## Abstract

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

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

## 1. Introduction

Chaos communication has attracted great attention for its advantage of physical-layer security enhancement [1–3]. Due to the external optical feedback-induced rich dynamic behaviors, external cavity semiconductor lasers (ECSLs) can easily generate wideband noise-like chaotic signal with excellent complexity [4–6], as such ECSLs are extensively applied as chaotic carrier sources for all-optical chaos communication. The experimental demonstrations of ECSLs-based chaos communication in the commercial networks in Athens (Greece) and recent literatures have confirmed the feasibility of this technology [7–10].

In a chaos communication system, message is encoded onto (or into) the noise-like chaotic carrier and decoded in virtue of chaos synchronization. Therefore, the achievement of chaos synchronization between two ECSLs is the basis of chaos communication. Up to present, most of the investigations on ECSLs-based chaos synchronization and communication are concentrated on the unidirectional coupling and mutual coupling chaotic systems composed of ECSLs subject to conventional optical feedback (COF). Here the so-called COF means the feedback light is a linear replica of the output light with a time delay that is determined by the length of feedback loop [7,11–13]. In the unidirectional coupling COF-ECSL systems, two types of chaos synchronization, namely the complete chaos synchronization and the generalized chaos synchronization, which are achieved respectively based on the mechanisms of symmetric operation and injection-locking effect, can be achieved [12,13]. In the mutual coupling COF-ECSL systems, the achievement of chaos synchronization is mainly based on the symmetric operation mechanism, and two types of synchronization referred to as the isochronal chaos synchronization and the leader-laggard chaos synchronization are achievable [14–16]. On the other hand, there is another type of optical feedback called as phase-conjugate feedback (PCF) by using a phase-conjugate mirror as the external cavity reflector [17–20]. Similar to the COF-ECSLs, ECSLs subject to PCF also show rich dynamic behaviors, such as the symmetry-breaking and restoring bifurcations, super-harmonic self-pulsation [17,19]. While with respect to the COF scenario, the PCF exhibits enhancements in the bandwidth and statistical complexity of chaos [20]. Regarding the chaos synchronization in PCF-ECSL systems, a brief demonstration of the complete chaos synchronization based on the symmetric operation mechanism is reported in [21]. However, with the best of our knowledge, the more practical injection-locking chaos synchronization in the PCF-ECSL system has never been studied, which motivates a thorough investigation on the theoretical conditions, the properties of mismatch robustness and detuning tolerance of the injection-locking chaos synchronization between PCF-ECSLs, as well as the performance of PCF-ECSL-based chaos communication.

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

## 2. Architecture and theoretical analysis

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

For the numerical purpose, the modified Lang-Kobayashi rate equations that take the PCF terms into consideration are adopted to describe the dynamic behaviors of MSL and SSL. The rate equations for the MSL are written as [20–22]:

*M*and

*S*stand for MSL and SSL, respectively.

*E*(

*t*) is the slowly-varying complex electronic field,

*N*(

*t*) is the carrier number in laser cavity. The optical gain in laser cavity is defined as

*G*(

*t*)=

*g*[

*N*(

*t*)-

*N*

_{0}]/[1+

*ɛ*|

*E*(

*t*)|

^{2}], where

*g*denotes the differential gain,

*N*

_{0}is the transparent carrier number, and

*ɛ*is the saturation factor. The other parameters are the laser bias current

*I*, the linewidth enhancement factor

*α*, the electron charge

*q*, the photon lifetime

*τ*, the carrier lifetime

_{p}*τ*, the spontaneous emission factor

_{e}*β*, the operation frequency

*ω*, and the detuning frequency Δ

*ω=ω*-

_{M}*ω*is a Gaussian white noise with zero mean and unity variance, which is introduced here to model the spontaneous emission noise. The second terms in the right hand of Eqs. (1) and (3) denote the phase-conjugate feedback of MSL and SSL, respectively. Therein,

_{S}. χ*k*is the feedback strength,

_{f}*τ*is the feedback delay and

_{f}*ϕ*is the phase change of PCM. It is worth mentioning that the PCF term

_{PCM}*k**(

_{f}E*t*-

*τ*

_{f}_{)}exp(

*i*

*ϕ*) is different from the COF term

_{PCM}*k*(

_{f}E*t*-

*τ*

_{f}_{)}exp(-

*iωτ*). Except the conjugation in the amplitude of feedback electronic field, the PCF term does not include the phase term exp(-

_{f}*iωτ*) because of the properties of phase-conjugate field [20]. Moreover, the third term in the right hand of Eq. (3) is the injection term, wherein

_{f}*k*is the injection strength and

_{i}*τ*is the corresponding flight time from MSL to SSL.

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

*A*(

*t*) is the amplitude of slowly-varying complex electronic field,

*ϕ*(

*t*) is the corresponding phase,

*θ*(

_{f}*t*) and

*θ*(

_{i}*t*)) are the phase changes induced by feedback and injection, respectively. It is worth mentioning that when the output of SSL is locked to that of MSL, the effect of the spontaneous emission noise is neglectable, as such for the sake of simplicity, the spontaneous emission noise terms are not considered here.

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

where*ϕ*is the locking phase difference between the master laser and the slave laser, and it is determined by [24]

_{L}*ϕ*=

_{PCM}*ϕ*-

_{PCMS}*ϕ*is the phase change difference of PCMs. This is the essential condition for the achievement of injection-locking chaos synchronization in the closed-loop PCF-ECSL system. Since the values of

_{PCSM}*ϕ*,

_{L}*τ*, and Δ

_{f}*ϕ*are constant, this condition is difficult to be satisfied, unless Δ

_{PCM}*ω*=0 and Δ

*ϕ*=2

_{PCM}*ϕ*. Therefore, it is necessary to add an additional phase shifter (PS) in the feedback loop of SSL to modify the feedback phase, and the phase change should meet

_{L}On the other hand, the phase shift can also be added in the injection light. In this case, the injection term in Eq. (3) should be modified as *k _{i}E_{M}*(

*t*-

*τ*)exp(-

_{i}*iω*)exp(

_{M}τ_{i}*i*Δ

*ωt*)exp(

*i*

*ϕ*

_{PS}_{2}). Under such a scenario, the injection-locking phase relation in Eq. (10) should be modified as

*ω/π*can be added in the feedback loop of SSL to act as the PS1. Similarly, a phase modulator driven by a periodic sawtooth signal with a frequency of Δ

*ω/*2

*π*can be added in the injection link to act as the PS2 for the second scenario defined by Eq. (17). In practice, the frequency detuning can be eliminated by properly tuning the operation temperatures of MSL and SSL, and then only a fixed optical phase shifter that provides a constant phase shift of

*ϕ*

_{PS}_{1}=2

*ϕ*-Δ

_{L}*ϕ*or

_{PCM}*ϕ*

_{PS}_{2}=-

*ϕ*+Δ

_{L}*ϕ*/2, can guarantee the phase locking conditions in Eqs. (15) and (17).

_{PCM}To quantitatively measure the synchronization quality between MSL and SSL, we adopt the cross-correlation function, which is defined as [25–27]

*P*=|

*A*(

*t*)|

^{2}is the photon number in the laser cavity, and

*τ*is the varying time shift.

In our simulations, the rate equations are solved by using the fourth-order Runge-Kutta algorithm. Unless otherwise stated, the frequency detuning between MSL and SSL is set as Δ*ω=*0, and the values of intrinsic parameters of MSL and SSL are identical, which are chosen to be those reported in [28]: the operation wavelength *λ*=1550 nm, the linewidth enhancement factor *α*=5, the gain saturation coefficient *ɛ*=5×10^{−7}, the carrier lifetime *τ _{e}*=2 ns, the photon lifetime

*τ*=2 ps, the differential gain coefficient

_{p}*g*=1.5×10

^{−8}ps

^{−1}, the transparency carrier density

*N*

_{0 }= 1.5×10

^{8}, the spontaneous emission factor

*β*=1.5×10

^{−6}ns

^{−1}, and the electron charge

*q*=1.602×10

^{−19}C. The bias currents of MSL and SSL are fixed at

*I=*2

*I*

_{th}, where

*I*

_{th}=14.7 mA is the threshold current, the feedback strength is fixed at

*k*= 15 ns

_{f}^{−1}and the feedback delay is set at

*τ*

_{f}_{ }= 3ns. For the sake of simplicity, the injection delay from MSL to SSL is set as

*τ*=0 ns, the phase change difference of PCMs is set as 0 rad. In the following sections, we numerically investigate the properties of injection-locking chaos synchronization and communication in the closed-loop PCF-ECSL system.

_{i}## 3. Results and analysis

We first investigate the phase locking phenomenon induced by the injection from MSL to SSL. When the emission frequency of SSL is locked to that of MSL, the phase relationship between them is determined by Eq. (10). Here we calculate the average phase difference Δ*ϕ=*〈*ϕ _{S}*(

*t*)-

*ϕ*(

_{M}*t*)-Δ

*ωt*〉 and compare it with the theoretical locking phase difference defined in Eq. (11). Figure 2 shows the simulation and analytical results for three exemplary frequency detuning cases with Δ

*f*(Δ

*ω*/2

*π*) = 1GHz, 0, and −1GHz, as a function of injection strength

*k*. It is obvious that the variations of numerical phase difference Δ

_{i}*ϕ*in all the three cases well agree with those of the analytical locking phase difference

*ϕ*. That is, stable phase locking for chaotic light injection is easily achieved. For the case of zero detuning, the locking phase difference keeps at 0.563π which precisely equals to the theoretical results calculated by Eq. (11). While for the nonzero frequency detuning cases, the locked phase asymptotically gets closer and closer to that of zero tuning case, as the increase of injection strength. This phenomenon qualitatively agrees with that occurring in the open-loop semiconductor laser system with COF [24]. On the other hand, the variations of the cross correlation between MSL and SSL versus the tuning phase shift

_{L}*ϕ*in the feedback loop and the injection link are presented in Fig. 2(b). It is indicated that for the case with PS in feedback loop, high-quality chaos synchronization only occurs when the value of

_{PS}*ϕ*

_{PS}_{1}=1.126π which is twice of the steady state locking phase

*ϕ*. For the case with PS in the injection link, high-quality chaos synchronization occurs when the value of

_{L}*ϕ*

_{PS}_{2}=0.437

*π*(

*π*-

*ϕ*) and 1.437

_{L}*π*(-

*ϕ*). These phenomena are perfectly in line with the theoretical conditions in Eqs. (15) and (17). It is worth noting that due to the synchronization quality is quantified by using Eq. (18) with

_{L}*τ*=0 to calculate the cross-correlation coefficient between the output intensities of MSL and SSL, the phase correlation between the outputs of MSL[

*E*(

_{M}*t*)) and SSL (

*E*(

_{S}*t*)] is ambiguous. In fact, for the case of

*ϕ*

_{PS}_{2}=0.437

*π*, there is a phase difference of

*π*between

*E*(

_{M}*t*) and

*E*(

_{S}*t*), this also agrees with the analytical results in Eq. (16). That is, under such a scenario, the synchronization between MSL and SSL is inverse-phase synchronization. Since the results of intensity chaos synchronization for the cases of

*ϕ*

_{PS}_{2}=0.437

*π*and

*ϕ*

_{PS}_{2}=1.437

*π*are quite similar, for the sake of simplicity,

*ϕ*

_{PS}_{2}is fixed at 1.437

*π*and

*ϕ*

_{PS}_{1}is fixed at 1.126

*π*in the following investigations on the injection-locking intensity chaos synchronization and chaotic communication in PCF-ECSLs.

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

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

On the other hand, the intrinsic parameter mismatch robustness property of the PCF-ECSL and COF-ECSL configurations are presented in Fig. 4(b). Here, the mismatch is introduced according to the method reported in [15,29]: the intrinsic parameters for MSL are fixed, while the SSL’s internal parameters *α*, *g* and *τ _{p}* are decreased, and accordingly the parameters

*N*

_{0},

*s*and

*τ*are increased by the same amount (ratio). It is indicated that the quality variation of chaos synchronization versus the mismatch in the PCF-ECSL configuration is similar to that of COF-ECSL configuration. Although the synchronization quality is degraded gradually as the increase of mismatch ratio, the injection-locking chaos synchronization in the closed-loop PCF-ECSL system is robust to a relatively large mismatch up to a few tens of percentages. The cross-correlation coefficient between the intensities of MSL and SSL can be maintained larger than 0.95 in the mismatch range from −20% to 30%. Overall, high-quality injection-locking chaos synchronization in closed-loop PCF-ECSL system is feasible, as long as the analytical conditions in Eqs. (15) and (17) are satisfied.

_{e}Finally, we turn to investigate the performance of chaos communication in the closed-loop PCF-ECSL system. The message is encrypted in the chaotic carrier by the method of chaos modulation [28], which can be mathematically described as *E _{mod}*(

*t*)=

*E*(

_{M}*t*)[1+

*Am*(

*t*)], where

*E*(

_{mod}*t*) denotes the modulated chaotic carrier, the message

*m*(

*t*) is a random binary sequence, and

*A*=0.07 is the modulation index, it is small enough to guarantee the message being efficiently hidden in the chaotic carrier. The message decryption is carried out by the way of direct subtraction decoding, which is described as

_{m}*m*'(

*t*)=

*LPF*[|

*E*(

_{Lmod}*t*)|

^{2}-|

*E*(

_{S}*t*)|

^{2}], where

*E*(

_{Lmod}*t*) denotes the modulated chaotic carrier after fiber link transmission from MSL to SSL, and the LPF operation uses a five-order Butterworth low-pass-filter with a cutoff frequency equal to the message bit rate. Figure 5 illustrates the message transmission from MSL to SSL, through a 60km dispersion-shifted fiber (DSF) link with parameters identical to that in [30]. Here we considered two transmission cases with different bit rates of 2.5Gbit/s and 5Gbit/s, respectively. It is demonstrated that the messages encrypted in the chaotic carrier of MSL can be successfully decrypted by SSL.

To further investigate the chaos communication performance of the PCF-ECSL system, Fig. 6(a) shows the influence of the message bit rate on the Q-factor of the decrypted message. Here the Q-factor is calculated by *Q*=(*P*_{1}-*P*_{0})/(*σ*_{1}-*σ*_{0}), where *P*_{1} and *P*_{0} are the average powers of bits “1” and “0”, respectively, and *σ*_{1} and *σ*_{0} are the corresponding standard deviations [30–32]. The results (square and triangle curves) indicate that acceptable communication performance with a Q-factor greater than 5 can be maintained in the PCF-ECSL system, when the bit rate is lower than 6.5Gbit/s, even though the Q-factor is gradually degraded as the bit rate increases. While for the COF configuration (circle curve), identical acceptable communication performance requires that the message bit rate should be lower than 5.5Gbit/s. Comparison indicates that, with respect to the COF-ECSL configuration, the PCF-ECSL system can provide a larger Q-factor in the case of identical bit rate transmission, and supports a higher transmission capacity for identical Q-factor requirement. Based on this, we can conclude that the communication performance of PCF-ECSL system is enhanced with respect to the COF-ECSL configuration. The improvement of communication performance is intuitively attributed to the effective bandwidth enhancement effect of PCF. Figure 6(b) shows the effective bandwidth of chaotic carriers generated by COF-ECSL and PCF-ECSL, as a function of the feedback strength. Here the definition of effective bandwidth of chaotic carrier is identical to those in [6,20]. It is apparently indicated that the effective bandwidth of chaotic carrier generated by PCF-ECSL is always lager than that of COF-ECSL. The results are well in line with those in [20]. It is well known that, the larger the bandwidth of carrier is, the lager the transmission capacity is. For this reason, as shown in Fig. 6(a), the transmission capacity of the chaotic communication in PCF-ECSL system is relatively improved with respect to that in COF-ECSL system. In addition, our repeating simulation results indicate that as long as the conditions Eqs. (15) and (17) are satisfied, high-quality injection chaos synchronization can be easily achieved, and the additional phase shifter do not influences the maximum transmission bit rate.

## 4. Conclusions

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

## Funding

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

## Disclosures

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

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