closed-loop chaos synchronization system based on ECSLs subject to common chaotic phase-modulated optical feedback (CCPMOF). Simulation results demonstrate that with respect to the conventional closed-loop ECSL system, simultaneous bandwidth enhancement and effective TDS suppression for chaotic carrier can be achieved, high quality chaos synchronization can be performed much more easily, and the communication performance can be improved.

2. Principles and theoretical model

The configuration of closed-loop ECSL system subject to CCPMOF is illustrated in Fig. 1. Similar to the conventional closed-loop configuration [21], two twin ECSLs, namely master semiconductor laser (MSL) and slave semiconductor laser (SSL), are unidirectionally coupled with optical fiber. While differing from the conventional configurations, a phase modulator is deployed in the feedback loops of MSL and SSL, and the driving signals of phase modulators are distributed from a third chaotic ECSL which is referred as the driving semiconductor laser (DSL). The output of DSL is equally split into two parts, both of which are photo-detected, amplified and then respectively used as the driving signals of the phase modulators in the feedback loops of MSL and SSL. Under such a scenario, both of MSL and SSL suffer common chaotic phase-modulated optical feedback.

Fig. 1 Schematic of closed-loop ECSL system subject to common chaotic phase modulated feedback. DSL: driving semiconductor laser, MSL: master semiconductor laser, SSL: slave semiconductor laser, PM: phase modulator, OI: optical isolator, FC: fiber coupler, OC: optical circulator, PD: photodetector, R: reflector, Amp: electronic amplifier.

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For the numerical purpose, we adopt the modified Lang-Kobayashi rate equations taking the feedback term and the injection term into consideration, to describe the dynamics of the ECSLs in the proposed scheme. The rate equations for DSL are written as [22]

\frac{d E_{d} (t)}{d t} = \frac{1}{2} (1 + i α) (G_{d} (t) - \frac{1}{τ_{p}}) E_{d} (t) + k_{d} E_{d} (t - τ_{d}) \exp (- i ω_{d} τ_{d}) + \sqrt{2 β N_{d} (t)} χ_{d} (t),

\frac{d N_{d} (t)}{d t} = \frac{I_{d}}{e} - \frac{N_{d} (t)}{τ_{e}} - G_{d} (t) {| E_{d} (t) |}^{2} .

While those for MSL and SSL are written as [21,23]

\begin{array}{l} \frac{d E_{m, s} (t)}{d t} = \frac{1}{2} (1 + i α) (G_{m, s} (t) - \frac{1}{τ_{p}}) E_{m, s} (t) + k_{f} E_{m, s} (t - τ_{f}) \exp (- i ω_{m, s} τ_{f}) \exp (i ϕ_{P M} (t)) \\ + σ E_{m} (t - τ_{i}) \exp (- i ω_{m} τ_{i}) \exp (i Δ ω t) + \sqrt{2 β N_{m, s} (t)} χ_{m, s} (t), \end{array}

\frac{d N_{m, s} (t)}{d t} = \frac{I_{m, s}}{e} - \frac{N_{m, s} (t)}{τ_{e}} - G_{m, s} (t) {| E_{m, s} (t) |}^{2} .

G_{d, m, s} (t) = \frac{g (N_{d, m, s} (t) - N_{0})}{1 + ε {| E_{d, m, s} (t) |}^{2}}

ϕ_{P M} (t) = \frac{A \cdot Ν [{| E_{d} (t - Δ t_{d}) |}^{2}]}{V_{π}} \cdot π

Ν (x) = \frac{x - x_{min}}{x_{m a x} - x_{min}}

In these equations, the subscripts d, m, and s stand for DSL, MSL, and SSL, respectively. E(t) denotes the slow-varying complex electric field, and N(t) denotes the carrier number in the laser cavity. e = 1.6 × 10⁻¹⁹ C is the electron charge. The optical gain G is defined in Eq. (5), wherein |E(t)|² denotes the photon number in the cavity. k_d and τ_d denote the feedback strength and time delay of DSL, respectively. k_f and τ_f denote the feedback strength and time delay of MSL and SSL, respectively. σ and τ_i denote the coupling strength and flight time from MSL to SSL, respectively.

ω = 2 π f

is the angle frequency, and

Δ ω = ω_{m} - ω_{s}

is the detuning frequency between MSL and SSL. χ(t) is Gaussian noise source with unity variance and zero mean, which is adopted to model the spontaneous emission noise. The additional phase induced by the phase modulation (PM) is presented in Eq. (6), wherein A is the amplitude of driving signal, Ν[·] denotes normalizing the values of [·] into the range from 0 to 1, Δt_d is the flight time from DSL to MSL and SSL, and V_π is the half-wave voltage of phase modulator. The phase modulation index β_PM is determined by A/V_π. The normalization Ν[·] is mathematically defined in Eq. (7), wherein “min” and “max” mean the minimum and maximum values, respectively.

To quantitatively investigate the bandwidth and TDS characteristics of chaotic carrier in the proposed system, the efficient bandwidth and auto-correlation function (ACF) are calculated. The efficient bandwidth is defined as the span between the DC and the frequency where 80% of the energy is contained in the RF spectrum, as those in [9,18,19,24]. The ACF [A(∆t)] that measures how well the time series matches its time-shifted replica is defined as [15,16,25]

A (Δ t) = \frac{〈 (I (t + Δ t)) - 〈 I (t + Δ t) 〉) \cdot (I (t) - 〈 I (t) 〉) 〉}{{(〈 {(I (t + Δ t) - 〈 I (t + Δ t) 〉)}^{2} 〉 \cdot 〈 {(I (t) - 〈 I (t) 〉)}^{2} 〉)}^{1 / 2}},

where I(t) = |E(t)|² is the intensity time series, I(t + ∆t) contains the time shift ∆t with respect to I(t), and <·> stands for the time averaging. Similarly, to quantify the chaos synchronization quality, the cross-correlation function (CCF) is defined as [6–8,26]

C (Δ t) = \frac{〈 (I_{m} (t + Δ t)) - 〈 I_{m} (t + Δ t) 〉) \cdot (I_{s} (t) - 〈 I_{s} (t) 〉) 〉}{{(〈 {(I_{m} (t + Δ t) - 〈 I_{m} (t + Δ t) 〉)}^{2} 〉 \cdot 〈 {(I_{s} (t) - 〈 I_{s} (t) 〉)}^{2} 〉)}^{1 / 2}}

In our simulations, the rate equations are handled with the fourth-order Runge-Kutta algorithm. The intrinsic parameters of the lasers are set as the typical parameters reported in [27]: the wavelength $λ = 1550$ nm, the linewidth enhancement factor α = 5, the differential gain parameter $g = 1.5 \times 1 0^{- 8}$ ps⁻¹, the gain saturation coefficient $ε = 5 \times 1 0^{- 7}$ , the spontaneous emission rate $β = 1.5 \times 1 0^{- 6}$ ns⁻¹, the photon lifetime $τ_{p} = 2$ ps, the carrier lifetime $τ_{e} = 2$ ns and the transparent carrier number $N_{0} = 1.5 \times 1 0^{8}$ . The operation current I is set as $I = 2 I_{t h}$ , where $I_{t h} = 14.7$ mA is the threshold current. The feedback strength of DSL is chosen as $k_{d} = 20$ ns⁻¹ and the feedback delay is set as τ_d = 3 ns. With these parameters, the DSL works in a chaotic regime. Unless otherwise stated, the feedback strength and delay of MSL and SSL are respectively set as $k_{f} = 20$ ns⁻¹ and $τ_{f} = 5$ ns, the time delay of driving signal is set as $Δ t_{d} = 1$ ns, and the flight time from MSL to SSL is set as $τ_{i} = 0$ ns for the sake of simplicity.

3. Influence of CCPMOF on characteristics of chaotic carrier

3.1 Influence of CCPMOF on dynamic behaviors of ECSL

Firstly, we investigate the influence of CCPMOF on the dynamic behaviors of MSL and SSL. Since the dynamic behaviors of MSL and SSL are much similar for the symmetric configuration, here we only show the results of MSL for the sake of simplicity. Figure 2 shows the bifurcation maps of MSL under the feedback scenarios of COF and CCPMOF with different PM indexes. For the COF case, an obvious state transition process can be observed as the increase of feedback strength. The state of MSL changes from stable to quasiperiodic (multiple periodic) at about 5ns⁻¹, and further changes to chaotic at about 7.5ns⁻¹. Nevertheless, under the proposed CCPMOF scenario, it is apparent that no transition process occurs. Even when the feedback is weak [see the insets of Figs. 2(b)-2(d)], the output of ECSL is always chaotic. That is, it is very easy to get chaotic carrier for chaos communication with CCPMOF.

Fig. 2 Bifurcation maps of MSL with peak-to-peak intensity versus feedback strength, under the scenarios of (a) COF and CCPMOF with PM indexes of (b) 0.5, (c) 1, and (d) 2. The insets in (b)-(d) show the detail of bifurcation during the feedback strength range from 0 to 2ns⁻¹.

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Figure 3 shows the possibility distribution function (PDF) of the intensity of chaotic signals under the COF and CCPMOF scenarios with different feedback strengths. It is obvious that the PDF of chaos generated with COF does not have a Gaussian profile and the PDF is not symmetric. While in the CCPMOF case with identical feedback strength, the profile of PDF is similar to the Gaussian distribution, especially when the feedback strength is not very strong, as those shown in Figs. 3(b1) and 3(b2). Therefore, it can be concluded that with respect to COF, the CCPMOF here can optimize the PDF of the intensity of chaos, which is beneficial to obtaining a balanced bit ratio of “1” and “0” in the application of random bit generation [15].

Fig. 3 PDF of chaos under the scenarios of COF (first row) and CCPMOF (second row) with β_PM = 1, for the cases with different feedback strengths of 10ns⁻¹, 20ns⁻¹, 30ns⁻¹ and 40ns⁻¹.

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3.2 Bandwidth enhancement and TDS suppression induced by CCPMOF

It is well known that optical phase modulation with sinusoidal signal would induce infinite sidebands in optical spectrum. The chaotic PM driving signal here can be regarded as a superposition of sinusoidal signals with different frequencies, hence the chaotic PM would induce a mass of new frequency components into the feedback light, and consequently, the bandwidth of generated chaos would be enhanced. Moreover, since PM is a type of nonlinear modulation, the feedback in the present scheme is no longer a linear delayed replica of output of ECSL, then the external-cavity resonance would be weakened, and consequently, the TDS in chaotic carrier would be suppressed correspondingly. In this subsection, we concentrate on the bandwidth enhancement and TDS suppression properties of CCPMOF.

Figure 4 shows the temporal waveforms of intensities, as well as the corresponding RF spectra and ACF curves of the chaotic signals generated by the COF and the CCPMOF, in the case of identical feedback strength and feedback delay. Apparently, the chaotic pulses in the CCPMOF case [Fig. 4(b1)] are denser than those in the COF case [Fig. 4(a1)]. That is, the bandwidth is enhanced. This phenomenon can also be observed in the corresponding RF spectra shown in Figs. 4(a2) and 4(b2), where it is indicated that in the CCPMOF case the energy distributed nearby f_RO is reduced, and the attenuation of frequency components higher than f_RO is obviously smoother than that in the COF case. The efficient bandwidth in the COF case is 9.6 GHz, while that in the CCPMOF case is 14.6 GHz, the bandwidth has been enhanced by 5 GHz. On the other hand, in the COF case, there is an apparent peak appearing at the feedback delay position in the ACF curve, which means the TDS can be clearly identified. While in the proposed case, the ACF values except that at ∆t = 0 always maintain at a low level nearby 0, no distinguishable peak appears at the feedback delay position. That is, the TDS is totally concealed, and the complexity of chaotic carrier is greatly enhanced [10,25].

Fig. 4 Time series, RF spectra and ACF curves of chaos generated by COF and CCPMOF with identical feedback strength and delay. Here the feedback strength and feedback time delay for both of COF and CCPMOF are set as 20ns⁻¹ and 5ns, respectively; the PM index is fixed as β_PM = 1.

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To further investigate the bandwidth enhancement and TDS suppression properties of CCPMOF, Fig. 5 shows the variations of efficient bandwidth and TDS value in ACF curve, as a function of the feedback strength and PM index. Here the TDS value is the value of the peak appearing at the feedback delay position in the ACF curve. As that shown in Fig. 5(a), for a fixed PM index, the efficient bandwidth gradually increases as the increase of feedback strength. Similarly, for a fixed feedback strength, a larger PM index can induce more significant bandwidth enhancement. On the other hand, as shown in Fig. 5(b), for a fixed PM index, the TDS value would gradually increase as the increase of feedback strength, and when the feedback strength is strong enough, the TDS may be revealed. While for a fixed feedback strength, the TDS value gets smaller and smaller, and finally it is suppressed at a low level nearby 0, as the increase of PM index. Repeating simulations with different PM indexes and different feedback strengths indicate that even though under strong feedback scenarios, the TDS can be totally concealed by properly enlarging the PM index. These phenomena are because that as the increase of PM index, more and more new frequency components are introduced into the feedback light, then the spectrum of feedback light would be gradually expanded, and the periodicity induced by external cavity resonance would be gradually weakened; consequently, the efficient bandwidth of chaotic carrier is correspondingly enhanced, and simultaneously the TDS is effectively suppressed.

Fig. 5 Efficient bandwidth (in GHz) and TDS value in ACF curve versus PM index β_PM and feedback strength k_f.

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Overall, under the CCPMOF scenario, significant bandwidth enhancement and effective TDS suppression for chaotic carrier can be achieved by properly increasing the PM index. This can be implemented by increasing the gain of RF amplifier or adopting low half-wave voltage optical phase modulators. That is, it is easy to generate broadband chaotic carrier with effective TDS suppression for chaos communication in the proposed closed-loop ECSL system.

4. Chaos synchronization in closed-loop ECSLs subject to CCPMOF

4.1 Influences of PM index and injection strength on chaos synchronization

Chaos synchronization is the basis of chaos-based communication, thus in this section we focus on the properties of chaos synchronization in the proposed closed-loop system consisting of ECSLs subject to CCPMOF. To quantitatively describe the level of chaos synchronization between MSL and SSL, we refer the case with a cross correlation coefficient [C(0) in Eq. (9)] larger than 0.95 as the high-quality chaos synchronization. Figure 6(a) shows the cross correlation between the chaotic signals generated by MSL and SSL in the space of PM index and injection strength. For the COF case (β_PM = 0), high-quality chaos synchronization requires that the injection strength should be larger than a threshold of 36 ns⁻¹. However, in the proposed system, in the case of β_PM = 1, high quality chaos synchronization can be achieved when the injection strength is larger than 14 ns⁻¹. Furthermore, as the increase of PM index, the threshold injection strength for high-quality synchronization would become smaller and smaller. When β_PM = 2, the threshold injection strength is reduced as 5 ns⁻¹, and when β_PM = 3, it is further reduced as 2 ns⁻¹. Apparently, in the proposed system, high-quality chaos synchronization is much easier to be achieved, and the operation region for the achievement of high-quality chaos synchronization is much wider, with respect to the COF closed-loop system. The prominent chaos synchronization property is due to that with the symmetric CCPMOF, an identical additional phase is added in both of the feedback light of MSL and SSL, with the affections of identical additional phases in the cavities of MSL and SSL, the phase difference between the outputs of MSL and SSL would be reduced, and stable phase locking between SSL and MSL can be more easily obtained, which then prompts the intensity locking and chaos synchronization in intensity [28]. As the increase of PM index, the phase difference between MSL and SSL would be further reduced, then the phase and intensity of SSL are more easily locked to those of MSL, and consequently, chaos synchronization is easier to be achieved.

Fig. 6 Cross correlations (CCs) for (a) MSL-SSL, (b) DSL-MSL, and (c) DSL-SSL versus the PM index β_PM and the injection strength σ.

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On the other hand, the cross correlation between the intensity of DSL [I_d(t-Δt_d)] and MSL (SSL) is also shown in Fig. 6(b) [Fig. 6(c)]. It is apparent that the correlation between DSL and MSL (SSL) always keeps at a very low level with a cross correlation coefficient smaller than 0.1. This is because the DSL chaos only affects the feedback phases of MSL and SSL, due to the nonlinearity of phase modulation, the feedback term in Eq. (1) shows a complex nonlinear relationship with the output of DSL, consequently, the evolutions of optical fields of MSL and SSL are incoherent with that of DSL. It is worth mentioning that this phenomenon is different from that in the scenario where the output of DSL is directly injected into MSL and SSL. Under that scenario, due to the injection-locking effect, the evolutions of MSL and SSL would be driven towards that of DSL, and consequently, the cross correlation between DSL and MSL (SSL) is rather high, they can even synchronize with each other, as long as the injection from DSL is sufficiently strong.

4.2 Mismatch robustness, detuning tolerance and phase fluctuation tolerance properties of chaos synchronization in CCPMOF closed-loop system

In practice, the intrinsic parameter mismatch, the frequency detuning and the phase fluctuations in injection and feedback paths would reduce the feasibility of chaos synchronization system, therefore it is essential to discuss their influences on the quality of chaos synchronization in the proposed system. Figure 7 shows the influences of intrinsic parameter mismatches of MSL and SSL on the chaos synchronization quality, for three indicative injection cases of σ = 20 ns⁻¹, σ = 40 ns⁻¹ and σ = 60 ns⁻¹. Here the intrinsic parameters of DSL are fixed to the initial values, while the mismatches of MSL and SSL are introduced by decreasing the internal parameters α, g, and τ_p and accordingly increasing the parameters N₀, ε, and τ_e of SSL by the same amount, as that reported in [27], which can be mathematically described as $α_{m, s} = (1 - μ_{m, s}) α$ , $g_{m, s} = (1 - μ_{m, s}) g$ , $τ_{p m, s} = (1 - μ_{m, s}) τ_{p}$ , $N_{0 m, s} = (1 + μ_{m, s}) N_{0}$ , $ε_{m, s} = (1 + μ_{m, s}) ε$ , $τ_{e m, s} = (1 + μ_{m, s}) τ_{e}$ , wherein μ stands for the mismatch ratio with respect to DSL. For the COF-based system, when the injection strength is weak, no chaos synchronization can be achieved as that shown in Figs. 7(a1), which has also been proved in Fig. 6; when the injection strength is nearby the threshold injection strength [see Fig. 7(b1)], high quality chaos synchronization can be achieved between MSL and SSL, but it is highly sensitive to mismatch between MSL and SSL; only when the injection strength is properly strong, as that shown in Fig. 7(c1), the chaos synchronization can be robust to certain mismatch. Nevertheless, in the CCPMOF-based system, high quality chaos synchronization can be always maintained in a relatively wide region where the mismatch of MSL is identical or similar to that of SSL. For the weak injection case shown in Figs. 7(a2)-7(a4), the robust mismatch range is above 10%. Moreover, the robust mismatch range would be gradually improved, as the increase of injection strength [Figs. 7(b2)-7(b4) and Figs. 7(c2)-7(c4)]. It is interesting to observe that positive mismatches for MSL and SSL would show better robustness with respect to the negative ones. This is because in a positive mismatch case, the parameters α, g, and τ_p of MSL and SSL are decreased, and the parameters N₀, s, and τ_e are accordingly increased, which induces that the variations of carrier change get slower, the phase variations induced by chaotic PM is dominated to affect the phase variation in the laser cavity. Consequently, with the identical dominated additional chaotic feedback phases, stable phase locking for the injection-locking conditions reported in [28] can be more easily achieved, and then chaos synchronization can be preserved.

Fig. 7 Influences of intrinsic parameter mismatches of MSL and SSL on the quality of chaos synchronization in the COF configuration (first column) and the proposed CCPMOF-based system with β_PM = 1 (second column), β_PM = 2 (third column) and β_PM = 3 (fourth column). The injection strengths in the first, second and third rows are set as 20 ns⁻¹, 40 ns⁻¹ and 60 ns⁻¹, respectively.

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Figure 8 shows the influence of frequency detuning of MSL and SSL on the quality of chaos synchronization, for the three indicative injection cases. The frequency detuning is introduced by fixing the operation frequency of DSL and tuning those of MSL and SSL. The frequency detuning between DSL and MSL is defined as $Δ f_{d m} = f_{d} - f_{m}$ , and that between DSL and SSL is defined as $Δ f_{d s} = f_{d} - f_{s}$ , then the frequency detuning between MSL and SSL in Eq. (3) is $Δ ω = ω_{m} - ω_{s} = 2 π (Δ f_{d s} - Δ f_{d m})$ . It is obvious that in the CCPMOF-based system, high quality chaos synchronization can be preserved in a certain region where the frequency detuning of MSL is identical or similar to that of SSL. Even when the injection strength is weak [Figs. 8(a2)-8(a4)], the chaos synchronization can tolerate a frequency detuning beyond 10GHz. As the injection strength increases, the detuning tolerance would be enhanced. In the strong injection cases, the chaos synchronization can tolerate several tens of GHz frequency detuning [Figs. 8(b2)-8(b4) and Figs. 8(c2)-8(c4)]. The frequency detuning tolerance property qualitatively agrees with that occurs in the COF-based closed-loop system reported in [29], for the same synchronization mechanism based on the injection-locking effect. Nevertheless, in the COF system (the first column), considerable frequency detuning tolerance for high-quality chaos synchronization requires much stronger injection.

Fig. 8 Influences of frequency detuning of MSL and SSL on the quality of chaos synchronization in the COF configuration (first column) and the proposed CCPMOF-based system with β_PM = 1 (second column), 2 (third column), and 3 (fourth column). The injection strengths in first row, second row, and third row are set as 20 ns⁻¹, 40 ns⁻¹ and 60 ns⁻¹, respectively.

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In practice, the phase fluctuations in injection light and feedback light of SSL are inevitable due to kinds of reasons, such as the imperfection of fiber, circumstance vibrations, etc, which may also degrade the quality of chaos synchronization. Here we summarize these phase fluctuations as an instantaneous phase noise in the injection for the sake of simplicity, and then the injection term in Eq. (3) is modified as $σ E_{m} (t - τ_{i}) \exp (- i ω_{m} τ_{i} (1 + β_{p} χ_{p} (t))) \exp (i Δ ω t)$ , wherein β_p is the phase fluctuation strength which is defined as the ratio of phase fluctuation with respect to the injection phase, and χ_p(t) is a Gaussian noise source with unity variance and zero mean. Figure 9 shows the influences of phase fluctuation on the cross correlation between MSL and SSL, for the abovementioned three indicative injection cases. It is indicated that high quality chaos synchronization can be preserved as the phase fluctuation strength is smaller than −1.3dB. Repeating simulations with other injection strengths and PM indexes indicate that in the synchronization region of CCPMOF system, similar phase fluctuation tolerance can also be preserved.

Fig. 9 Influences of phase fluctuation between the injection and feedback of SSL on the cross correlation between MSL and SSL in the COF configuration and the proposed CCPMOF system with different PM indexes, for the injection cases of (a) σ = 20 ns⁻¹, (b) σ = 40 ns⁻¹, and (c) σ = 60 ns⁻¹.

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In general, with respect to the COF-based system, it is much easier to achieve stable high-quality chaos synchronization in the CCPMOF-based system, and the mismatch robustness and detuning tolerance properties would be enhanced as the increase of injection strength. In addition, it is worth noting that the results for the cases with different PM indexes are very similar, which indicates that the PM index does not obviously affect the mismatch robustness, the detuning tolerance, and the phase fluctuation tolerance of chaos synchronization.

5. Chaos communication and security discussion

5.1 Chaos communication in closed-loop ECSLs subject to CCPMOF

On the basis of the high-quality chaos synchronization, in this subsection we investigate the chaos communication in the proposed CCPMOF closed-loop system. Here the message is a random binary sequence, it is embedded into the chaotic carrier by the way of chaos modulation which is mathematically described as $| E_{m m} (t) | = | E_{m} (t) | [1 + M m (t)]$ , wherein E_mm(t) denotes the modulated carrier, m(t) is the original message, M is the modulation index which determines the message amplitude in the modulated chaotic carrier, and it is chosen as 0.05 to guarantee that the message is well hidden in the chaotic carrier [21]. The message is recovered by the method of direct subtraction decoding which is described as $m' (t) = LPF [{| E_{m m} (t) |}^{2} - {| E_{s} (t) |}^{2}]$ , where LPF means the recovered message is filtered by a low-pass five-order Butterworth filter with a cut-off frequency equaling the message bit rate R. The communication performance is quantified by calculating the Q-factor of recovery message, which is defined as

Q = \frac{< I_{1} > - < I_{0} >}{ε_{1} + ε_{0}}

where <I₁> and <I₀> stand for the average power of bits “1” bit and “0”, respectively; ε₁ and ε₀ are the corresponding standard deviations. A Q-factor value greater than 6 corresponds to a satisfactory BER about 10⁻⁹ [30]. Figure 10(a) shows the distribution of Q-factor of recovery message in the space of PM index β_PM and message bit rate R. The results indicate that the range of satisfactory communication performance in the proposed system is obviously wider than that in conventional closed-loop system (β_PM = 0), and moreover, this range gradually gets wider and wider as the increase of PM index. This phenomenon is attributed to that the CCPMOF simultaneously enhances the bandwidth of transmission chaotic carrier generated by MSL and that of the synchronous local chaotic carrier generated by SSL, then wider transmission bandwidth can be provided, and consequently, messages with higher rates can be successfully recovered with satisfactory Q-factors. The improvement of communication performance can also be directly observed by comparing the eye diagrams of recovery messages in the conventional system and the proposed system. As those shown in Figs. 10(b1)-10(c3), in the cases of identical bit rates, the eye diagrams in the proposed system open wider than those in the conventional system, especially when higher bit rate message is transmitted.

Fig. 10 (a) Q-factor of recovery message versus PM index β_PM and message bit rate R. (b1)-(b3) show the eye diagrams of recovery message in the conventional closed-loop system, while (c1)-(c3) present those in the proposed scheme with β_PM = 1. T_B = 1/R is the bit period of message. The injection strength is σ = 40ns⁻¹.

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5.2 Security discussion

Finally, we turn to discuss the security of chaos communication in the CCPMOF closed-loop system. As aforementioned, the TDS can be effectively suppressed in the CCPMOF system. Therefore, it is difficult for the eavesdropper to guess the feedback delay of MSL and SSL from the transmission signal on the public channel. Figure 11(a) presents the influence of the feedback delay mismatch of SSL (with respect to the feedback delay of MSL) on the Q-factor of recovery message for the case of R = 4 Gbit/s. It is obvious the communication performance is extremely sensitive to the feedback delay mismatch between MSL and SSL. Similar results are also observed in repeating simulations with different values for R. It is indicated that a tiny feedback delay mismatch about 5ps that corresponds to a fiber length of 10⁻³m would induce the Q-factor decreasing from a satisfactory value 8.8 to 1.7 which indicates no useful message being recovered. Since the variation range of feedback delay can be rather large in practice, if the feedback delay is used as the private key, it is extremely difficult for the eavesdropper to intercept correct message, even if he holds a laser with parameters identical to MSL and SSL.

Fig. 11 (a) Variation of Q-factor versus the mismatch of feedback delay of SSL for the case of R = 4 Gbit/s and σ = 80ns⁻¹. (b) Q-factors of legal recovery message and illegal intercepted messages under the typical attacks of DDLF and US, as a function of the message rate R. The third column shows the eye diagrams of (c1) the legal recovery messages, (d1) the DDLF attack-intercepted massages and (e1) the US attack-intercepted massages for the case of R = 2 Gbit/s, respectively; while the fourth column shows those for the case of R = 4 Git/s. The PM index is set as β_PM = 1, and the injection strength for legal recovery is σ = 40ns⁻¹, while that for the US attack is set as 100ns⁻¹ to achieve chaos synchronization.

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Furthermore, we consider two typical attack scenarios in chaos communication systems, namely the direct detection with linear filtering (DDLF) and the utilization of synchronization (US) [23,26,31]. For the first attack scenario, the eavesdropper uses a photodetector to capture the transmission chaotic carrier, and then adopts a five-order Butterworth low-pass filter with a cutoff frequency equaling the bit rate of message, to directly extract message from the public link from MSL to SSL. While under the US attack scenario, the eavesdropper firstly captures a transmission chaotic signal by splitting the public link, and then amplifies it and injects it into an attack semiconductor laser (ASL) to obtain chaos synchronization for illegal message interception. Here we consider the extremely serious case that the eavesdropper is equipped with an ASL that is totally identical to MSL and SSL (including the feedback parameters), except that there is no PM in the feedback loop. Figure 11(b) presents the Q-factor of legal recovery message and those of the messages intercepted by DDLF and US, versus the message bit rate R. Apparently, the Q-factors of intercepted messages fastly decrease as the increase of message rate. Under the DDLF scenario, the eavesdropper cannot intercept the message with a satisfactory Q-factor of 6, and when the bit rate is larger than 1.5 Gbit/s, no useful message can be recovered as those shown in Figs. 11(d1) and 11(d2). Repeating simulations with smaller message modulation indexes indicate that, by properly reducing the message modulation index, the Q-factor of message intercepted by DDLF would be further reduced. For the US attack scenario, when the message bit rate is lower than 4 Gbit/s, the eavesdropper may intercept partial message such as that shown in Fig. 11(e1), and even intercept the message with a satisfactory Q-factor larger than 6. However, when the bit rate is higher than 4 Gbit/s, no useful message can be recovered as that shown in Fig. 11(e2). Comparing with the legal recovery message, the intercepted messages show much higher BER. From this point of view, the security of message can be guaranteed by properly increasing the bit rate of message.

In practice, MSL and SSL are twin lasers manufactured on one chip, it is difficult for the eavesdropper to obtain an ASL identical to MSL and SSL, and hence more difficult for him to intercept message. Overall, without precise knowledge of the feedback time delay and the PM details in the feedback loop, the eavesdropper is not able to correctly intercept the message. In addition, by properly increasing the bit rate of message, the BER of intercepted messages would sharply increase, and the interception difficulty is further increased.

6. Conclusions

In summary, we have proposed a closed-loop chaos system composed of ECSLs subject to CCPMOF. The proposed system supports simultaneous bandwidth enhancement and effective TDS suppression for chaotic carrier, easy-to-implement chaos synchronization and high-speed chaos communication. It is demonstrated that with CCPMOF the chaotic carrier can be easily obtained even when the feedback strength is small, the efficient bandwidth of chaotic carrier can be significantly enhanced, and the TDS can be easily suppressed at a non-distinguishable level with proper selection of the PM index and the feedback strength. With respect to the conventional closed-loop ECSL system, high-quality chaos synchronization can be more easily achieved in a wide range of PM index and injection strength, a weak injection can achieve high-quality chaos synchronization with properly large PM indexes, and in the wide synchronization operation region, the chaos synchronization always shows considerable mismatch robustness, detuning tolerance and injection phase fluctuation tolerance. Moreover, the performance of chaos communication can be obviously improved, a higher bit rate transmission is available. Based on the prominent TDS suppression and communication performance improvement, the communication security can be guaranteed by using the feedback time delay as the private key or properly increasing the bit rate of message.

Funding

National Natural Science Foundation of China (NSFC) (61671119, 61471087); 111 Project (B14039).

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Chaos synchronization and communication in closed-loop semiconductor lasers subject to common chaotic phase-modulated feedback

Abstract

1. Introduction

2. Principles and theoretical model

3. Influence of CCPMOF on characteristics of chaotic carrier

3.1 Influence of CCPMOF on dynamic behaviors of ECSL

3.2 Bandwidth enhancement and TDS suppression induced by CCPMOF

4. Chaos synchronization in closed-loop ECSLs subject to CCPMOF

4.1 Influences of PM index and injection strength on chaos synchronization

4.2 Mismatch robustness, detuning tolerance and phase fluctuation tolerance properties of chaos synchronization in CCPMOF closed-loop system

5. Chaos communication and security discussion

5.1 Chaos communication in closed-loop ECSLs subject to CCPMOF

5.2 Security discussion

6. Conclusions

Funding

References

Cited By

Figures (11)

Equations (10)

Optics Express