Time-Delay Signature Suppression and Communications of Nanolaser Based on Phase Conjugate Feedback

Abstract

The nonlinear dynamics of nanolasers (NLs), an important component of optical sources, has received much attention. However, there is a lack of in-depth research into the high-quality chaotic output of NLs and their applications in chaotic secure communications. In this paper, we make the NLs generate broadband chaotic signals whose time-delay signatures (TDS) are completely hidden by a phase conjugate feedback structure. And in the two-channel communication scheme, we make the NLs achieve a combination of a low-latency high degree of synchronization and two-channel transmission technique, which enhances the security of message encryption and decryption. We also investigate the effects of system parameters, Purcell factor $F$, spontaneous emission coupling factor $\beta$, and bias current $I$ on the TDS, as well as the effects of parameter mismatch and injection parameters on chaos synchronization and message recovery. The results show that the phase conjugate feedback-based NLs can achieve the suppression of the TDS within a certain parameter range, and it can achieve high-quality synchronization and enhance the security of chaotic communication under appropriate injection conditions.

1. Introduction

Optical chaos has attracted a lot of attention in the last two decades, especially in terms of potential applications in chaotic secure communications [1,2], high-speed physical random numbers [3,4,5], compressed sensing [6], and chaotic lidar/radar [7,8]. However, the feedback action of the external cavity between the laser and the external reflector generates a chaotic time-delay signature (TDS) [9,10], and when such chaotic signals are used as information carriers, they may be extracted by eavesdroppers in order to obtain the key parameters of the communication system and reconfigure them, which, in turn, jeopardizes the security of the chaotic communication system [11]. We also need to focus on the critical issue of laser chaotic bandwidth, which may affect the transmission capacity in chaotic communication, the distance resolution of chaotic lidar, and the rate of random number generation. However, the large size problem arising from most systems consisting of conventional semiconductor lasers and associated auxiliary components may not be suitable for photonic integrated circuits required for practical applications. Therefore, we need to explore a small and novel approach involving using NLs as chaotic entropy sources to generate high-quality broadband chaotic signals and achieve high-quality synchronization between them [12,13].

Currently, time series with specific time-delay can be extracted using standard statistical measurement analysis techniques, which mainly include autocorrelation function (ACF) [14], permutation entropy (PE) [15], and delay mutual information (DMI) [16]. In recent years, how to suppress TDS has become a popular topic of research, for which many published papers have proposed various schemes [17,18,19,20,21,22,23,24,25,26,27]. For example, introducing multiple feedback loops into the system, exploring complex feedback schemes, and adding multiple injection paths have all been considered. NLs have a very small size and volume, and they are able to operate efficiently at low energy inputs with high quantum efficiency. In addition, NLs have fast modulation response speeds, making them suitable for applications such as high-speed communications and data processing. In order to explore the effects of NLs in the integration of photonic circuits, related scholars have conducted experimental studies of their different structures. such as micro-post [28], nano-pillar [29] and bowtie [30], Fabry–Perot [31], nanowire [32], and nano-patch lasers [33], where continuous wave lasing is observed via optical pumping [34] and electrical pumping [35]. The two parameters of the system, Purcell factor $F$ and spontaneous radiation coupling factor $\beta$, have a very great influence on the dynamic characteristics of the NLs. K.A. Shore et al.’s studies analyzed the dynamic behavior of the NL under current modulation [36], optical feedback [37], optical injection [38], and inter-injection [39], respectively, and described in detail the effects of the relevant parameters, such as the Purcell factor F, and the effects of relevant parameters, such as spontaneous radiation coupling factor β, on the laser output are elaborated in detail. In addition, Han et al. further investigated the dynamics of mutually coupled semiconductor NLs and observed a rich dynamic output [40,41,42,43]. Elsonbaty et al. investigated the TDS of a single NL using a hybrid all-optical and electro-optical feedback scheme and compared the chaotic characteristics generated under normal optical feedback, phase conjugate optical feedback, and grating-mirror optical feedback [44]. Xiang [45] et al. demonstrated that chaotic signals with completely hidden TDS can be output from NLs in a dual chaotic injection system over a wide range of parameters. Based on the above studies, we can show that the laser properties are different to those of conventional mirror feedback. It is due to the property of a phase conjugate mirror, which reverses the phase of the reflected light with respect to the incident light, resulting in zero external cavity round trip phase change. From this, it can be predicted that NLs can be well implemented for TDS hiding by means of phase conjugate feedback, and we further investigate the synchronization characteristics and communication security under the corresponding structure.

The aim of this paper is that the master nanolaser (MNL) is fed back by the phase conjugate mirror and then further injected into the two slave nanolasers (SNL₁ and SNL₂) in parallel. In the simulation, we will quantify the TDS of the signals by introducing the autocorrelation function (ACF) and systematically analyze the dynamics of the MNL and the TDS of the output signals of the system under different feedback parameters. At the same time, we will study the effect of internal parameter mismatch and the differences between injection parameters on chaotic synchronization and communication for SNL₁ and SNL₂ [46]. The paper is organized as follows: Section 2 describes the delay suppression and synchronous communication model of an NL through phase conjugate feedback and parallel injection of SNL₁ and SNL₂, as well as the corresponding rate equations and parameter definitions. Section 3 presents the simulation results related to the above study. Finally, Section 4 provides the basic conclusions based on the simulation results.

2. Theoretical Model

The structure of the NL chaotic system is shown in Figure 1, including the master nanolaser (MNL) and two slave nanolasers (SNL₁ and SNL₂). The MNL is fed back by a phase conjugate mirror, and under appropriate feedback conditions, the MNL exhibits chaotic oscillation phenomena and generates rich nonlinear dynamics, as shown in Figure 1. The feedback intensity of the MNL and the injection intensity of the MNL into SNL₁ and SNL₂ are regulated by variable attenuators (VA₁, VA₂, and VA₃), and the optical isolators (OI₁ and OI₂) ensure that the MNL can be unidirectionally injected into SNL₁ and SNL₂, respectively. The control parameters in this system include the feedback parameter, injection parameter, bias currents I, Purcell factor F, and spontaneous emission coupling factor β. The L-K rate equation of the NL system studied in this paper is modeled as shown in [38].

$$\begin{array}{ll}\frac{d{I}_M\left(t\right)}{dt}& =\mathsf{\Gamma }\left[\frac{F\beta N_M\left(t\right)}{{\tau }_n}+\frac{g_n\left(N_M\left(t\right)-N_0\right)}{1+\epsilon I_M\left(t\right)}{I}_M\left(t\right)\right] -\frac{1}{{\tau }_p}{I}_M\left(t\right)\\ & +2k_d\sqrt{I_M\left(t\right)I_M\left(t-{\tau }_d\right)}\cos \left({\theta }_1\left(t\right)\right)+F_I\left(t\right)\end{array}$$

(1)

$$\frac{d{\varphi }_M\left(t\right)}{dt}=\frac{\alpha }{2}\mathsf{\Gamma }{g}_n\left(N_M\left(t\right)-N_{th}\right)-k_d\frac{\sqrt{I_M\left(t-{\tau }_d\right)}}{\sqrt{I_M\left(t\right)}}\sin \left({\theta }_1\left(t\right)\right)+F_{\varphi }\left(t\right)$$

(2)

$$\frac{d{N}_M\left(t\right)}{dt}=\frac{I_{dc}}{e{V}_a}-\frac{N_M\left(t\right)}{{\tau }_n}\left(F\beta +1-\beta \right)-\frac{g_n\left(N_M\left(t\right)-N_0\right)}{1+\epsilon I_M\left(t\right)}{I}_M\left(t\right)+F_N\left(t\right)$$

(3)

$$\begin{array}{ll}\frac{d{I}_{S1|S2}\left(t\right)}{dt}& =\mathsf{\Gamma }\left[\frac{F\beta N_{S1|S2}\left(t\right)}{{\tau }_n}+\frac{g_n\left(N_{S1|S2}\left(t\right)-N_0\right)}{1+\epsilon I_{S1|S2}\left(t\right)}{I}_{S1|S2}\left(t\right)\right]-\frac{1}{{\tau }_p}{I}_{S1|S2}\left(t\right)\\ \hspace{0.33em}& +2k_{ri}\sqrt{I_{S1|S2}\left(t\right)I_{M|M}\left(t-{\tau }_{ri}\right)}\cos \left({\theta }_j\left(t\right)\right)\end{array}$$

(4)

$$\frac{d{\varphi }_{S1|S2}\left(t\right)}{dt}=\frac{\alpha }{2}\mathsf{\Gamma }{g}_n\left(N_{I|S}\left(t\right)-N_{th}\right)-2\pi \mathsf{\Delta }{f}_i-k_{ri}\frac{\sqrt{I_{M|M}\left(t-{\tau }_{ri}\right)}}{\sqrt{I_{S1|S2}\left(t\right)}}\sin \left({\theta }_j\left(t\right)\right)$$

(5)

$$\frac{d{N}_{S1|S2}\left(t\right)}{dt}=\frac{I_{dc}}{e{V}_a}-\frac{N_{S1|S2}\left(t\right)}{{\tau }_n}\left(F\beta +1-\beta \right)-\frac{g_n\left(N_{S1|S2}\left(t\right)-N_0\right)}{1+\epsilon I_{S1|S2}\left(t\right)}{I}_{S1|S2}\left(t\right)$$

(6)

$${\theta }_1\left(t\right)={\varphi }_M\left(t\right)-{\varphi }_M\left(t-{\tau }_d\right)$$

(7)

$${\theta }_j\left(t\right)=2\pi f_{M|M}{\tau }_{ri}+{\varphi }_{S1|S2}\left(t\right)-{\varphi }_{M|M}\left(t-{\tau }_{ri}\right)-2\pi \mathsf{\Delta }{f}_{i}t\left(i=1,2;j=2,3\right)$$

(8)

$$F_I\left(t\right)=\sqrt{\frac{2I\left(t\right)\beta N\left(t\right)}{{\tau }_n\Delta t}}{\mathrm{x}}_I$$

(9)

$$F_n\left(t\right)=-\sqrt{\frac{2I\left(t\right)\beta N\left(t\right)}{{\tau }_n\Delta t}}{\mathrm{x}}_s+\sqrt{\frac{2N\left(t\right)}{{\tau }_n\Delta t{V}_a}}{\mathrm{x}}_n$$

(10)

$$F_{\varphi }\left(t\right)=\frac{1}{I\left(t\right)}\sqrt{\frac{2I\left(t\right)\beta N\left(t\right)}{2{\tau }_n\Delta t}}{\mathrm{x}}_{\varphi }$$

(11)

In the above rate equations, the subscripts ‘M’, ’S1’, and ’S2’ denote MNL, SNL₁, and SNL₂, respectively. $I\left(t\right)$ is the photon density, $\varphi \left(t\right)$ is the phase, and $N\left(t\right)$ is the carrier density. $F$ is the Purcell factor, $\beta$ is the spontaneous radiation coupling factor, $\mathsf{\Gamma }$ is the confinement factor, and ${\tau }_n$ and ${\tau }_p$ are the carrier lifetime and the photon lifetime. $g_n$ is the differential gain, $N_0$ is the transparency carrier density, $\epsilon$ is the gain saturation factor, and $\alpha$ is the linewidth enhancement factor. $I_{dc}=q{I}_{th}$ is the laser operating current, $I_{th}$ is the threshold current with respect to $F$ and $\beta$, $V_a$ is the volume of the active region, $e$ is the electron charge, and $N_{th}$ is the threshold carrier density.

In the optical feedback parts of Equations (1) and (2), the feedback strength of the feedback path $k_d$ and the feedback delay ${\tau }_d$ are included. $k_d$ can be expressed as follows [38]

$$k_d=f\left(1-R\right)\sqrt{\frac{R_{ext}}{R}}\frac{c}{2nL}$$

(12)

where $f$ is the feedback coupling factor, which can be adjusted in the model to determine the feedback strength $k_d$; $c$ is the speed of light in free space; $n$ is the refractive index; and $L$ is the feedback cavity length. $R$ is the reflectivity of the NL and $R_{ext}$ is the specular reflectivity. Optical injection into both from the NL is controlled by the injection rate $k_{ri}$, while $k_{ri}$ and ${\tau }_{ri}$ are the injection rate and injection delay, respectively. $R_{inj}$ is the injection coefficient, and $k_{ri}$ can be expressed as follows [38]:

$$k_{ri}=\left(1-R\right)\sqrt{\frac{R_{inj}}{R}}\frac{c}{2nL}$$

(13)

In Equations (9)–(11), the symbols ${\mathrm{x}}_I$, ${\mathrm{x}}_n$, and ${\mathrm{x}}_{\varphi }$ are Gaussian distributed random variables with zero mean and unit variance. The rate equations with phase conjugate feedback and Langevin noise terms are solved numerically using the Runge–Kutta algorithm. The Langevin noise is modeled by applying the random noise as computed, where the noise interval $\Delta t$ is 0.5 ps.

Table 1. Parameters used in the numerical simulations [38].

Parameter	Description	Value
$\mathsf{\Gamma }$	Confinement factor	0.645
${\tau }_n$ ${\tau }_p$	Carrier lifetime Photon lifetime	1 ns 0.36 ps
td	Feedback delay	0.2 ns
$g_n$	Differential gain	$1.65\times 10^{-6}\hspace{0.33em}\mathrm{c}{\mathrm{m}}^3/\mathrm{s}$
$N_0$	Transparency carrier density	$1.1\times 10^{-18}\hspace{0.33em}\mathrm{c}{\mathrm{m}}^{-3}$
$\epsilon$	Gain saturation factor	$2.3\times 10^{-17}\hspace{0.33em}\mathrm{c}{\mathrm{m}}^3$
$\alpha$	Linewidth enhancement factor	5
$V_{\mathrm{a}}$	Volume of active region	$3.96\times 10^{-13}\hspace{0.33em}\mathrm{c}{\mathrm{m}}^3$
${\lambda }_{}$	Wavelength of NL	1591 nm
$R$	Laser facet reflectivity	0.85
$R_{ext}$	External facet power reflectivity	0.95
$R_{inj}$	Injection ratio	0–0.1
n	Refractive index	3.4
L	Cavity length	$1.39\hspace{0.33em}\mathsf{\mu }\mathrm{m}$
Q	Quality factor	428
f	Feedback coupling fraction	0–0.9

lists some of the important parameters used in the simulation of the NL system and the corresponding parameter annotations.

In order to quantify the TDS of the signal, we used the autocorrelation function (ACF), and in the following study, the TDS is considered to be hidden when the peak value of the ACF is less than 0.2. The ACF is defined as follows [43,44]:

$$C\left(\Delta t\right)=\frac{〈\left[I\left(t+\Delta t\right)-〈I\left(t+\Delta t\right)〉\right]\left[I\left(t\right)-〈I\left(t\right)〉\right]〉}{\sqrt{〈{\left[I\left(t+\Delta t\right)-〈I\left(t+\Delta t\right)〉\right]}^2〉〈{\left[I\left(t\right)-〈I\left(t\right)〉\right]}^2〉}}$$

(14)

where <> denotes the average value of the time series $I\left(t\right)$, $\mathsf{\Delta }t$ denotes the time shift, ACF denotes a comparison between the similarity of a signal and itself at different points in time, when the lower the autocorrelation value, the higher the unpredictability of the chaotic stochastic sequences, as well as the more difficult it is to be extracted into useful information.

3. Simulation Results

In this section, we simulate the laser rate equations of the system using the fourth-order Runge–Kutta algorithm. Firstly, the nonlinear dynamics of the MNL is investigated, and suitable parameters are chosen so that the two parallel injected SNLs operate in a chaotic state. Then, the focus is on the synchronization performance of the two SNLs under equal and unequal frequency detuning as the injection intensity varies. Through these studies, we identify the range of parameters that enable the SNLs to be highly synchronized and further apply them to relevant chaotic secure communications.

The output of the NL can be controlled by varying the feedback parameter to control the nonlinear dynamical system. Figure 2a,b show the intensity time series and ACF plots of the MNL at a feedback parameter of 0.02, while (c) and (d) represent the intensity time series and ACF plots of the MNL at a feedback parameter of 0.2. Here, the relevant parameters are set as $F$ = 14, $\beta$ = 0.05, $I=2I_{th}$, and $I_{th}$ = 1.127 mA. It can be seen from the Figure 2 that the MNL works in chaotic state when the feedback parameters are 0.02 and 0.2. At the same time, their corresponding ACF values at the time-delay of 0.2 ns are both less than 0.2, indicating that the TDS of the output chaotic signal is completely hidden. Then, the ACF line graphs of the MNL are investigated according to the variation in the feedback parameters under $F$ = 14, $\beta$ = 0.05, $F$ = 14, $\beta$ = 0.1, and $F$ = 30, $\beta$ = 0.1, respectively. From Figure 3, it can be observed that as the feedback parameters increase, the ACF value of the MNL gradually increases in all three different cases, but the corresponding ACF values in this interval range are less than 0.2. This indicates that the delay of the MNL can be well suppressed and is conducive to the subsequent study of the synchronous communication of the two SNLs. When $F$ = 30 and $\beta$ = 0.1, the ACF of MNL greatly increases with the increase in the feedback parameters because the larger the values of $F$ and $\beta$, the greater the damping of the relaxation oscillation of NL [47]. Therefore, we can use medium $F$ and small $\beta$ to better hide the TDS of NLs in a wider parameter space.

In practice, parameter mismatch is inevitable. We proceed to study the synchronization performance in the presence of mismatched parameters between the two SNLs.

There are two different cases: Case 1 is that MNL is equal to the frequency detuning between SNL₁ and SNL₂, and Case 2 is that MNL is unequal to the frequency detuning between SNL₁ and SNL₂.

3.1. Case 1: MNL Is Equal to the Frequency Detuning between SNL₁ and SNL₂

We investigate the correlation coefficient C_TR of the two SNLs with respect to the variation in the injected intensity $k_{inj1}$ and $k_{inj2}$ when the MNL is equal to the frequency detuning of the two SNLs. In the simulation, we choose different parameter values as $\mathsf{\Delta }{f}_1$ = ${\mathsf{\Delta }f}_2$= 10 GHz, $\mathsf{\Delta }{f}_1$=${\mathsf{\Delta }f}_2$ = 15 GHz, and $\mathsf{\Delta }{f}_1$ = ${\mathsf{\Delta }f}_2$ = −15 GHz. It can be seen in Figure 4a that when the frequency detuning $\mathsf{\Delta }{f}_1$ = ${\mathsf{\Delta }f}_2$ = 10 GHz, the region of highly synchronous SNL₁ and SNL₂ is almost in the range of a 45° angle and will be larger and larger with the increase in the injection intensity. When the frequency detuning is increased to 15 GHz in Figure 4b, it is clearly observed that a higher injection strength is required to enable the two SNLs to be highly synchronized. In Figure 4c, when the frequency detuning is −15 GHz, it can be clearly observed that the range of achieving high synchronization between the two SNLs is much wider, and a wider region can be achieved with good synchronization at a very small range of injection strength.

3.2. Case 2: MNL Is Unequal to the Frequency Detuning between SNL₁ and SNL₂

Then, we investigated the case when the MNL is not equal to the frequency detuning of the two SNLs. In Figure 5a, when the frequency detuning $\mathsf{\Delta }{f}_1$ = 10 GHz, ${\mathsf{\Delta }f}_2$ = 15 GHz, the synchronization effect of the two SNLs is obviously weakened, and a larger injection intensity is required to achieve a highly synchronized effect. In Figure 5b,c, when the frequency detuning is $\mathsf{\Delta }{f}_1$ = −15 GHz, ${\mathsf{\Delta }f}_2$ = 10 GHz and $\mathsf{\Delta }{f}_1$ = −15 GHz, and ${\mathsf{\Delta }f}_2$ = 15 GHz, respectively, the synchronization effect achieved is similar to that of Figure 5a. At high injection intensity, the two SNLS will be locked via MNL injection so that the synchronization effect between them is less affected by the change in the injection rate. Finally, in Figure 5d, when the frequency detuning $\mathsf{\Delta }{f}_1$ = −15 GHz and ${\mathsf{\Delta }f}_2$ = −5 GHz, it can be seen that when the frequency detuning is negative, the synchronization performance of the two SNLs is better than that shown in the above figure. This is because when the frequency detuning of the two SNLs is negative, their resonant frequencies will be closer to each other, which is more helpful for promoting synchronization between them.

Subsequently, we investigate the ACF of SNL₁ at frequency detuning of −15 GHz, 10 GHz, and 15 GHz, respectively. From Figure 6, it can be found that the TDS of SNL₁ are all well suppressed at an injection strength of 300 ns⁻¹. This is mainly due to the fact that the high injection rate causes SNL₁ to be locked by the MNL injection, and, thus, the TDS of MNL are shown to be well hidden. The TDS of SNL₂ is similar to those of SNL₁ due to the injection strength $k_{inj1}$ = $k_{inj2}$ in the subsequent study.

To further investigate the effect of internal parameter mismatch on the synchronization of the two SNLs, we fixed the parameters of SNL₁ and varied the parameters of SNL₂ according to the associated mismatch ratio u, where $u=\frac{{\mathsf{\Pi }}^{{\mathrm{S}\mathrm{N}\mathrm{L}}_1}-{\mathsf{\Pi }}^{{\mathrm{S}\mathrm{N}\mathrm{L}}_2}}{{\mathsf{\Pi }}^{{\mathrm{S}\mathrm{N}\mathrm{L}}_1}}$. We also fixed the external parameters, such as injection strength and frequency detuning, to be the same for both SNLs. The variation in the associated peak C_TR with the mismatch ratio u when $k_{inj1}$ = $k_{inj2}$ = 300 ns⁻¹ is shown in Figure 7a. It can be observed from the figure that the synchronization performance corresponding to the five mismatched parameters is also different at moderate injection intensity. As the parameter mismatch ratio increases, the corresponding correlation peak C_TR decreases, but the overall value of C_TR is greater than 0.95, indicating that the system can still achieve high-quality chaotic synchronization. When the injection rate is very large, i.e., $k_{inj1}$ = $k_{inj2}$ = 600 ns⁻¹, it can be seen from Figure 7b that the variation in the parameter mismatch u has less effect on the chaotic synchronization. Therefore, the difference in the synchronization and decoding characteristics of the system when the parameters are mismatched is not significant, which also indicates that that the system is robust to parameter mismatch.

Before the message encoding and decoding process, we fixed the frequency detuning of both SNLs to −15 GHz and investigated the corresponding transmission bandwidths. Figure 8 shows the spectrograms of the chaotic carriers generated by the SNL₁ by varying $F$ and $\beta$ at injection currents of $I=2\mathrm{I}\mathrm{t}\mathrm{h}$ and $I=4\mathrm{I}\mathrm{t}\mathrm{h}$, respectively. We define the effective bandwidth as the range between DC and the frequency containing 80% of the spectral power. When the injection current is set at $I=2\mathrm{I}\mathrm{t}\mathrm{h}$, the effective bandwidths are 36.6 GHz, 22.8 GHz, and 13.3 GHz for ($F$, $\beta$) values of (14, 0.05), (14, 0.1), and (30, 0.1). In comparison, the effective bandwidths for the same ($F$, $\beta$) settings are 35.8 GHz, 21.8 GHz, and 11.8 GHz when the injection current is set at $I=4\mathrm{I}\mathrm{t}\mathrm{h}$. These results show that the system is capable of achieving broadband chaos and, hence, high-speed message transmission when all factors are considered. However, we also need to ensure that the TDS can be completely hidden to ensure the security of the communication and prevent information leakage. By choosing the appropriate $F$, $\beta$, and injection current, i.e., $F$ = 14, $\beta$ = 0.05, and $I=2\mathrm{I}\mathrm{t}\mathrm{h}$, the system can be made to stably output chaotic signals with time-delay hiding, bandwidth enhancement, and a high degree of unpredictability.

Finally, we conducted a related study of the encoding and decoding process of the messages. Based on the above studies, the frequency detuning and injection rate parameters were fixed at −15 GHz and 300 ns⁻¹, enabling the two SNLs to have low latency, high bandwidth, and high synchronization performance. The CMO encryption method is used in Figure 9, where a small-amplitude message is used to modulate the laser signal output. The mathematical expression is $I_{ext}(t)=I(t)[1+h_{CMO}m(t)]$, where $h_{CMO}$ denotes the modulation depth and $m(t)$ denotes the message to be transmitted. In a two-channel communication system, the drive signal from the MNL is transmitted through channel 1 for high-quality chaotic synchronization between SNL₁ and SNL₂, while the modulation signal $I_{ext}(t)$ is transmitted through channel 2. Therefore, at the SNL₁ end, we can obtain the differential signal by subtracting the output of SNL₁ from the transmitted signal and then filter the differential signal using a fifth-order Butterworth filter with a cutoff frequency equal to 0.6 × B_S to obtain the recovered message (where B_S is the bit rate of the message, and B_S = 8 Gbit/s is taken as an example in this study).

Figure 9 shows the results of the message encryption and decryption process. Firstly, we illustrate the quality of the recovered messages without considering parameter mismatches, using the quality factor $Q=(I_1-I_0)/({\sigma }_1+{\sigma }_0)$ and the eye diagram, where $I_1$ and $I_0$ are the average optical power values of bits “1” and “0”. ${\sigma }_1$ and ${\sigma }_0$ are the corresponding standard deviations. We stipulate that the quality factor Q > 6 is sufficient for communication, i.e., BER < 10⁻⁹. The original message $m(t)$, the recovered message $m\prime (t)$, and the corresponding eye diagrams are depicted for a $h_{CMO}$ = 0.6 modulation depth. From Figure 9(a1–c1), it can be seen that the message can be fully recovered at a moderate injection rate without parameter mismatch, corresponding to Q = 10.41, and the eye diagram is clear and widely open. Then, parameter mismatches are introduced for SNL₂: ${{\displaystyle {\tau }_{\mathrm{n}}}}^{{{\displaystyle SNL}}_2}=(1+u){{\displaystyle {\tau }_{\mathrm{n}}}}^{{{\displaystyle SNL}}_1}$, ${{\displaystyle g_{\mathrm{n}}}}^{{{\displaystyle SNL}}_2}=(1-u){{\displaystyle g_{\mathrm{n}}}}^{{{\displaystyle SNL}}_1}$, ${{\displaystyle {\tau }_{\mathrm{p}}}}^{{{\displaystyle SNL}}_2}=(1-u){{\displaystyle {\tau }_p}}^{{{\displaystyle SNL}}_1}$, ${{\displaystyle {\alpha }_{\mathrm{n}}}}^{{{\displaystyle SNL}}_2}=(1-u){{\displaystyle {\alpha }_{\mathrm{n}}}}^{{{\displaystyle SNL}}_1}$, and ${{\displaystyle N_0}}^{{{\displaystyle SNL}}_2}=(1+u){{\displaystyle N_0}}^{{{\displaystyle SNL}}_1}$. This is shown in Figure 9(a2–c2), where the mismatch ratio u = 0.1 is set and the injection rate is constant. We can clearly see that the value of the quality factor Q of SNL₂ decreases to 9.99, but it still satisfies the requirement that Q is greater than 6, and, thus, the eye diagram is still able to demonstrate openness, i.e., it is better able to recover the same message. This corroborates with the effect of parameter mismatch on synchronization performance described above and proves that the system is robust to parameter mismatch. Finally, keeping the mismatch ratio u = 0.1 and increasing the injection rate to a reasonable 600 ns⁻¹ is used to compensate for the degradation due to mismatch. The corresponding results are shown in Figure 9(a3–c3), where the value of quality factor Q is 10.24, which further improves the message recovery. However, the security of the system may be affected when the injection rate is too large, so we should choose appropriate injection parameters to enable the system to achieve a high synchronization performance and enhance the security of chaotic communication.

4. Conclusions

In summary, the NL is capable of generating high-quality chaotic signals with low time-delay, wide bandwidth, and high synchronization signature through phase conjugate feedback and then parallel injection into SNL₁ and SNL₂. The effects of system parameters, Purcell factor F, spontaneous radiation coupling factor β, and bias current I on the TDS hiding and bandwidth of chaotic output from the NL are investigated in detail. Then, we applied the output chaotic signals to two-channel communication and further investigated the effects of injection parameters, as well as internal parameter mismatch, on chaotic synchronous communication. According to this study, the selection of appropriate injection parameters can make the NL stably output chaotic signals with time-delay hiding, bandwidth enhancement, and high unpredictability, as well as enhance the security of chaotic communication. The research presented in this paper offers important theoretical guidance for the practical application of NL chaotic systems.