# Exploration of Four-Channel Coherent Optical Chaotic Secure Communication with the Rate of 400 Gb/s Using Photonic Reservoir Computing Based on Quantum Dot Spin-VCSELs

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Framework and Simulation Experiment Setup

_{x}and GS-PC

_{y}, respectively. Interestingly, its excited state yields two additional photonic currents recognizable as ES-PC

_{x}and ES-PC

_{y}. Each of the QPSK, 4QAM, 8QAM and 16QAM is IQ modulated with a group of bit sequences (baseband signal). In this scheme, there are four distinct groups of bit sequence signals, as depicted as ${b}^{1}$–${b}^{4}$, individually. For the convenience of discussion, the temporal dynamics of the QPSK, 4QAM, 8QAM and 16QAM are described by ${S}_{1}$(t), ${S}_{2}$(t), ${S}_{3}$(t) and ${S}_{4}$(t), respectively. The QPSK, 4QAM, 8QAM and 16QAM are masked within the chaotic GS-PC

_{x}, GS-PC

_{y}, ES-PC

_{x}and ES-PC

_{y}, respectively. These four channels of chaotic masked signals are integrated into a single optical fiber utilizing a wavelength division multiplexer (WDM Mux). In the RCM, after fiber transmission, the combined signals are partitioned into four-channel chaotic masked signals via a wavelength division demultiplexer (WDM DeMux). Each channel of chaotic masked signal is subsequently bisected into dual beams. A singular beam of chaotic masked signal is introduced to a photonic RC. Here, the predicted outputs from the RC

_{1}–RC

_{4}are denoted as the GS-${\mathrm{PC}}_{x}^{\prime}$, GS-${\mathrm{PC}}_{y}^{\prime}$, ES-${\mathrm{PC}}_{x}^{\prime}$ and ES-${\mathrm{PC}}_{y}^{\prime}$, respectively. Once output weights are precisely trained within the non-linear node states of each photonic RC, the GS-PC

_{x}, GS-PC

_{y}, ES-PC

_{x}and ES-PC

_{y}can be perfectly synchronized with GS-${\mathrm{PC}}_{x}^{\prime}$, GS-${\mathrm{PC}}_{y}^{\prime}$, ES-${\mathrm{PC}}_{x}^{\prime}$ and ES-${\mathrm{PC}}_{y}^{\prime}$ correspondingly. In this scenario, signal types QPSK, 4QAM, 8QAM and 16QAM can be demodulated by applying synchronous subtraction between the chaotic masked signal and each RC’s predicted output. These demodulated messages, noted as ${S}_{1}^{\prime}$(t), ${S}_{2}^{\prime}$(t), ${S}_{3}^{\prime}$(t) and ${S}_{4}^{\prime}$(t), are then channeled into their respective coherent demodulation units (CDUs with the subscripts of 1–4). Post coherent demodulation and DSP, four sets of bit sequence signals are further decoded. These reinstated signal bits are referred to as ${b}^{\prime 1}$–${b}^{\prime 4}$, respectively.

_{x}and GS-PC

_{y}) using the FPBS

_{1}, with their amplitudes represented as ${E}_{Gx}$(t) and ${E}_{Gy}$(t), respectively. Likewise, the light emitted from its excited state (ES) is separated into two chaotic polarization components (ES-PC

_{x}and ES-PC

_{y}) using the FPBS

_{2}, and their amplitudes are indicated by ${E}_{Ex}$(t) and ${E}_{Ey}$(t), respectively. The QPSK and 4QAM signals are concealed within the chaotic GS-PC

_{x}and GS-PC

_{y}using the power combiners 1 and 2 (PC

_{1}and PC

_{2}), respectively. These two chaotic hidden signals can be described as (${E}_{Gx}$(t) + ${S}_{1}$(t)) and (${E}_{Gy}$(t) + ${S}_{2}$(t)), respectively, and are combined into a single beam through the fiber polarization coupler 1 (FPC

_{1}). The 8QAM and 16QAM signals are masked within the ES-PC

_{x}and ES-PC

_{y}using the PC

_{3}and PC

_{4}, respectively. These two chaotic masked signals are represented as (${E}_{Ex}$(t) + ${S}_{3}$(t)) and (${E}_{Ey}$(t) + ${S}_{4}$(t)), respectively, and merged into a single beam via FPC

_{2}. The mixed light-waves from FPC

_{1}and FPC

_{2}are coupled into an optical fiber through the WDM Mux. After fiber transmission, the multiplexed light-waves are split into two beams with different wavelengths via the WDM DeMux. One beam of light from the WDM DeMux is divided into GS-PC

_{x}and GS-PC

_{y}, which contain hidden messages, via the FPBS

_{3}. The GS-PC

_{x}, carrying the QPSK signal, is further split into two parts using the fiber beam splitter 1 (FBS

_{1}). One part is injected into input layer 1, and the other is converted into a current signal by the PD

_{5}. The GS-PC

_{y}with 4QAM, ES-PC

_{x}with 8QAM and ES-PC

_{y}with 16QAM are processed similarly.

_{x}, including QPSK and the GS-PC

_{y}with 4QAM, are transformed into two distinct current signals via the PD

_{1}and PD

_{2}, amplified using electric amplifiers EA

_{1}and EA

_{2}, and eventually sampled as separate input data series through the discrete modules DM

_{1}and DM

_{2}, respectively. These data series are designated as ${u}_{Gx}$(n−${L}_{Gx}$) and ${u}_{Gy}$(n−${L}_{Gy}$). Moreover, the sampled time series of the QPSK, 4QAM, 8QAM and 16QAM are respectively described as ${I}_{1}$(n), ${I}_{2}$(n), ${I}_{3}$(n) and ${I}_{4}$(n), where ${I}_{1}$(n) = $|{S}_{1}{(n)|}^{2}$, ${I}_{2}$(n) = $|{S}_{2}{(n)|}^{2}$, ${I}_{3}$(n) = $|{S}_{3}{(n)|}^{2}$ and ${I}_{4}$(n) = $|{S}_{4}{(n)|}^{2}$. As a result, ${u}_{Gx}$(n−${L}_{Gx}$) = (${C}_{Gx}$(n−${L}_{Gx}$) + ${I}_{1}$(n−${L}_{Gx}$)), ${u}_{Gy}$(n−${L}_{Gy}$) = (${C}_{Gy}$(n−${L}_{Gy}$) + ${I}_{2}$(n−${L}_{Gy}$)), where ${C}_{Gx}$(n−${L}_{Gx}$) = $|{E}_{Gx}(n-{L}_{Gx}){|}^{2}$ and ${C}_{Gy}$(n−${L}_{Gy}$) = $|{E}_{Gy}(n-{L}_{Gy}){|}^{2}$. The term n denotes the discrete time index, while ${L}_{Gx}$ and ${L}_{Gy}$ signify the discrete channel delay lengths for GS-PC

_{x}and GS-PC

_{y}, respectively. Input layers 3 and 4 process ES-PC

_{x}containing 8QAM and ES-PC

_{y}carrying 16QAM in a similar manner, yielding respective input data as ${u}_{Ex}$(n−${L}_{Ex}$) and ${u}_{Ey}$(n−${L}_{Ey}$). Here, ${u}_{Ex}$(n−${L}_{Ex}$) equals (${C}_{Ex}$(n−${L}_{Ex}$) + ${I}_{3}$(n−${L}_{Ex}$)) and ${u}_{Ey}$(n−${L}_{Ey}$) corresponds to (${C}_{Ey}$(n−${L}_{Ey}$) + ${I}_{4}$(n−${L}_{Ey}$)), where ${C}_{Ex}$(n−${L}_{Ex}$) = $|{E}_{Ex}(n-{L}_{Ex}){|}^{2}$ and ${C}_{Ey}$(n−${L}_{Ey}$) = $|{E}_{Ey}(n-{L}_{Ey}){|}^{2}$. ${L}_{Ex}$ represents the discrete channel delay length for the ES-PC

_{x}and ${L}_{Ey}$ illustrates that of the ES-PC

_{y}. Importantly, ${C}_{Gx}$(n−${L}_{Gx}$), ${C}_{Gy}$(n−${L}_{Gy}$), ${C}_{Ex}$(n−${L}_{Ex}$) and ${C}_{Ey}$(n−${L}_{Ey}$) are considered four distinct prediction targets. The sampled data, ${u}_{Gx}$(n−${L}_{Gx}$) and ${u}_{Ex}$(n−${L}_{Ex}$), are multiplied by the mask signal, Mask

_{x}, while ${u}_{Gy}$(n−${L}_{Gy}$) and ${u}_{Ey}$(n−${L}_{Ey}$) are multiplied by Mask

_{y}. Both Mask

_{x}and Mask

_{y}are chaotic signals, as illustrated in [24]. Post scaling with a scaling factor $\gamma $ through the scaling operation circuits (SC

_{1}–SC

_{4}), the four input layers yield output signals denoted as ${S}_{Gx}$(n), ${S}_{Gy}$(n), ${S}_{Ex}$(n) and ${S}_{Ey}$(n), respectively. These are respectively modulated with the optical-field phases of CW

_{1}–CW

_{4}. The FPC

_{3}first couples the modulated ${S}_{Gx}$(n) and ${S}_{Gy}$(n) into a single beam, which is then injected into the ground state of the reservoir QD spin-VCSEL. Similarly, the FPC

_{4}couples the modulated ${S}_{Ex}$(n) and ${S}_{Ey}$(n) into a single beam, which is subsequently injected into the excited state of the reservoir QD spin-VCSEL.

_{1}–DL

_{4}) is defined as $\tau $. In the output layers (OLs), the GS-${\mathrm{PC}}_{x}^{\prime}$ and GS-${\mathrm{PC}}_{y}^{\prime}$ emissions from the QD spin-VCSEL are bifurcated using FPBS

_{5}. Similarly, the ES-${\mathrm{PC}}_{x}^{\prime}$ and ES-${\mathrm{PC}}_{y}^{\prime}$ emissions are split through FPBS

_{6}. The intensity values of GS-${\mathrm{PC}}_{x}^{\prime}$, GS-${\mathrm{PC}}_{y}^{\prime}$, ES-${\mathrm{PC}}_{x}^{\prime}$ and ES-${\mathrm{PC}}_{y}^{\prime}$ are sampled at intervals of $\theta $ and are considered as virtual nodes. Accordingly, the total number N of virtual nodes along each delay line is determined by the ratio N = $\tau $/$\theta $. The states of the N virtual nodes along the DL

_{1}–DL

_{4}are weighted and linearly summed up. The combined weighted states from the DL

_{1}and DL

_{2}are represented as ${\mathrm{y}}_{\mathit{Gx}}^{\prime}$(n) and ${\mathrm{y}}_{\mathit{Gy}}^{\prime}$(n), respectively, while those from the DL

_{3}and DL

_{4}are signified as ${\mathrm{y}}_{\mathit{Ex}}^{\prime}$(n) and ${\mathrm{y}}_{\mathit{Ey}}^{\prime}$(n). In this setup, by calibrating the output weights, ${\mathrm{y}}_{\mathit{Gx}}^{\prime}$(n) and ${\mathrm{y}}_{\mathit{Gy}}^{\prime}$(n) can achieve synchronization with ${C}_{Gx}$(n−${L}_{Gx}$) and ${C}_{Gy}$(n−${L}_{Gy}$), respectively. Likewise, ${\mathrm{y}}_{\mathit{Ex}}^{\prime}$(n) and ${\mathrm{y}}_{\mathit{Ey}}^{\prime}$(n) can be attuned to synchronize with ${C}_{Ex}$(n−${L}_{Ex}$) and ${C}_{Ey}$(n−${L}_{Ey}$). Under these synchronization conditions, the concealed messages QPSK and 4QAM are decoded by the synchronous subtraction of ${\mathrm{y}}_{\mathit{Gx}}^{\prime}$(n) from ${C}_{Gx}$(n−${L}_{Gx}$) and ${\mathrm{y}}_{\mathit{Gy}}^{\prime}$(n) from ${C}_{Gy}$(n−${L}_{Gy}$), with the retrieved messages designated as ${S}_{1}^{\prime}$(t) and ${S}_{2}^{\prime}$(t), respectively. In a similar fashion, the messages 8QAM and 16QAM are decoded by the synchronous subtraction of ${\mathrm{y}}_{\mathit{Ex}}^{\prime}$(n) from ${C}_{Ex}$(n−${L}_{Ex}$) and ${\mathrm{y}}_{\mathit{Ex}}^{\prime}$(n) from ${C}_{Ey}$(n−${L}_{Ey}$), with their decoded equivalents presented as ${S}_{3}^{\prime}$(t) and ${S}_{4}^{\prime}$(t), correspondingly.

_{5}–CW

_{8}using intensity modulators (IM

_{1}–IM

_{4}) and then each injected into its corresponding coherent demodulation unit (CDU). Each CDU comprises a polarization-diversity digital coherent receiver (PDDCR), a submatrix finder (SF), a set of five digital signal processors (DSPs), and a bit error rate estimation module (BEREM). The PDDCR, depicted in VPI [23], models an optical coherent quadrature receiver that encompasses a local oscillator, optical hybrids, post-detection electrical filters, and analog-to-digital converters. The SF is used to extract specified elements of the input matrix. The DSPs with subscripts 1, 6, 11 and 16 address the compensation of group velocity dispersion and nonlinear effects within the optical fiber, whereas the DSPs labeled with subscripts 2, 7, 12 and 17 are designated to down-sample the in-phase and quadrature signals to match the baud rate. The DSPs marked with subscripts 3, 8, 13 and 18 are dedicated to estimating and correcting frequency discrepancies between the received optical signal and the local oscillator. The DSPs inscribed with subscripts 4, 9, 14 and 19 adjust and align the clock phase of both transmitter and receiver. The DSPs tagged with the subscripts 5, 10, 15 and 20 are dedicated to estimating and correcting phase discrepancies between the received optical signal and the local oscillator. The BEREMs labeled with subscripts 1-4, as four-dimensional bit error rate modules, are capable of generating $BER$s for the baseband signals and facilitating their demodulation. After processing through the four CDUs, four sets of baseband signals (or bit streams) encapsulated within the decoded modulation messages, ${S}_{1}^{\prime}$(t), ${S}_{2}^{\prime}$(t), ${S}_{3}^{\prime}$(t) and ${S}_{4}^{\prime}$(t) are effectively reconstructed. These recovered bit streams are denoted as ${b}^{\prime 1}$-${b}^{\prime 4}$, respectively.

_{1}(which serves as the driving QD spin-VCSEL) are delineated as follows [19,26]:

_{2}(the reservoir QD spin-VCSEL) under the influence of optical feedback and optical injection are revised as follows [19,26]:

_{1}–DL

_{4}) shown in Figure 2; ${\omega}_{G}$ is the resonant frequency of the light emitted from the ground state; and ${\omega}_{E}$ is the resonant frequency of light emitted from the excited state. $\Delta {\omega}_{G}$ represents the frequency detuning between CW

_{1}(CW

_{2}) and the ground state emission of the reservoir QD Spin-VCSEL; $\Delta {\omega}_{E}$ denotes the frequency detuning between CW

_{3}(CW

_{4}) and the excited state emission of the reservoir QD Spin-VCSEL. ${\beta}_{sp}$ is the rate of spontaneous emission, also viewed as an indicator of noise strength. The terms ${\xi}_{D,GS}^{\pm}$, ${\xi}_{D,ES}^{\pm}$, ${\xi}_{DS}^{\pm}$ and ${\xi}_{ES}^{\pm}$ embody independent Gaussian white noise sources with zero mean and unit variance. ${k}_{f}$ is the feedback coupling strength; ${k}_{inj}$ stands for the strength of optical injection. ${E}_{inj}^{1}$ and ${E}_{inj}^{2}$ are the slowly varying complex amplitudes of the CW

_{1}and CW

_{2}, which are converted to RCP and LCP by the PCL

_{2}and PCL

_{3}; ${E}_{inj}^{3}$ and ${E}_{inj}^{4}$ are the injected optical fields for the CW

_{3}and CW

_{4}, likewise converted by the PCL

_{4}and PCL

_{5}. ${E}_{inj}^{1}$ and ${E}_{inj}^{2}$ account for the light fields ${E}_{GS}^{+}$ and ${E}_{GS}^{-}$, respectively, while ${E}_{inj}^{3}$ and ${E}_{inj}^{4}$ are charged with generating the optical fields ${E}_{ES}^{+}$ and ${E}_{GS}^{-}$. The total pump strengths $\eta $ = ${\eta}^{+}$ + ${\eta}^{-}$, ${\eta}^{+}$ and ${\eta}^{-}$ are the pump intensities for the RCP and LCP components, respectively.

_{1}–CW

_{4}, respectively. The masked input signals ${S}_{Gx}(t)$, ${S}_{Gy}(t)$, ${S}_{Ex}(t)$ and ${S}_{Ey}(t)$ can be expressed as

_{x}or ES-PC

_{x}, and ${\tau}_{y}$ is the channel delay of the GS-PC

_{y}or ES-PC

_{y}.

_{x}and GS-${\mathrm{PC}}_{x}^{\prime}$, GS-PC

_{y}and GS-${\mathrm{PC}}_{y}^{\prime}$, ES-PC

_{x}and ES-${\mathrm{PC}}_{x}^{\prime}$, and ES-PC

_{y}and ES-${\mathrm{PC}}_{y}^{\prime}$) plays a key role in in security and encrypted message recovery. In the following, we use four parallel RCs to address chaos synchronization between each pair of PCs. According to lag chaotic synchronization theory, the lag synchronization solution is obtained as follows.

_{x}, GS-PC

_{y}, ES-PC

_{x}and ES-PC

_{y}such that

## 3. Results and Discussions

_{x}and GS-PC

_{y}, as well as the ES-PC

_{x}and ES-PC

_{y}emitted by the driving QD spin-VCSEL. According to the representations in Figure 3, the PSD distributions for these PCs consistently demonstrate chaotic behavior. The effective 3 dB bandwidths for both the GS-PC

_{x}and GS-PC

_{y}are calculated to be 180 GHz, whereas the ES-PC

_{x}and ES-PC

_{y}are slightly higher at 200 GHz. Correspondingly, the effective 3 dB bandwidths for the GS-${\mathrm{PC}}_{x}^{\prime}$, GS-${\mathrm{PC}}_{y}^{\prime}$, ES-${\mathrm{PC}}_{x}^{\prime}$ and ES-${\mathrm{PC}}_{y}^{\prime}$ of the reservoir system exhibit similar characteristics to those of their driving system counterparts (GS-PC

_{x}, GS-PC

_{y}, ES-PC

_{x}and ES-PC

_{y}, respectively). These outcomes suggest that our system is capable of achieving high-speed, four-channel coherent optical chaotic secure communications.

_{x}, GS-PC

_{y}, ES-PC

_{x}and ES-PC

_{y}, respectively. We collect 5096 samples of these delayed outputs at a sampling interval of 10 ps. After discarding the initial 1000 samples to remove transients, we allocate 2048 samples for training each of the four reservoirs, and an equivalent number of subsequent points for testing the corresponding reservoir. Moreover, the prediction performance is bolstered by implementing chaotic mask signals derived from two coupled semiconductor lasers, detailed in [24]. These mask signals are normalized with standard deviations set to 1 and mean values calibrated to 0. Each reservoir’s virtual node interval, denoted by $\theta $, is fixed at 40 fs. Here, all rates for the QPSK, 4QAM, 8QAM and 16QAM are 100 Gb/s. The input data sampling period T is maintained at 10 ps, resulting in a data processing rate of 100 Gb/s. We establish the number of virtual nodes, N, at 250, where N = $\tau $/$\theta $ and $\tau $ = T. We maintain the scale factor $\gamma $, at a value of 1. To assess the predictions for the GS-PC

_{x}, GS-PC

_{y}, ES-PC

_{x}and ES-PC

_{y}made by these four parallel reservoirs using the reservoir QD spin-VCSEL, we introduce the normalized mean square error ($NMSE$) as a metric to compare the delayed predictive targets against their associated reservoir outputs, which is given as follows:

_{x}, ES-PC

_{x}, GS-PC

_{y}and ES-PC

_{y}, respectively. ${L}_{Gx}$, ${L}_{Gy}$, ${L}_{Ex}$ and ${L}_{Ey}$ are the defined lengths of the testing data set for each variable. L represents the total number of data points in the testing data set. The term “var” denotes the variance of the data. When $NMS{E}_{jx}$ and $NMS{E}_{jy}$ are both 0, it means that the outputs of the reservoirs (GS-PC

_{x}, GS-PC

_{y}, ES-PC

_{x}and ES-PC

_{y}) perfectly match with their corresponding predicted targets (${C}_{Gx}$(n−${L}_{Gx}$), ${C}_{Gy}$(n−${L}_{Gy}$), ${C}_{Ex}$(n−${L}_{Ex}$) and ${C}_{Ey}$(n−${L}_{Ey}$), respectively). On the other hand, if $NMS{E}_{jx}$ and $NMS{E}_{jy}$ both are 1, it means that the reservoir outputs are completely different from the predicted targets. When $NMS{E}_{jx}$ and $NMS{E}_{jy}$ are both less than 0.1, it indicates that each reservoir is able to accurately infer the chaotic dynamics of its corresponding predicted target, which is the PC of the driving QD Spin-VCSEL output.

_{x}, ES-PC

_{x}, GS-PC

_{y}and ES-PC

_{y}in our system, Figure 4 presents their predictive results. In this figure, T = 10 ps, $\theta $ = 40 fs, and N = 250. The samples of the delayed GS-PC

_{x}, GS-PC

_{y}, ES-PC

_{x}and ES-PC

_{y}from the driving QD Spin-VCSEL output are denoted as ${C}_{Gx}$(n−${L}_{Gx}$), ${C}_{Gy}$(n−${L}_{Gy}$), ${C}_{Ex}$(n−${L}_{Ex}$) and ${C}_{Ey}$(n−${L}_{Ey}$), respectively. The samples of the trained GS-${\mathrm{PC}}_{x}^{\prime}$, GS-${\mathrm{PC}}_{y}^{\prime}$, ES-${\mathrm{PC}}_{x}^{\prime}$ and ES-${\mathrm{PC}}_{y}^{\prime}$ from the reservoir QD spin-VCSEL output are denoted as ${y}_{Gx}^{\prime}(n)$, ${y}_{Gy}^{\prime}(n)$, ${y}_{\mathit{Ex}}^{\prime}(n)$ and ${y}_{\mathit{Ey}}^{\prime}(n)$, respectively. As observed from Figure 4, the chaotic trajectories of the ${C}_{Gx}$(n−${L}_{Gx}$), ${C}_{Gy}$(n−${L}_{Gy}$), ${C}_{Ex}$(n−${L}_{Ex}$) and ${C}_{Ey}$(n−${L}_{Ey}$) are almost identical to those of the ${y}_{\mathit{Gx}}^{\prime}(n)$, ${y}_{\mathit{Gy}}^{\prime}(n)$, ${y}_{\mathit{Ex}}^{\prime}(n)$ and ${y}_{\mathit{Ey}}^{\prime}(n)$, respectively. In Figure 5a, when T = 10 ps, $\theta $ = 40 fs, and N = 250, the prediction errors ($NMS{E}_{Gx}$ and $NMS{E}_{Gy}$) of the GS-PC

_{x}and GS-PC

_{y}are 0.0359 and 0.0375, respectively. The $NMS{E}_{Ex}$ and $NMS{E}_{Ey}$ for the ES-PC

_{x}and ES-PC

_{y}are 0.0995 and 0.0865, respectively. These indicate that the four parallel reservoirs based on the reservoir QD spin-VCSEL can accurately predict the chaotic dynamics of the GS-PC

_{x}, GS-PC

_{y}, ES-PC

_{x}and ES-PC

_{y}, respectively.

_{x}, GS-PC

_{y}, ES-PC

_{x}and ES-PC

_{y}, Figure 5a illustrates the relationship between the prediction errors ($NMS{E}_{Gx}$, $NMS{E}_{Gy}$, $NMS{E}_{Ex}$ and $NMS{E}_{Ey}$) and the sampling period T when $\theta $ is 40 fs. As shown in Figure 5a, $NMS{E}_{Gx}$ and $NMS{E}_{Gy}$ exhibit an almost linear decrease from 0.0362 to 0.0350 and from 0.0376 to 0.0366, respectively, as T increases from 2 ps to 128 ps. Similarly, the $NMS{E}_{Ex}$ and $NMS{E}_{Ey}$ also reveal a linear decrease from 0.0998 to 0.0961 and from 0.0867 to 0.0836, respectively. The reason why a longer sampling period T leads to reduced training error might be explained as follows. In this work, $\theta $ = T/N is fixed at 40 fs, and a smaller N is associated with a smaller T, resulting in a lower-dimensional state space. This situation can make the training of the four parallel reservoirs based on the reservoir QD spin-VCSEL become unstable and more difficult, consequently leading to a larger $NMSE$. Additionally, when T is fixed at a certain value, the $NMS{E}_{Ex}$ and $NMS{E}_{Ey}$ are significantly larger than $NMS{E}_{Gx}$ and $NMS{E}_{Gy}$. This may be explained by the fact that ES-PC

_{x}and ES-PC

_{y}have more complex chaotic dynamics than GS-PC

_{x}and GS-PC

_{y}, respectively, making the predictions of ES-PC

_{x}and ES-PC

_{y}more challenging compared to those of GS-PC

_{x}and GS-PC

_{y}. Figure 5b shows the relationship between the prediction errors ($NMS{E}_{Gx}$, $NMS{E}_{Gy}$, $NMS{E}_{Ex}$ and $NMS{E}_{Ey}$) and the virtual node interval $\theta $ when T is fixed at 10 ps. From the observations in Figure 5, it can be seen that as $\theta $ increases from 1 fs to 320 fs, the $NMS{E}_{Ex}$ and $NMS{E}_{Ey}$ slowly increase from 0.0979 to 0.0998 and from 0.0853 to 0.0868, respectively. Then, they gradually stabilize at 0.0998 and 0.0865. On the other hand, the $NMS{E}_{Gx}$ and $NMS{E}_{Gy}$ remain nearly constant at 0.0363 and 0.0376, respectively. The results indicate that when T = 10 ps, the choice of the virtual node interval $\theta $ has a slight impact on the prediction accuracy for the GS-PC

_{x}and GS-PC

_{y}. However, for the ES-PC

_{x}and ES-PC

_{y}, the prediction errors slightly increase with an increase in $\theta $, suggesting a potential sensitivity to the chosen $\theta $.

_{x}, GS-PC

_{y}, ES-PC

_{x}and ES-PC

_{y}emitted by the driving QD spin-VCSEL. This indicates that the delayed GS-PC

_{x}, GS-PC

_{y}, ES-PC

_{x}and ES-PC

_{y}can successfully synchronize with the GS-${\mathrm{PC}}_{x}^{\prime}$, GS-${\mathrm{PC}}_{y}^{\prime}$, ES-${\mathrm{PC}}_{x}^{\prime}$ and ES-${\mathrm{PC}}_{y}^{\prime}$ outputs by the reservoir QD spin-VCSEL, respectively. To further analyze the qualities of their chaos synchronizations, the correlation coefficients are introduced and defined as follows.

_{x}, GS-PC

_{y}, ES-PC

_{x}and ES-PC

_{y}, respectively. Notably, ${\rho}_{Gx}$ and ${\rho}_{Gy}$ are higher than ${\rho}_{Ex}$ and ${\rho}_{Ey}$, respectively. This is attributed to the fact that the $NMS{E}_{Gx}$ and $NMS{E}_{Gy}$ for the GS-PC

_{x}and GS-PC

_{y}are lower compared to the $NMS{E}_{Gx}$ and $NMS{E}_{Gy}$ for the ES-PC

_{x}and ES-PC

_{y}, respectively.

_{x}and GS-${\mathrm{PC}}_{x}^{\prime}$, GS-PC

_{y}and GS-${\mathrm{PC}}_{x}^{\prime}$, ES-PC

_{x}and ES-${\mathrm{PC}}_{x}^{\prime}$, and ES-PC

_{y}and ES-${\mathrm{PC}}_{y}^{\prime}$) using the reservoir QD spin-VCSEL, one of the messages QPSK, 4QAM, 8QAM and 16QAM can be decoded by synchronously dividing a reservoir-generated chaos and a delayed chaos masked message. The temporal traces of the delayed encoding message (${S}_{1}$(n−${L}_{Gx}$), or QPSK), the delayed chaos masked message (${U}_{Gx}$(n−${L}_{Gx}$)), and the decoding message (${S}_{1}^{\prime}$(n)) are displayed in Figure 7(a

_{1}–a

_{3}). As observed from Figure 7(a

_{1}–a

_{3}), the temporal trajectory of ${S}_{1}$(n−${L}_{Gx}$) is very similar to that of (${S}_{1}^{\prime}$(n)). Furthermore, ${U}_{Gx}$(n−${L}_{Gx}$) exhibits a chaotic state. Figure 7(a

_{4}–a

_{6}) present the temporal trajectories of ${S}_{2}$(n−${L}_{Gy}$) (4QAM), ${U}_{Gy}$(n−${L}_{Gy}$) and (${S}_{2}^{\prime}$(n)). As seen from these figures, the temporal trajectory of ${S}_{2}$(n−${L}_{Gy}$) is basically identical to that of (${S}_{2}^{\prime}$(n)), while ${U}_{Gy}$(n−${L}_{Gy}$) shows a chaotic state. Moreover, as displayed in Figure 7(a

_{7}–a

_{12}), the temporal trajectories of ${S}_{3}$(n−${L}_{Ex}$) (8QAM) and ${S}_{4}$(n−${L}_{Ey}$) (16QAM) are almost the same as those of (${S}_{3}^{\prime}$(n)) and (${S}_{4}^{\prime}$(n)), respectively. ${U}_{Ex}$(n−${L}_{Ex}$) and ${U}_{Ey}$(n−${L}_{Ey}$) both exhibit a chaotic state. Moreover, as in Figure 8, we present the eye-diagrams for these four decoded messages (${S}_{1}^{\prime}$(n), ${S}_{2}^{\prime}$(n), ${S}_{3}^{\prime}$(n) and ${S}_{4}^{\prime}$(n)). One sees from this figure that the “eyes” sizes of the eye-diagrams of these decoded messages are enough large, indicating that the decoded messages of the system have a relatively large tolerance error for noise and jitter and have good quality. However, the superposition of multiple decoded messages causes the signal line of each eye-diagram to become thicker and appear fuzzy. The reason is that very small synchronization errors may be converted into noise and superimposed on the signal line of the eye-diagram. These results indicate that the encoding messages QPSK, 4QAM, 8QAM and 16QAM can be effectively masked in a chaotic carrier and successfully recovered using reservoir computing.

_{1},a

_{2}), the $BER$s for ${S}_{1}^{\prime}$(t), ${S}_{2}^{\prime}$(t), ${S}_{3}^{\prime}$(t) and ${S}_{4}^{\prime}$(t) exhibit oscillatory behavior as ${k}_{inj}$ is adjusted within the range of 0.1 ${\mathrm{ns}}^{-1}$ to 50 ${\mathrm{ns}}^{-1}$. Their $BER$ values, respectively, fluctuate within the following ranges: from 1.02 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ to 1.22 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ for ${S}_{1}^{\prime}$(t), from 6.1 $\times {10}^{-3}$ to 7.5 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ for ${S}_{2}^{\prime}$(t), from 3.4 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ to 6.1 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ for ${S}_{3}^{\prime}$(t), and from 7.1 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ to 9.2 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ for ${S}_{4}^{\prime}$(t). Within this ${k}_{inj}$ range, all four decoded messages demonstrate minor oscillatory fluctuations in their $BER$s. The $BER$s cap at 1.5 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ for ${S}_{1}^{\prime}$(t) and at 8.7 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ for ${S}_{2}^{\prime}$(t), while those for ${S}_{3}^{\prime}$(t) and ${S}_{4}^{\prime}$(t) do not surpass 3.4 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ and 9.5 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$, respectively. Based on findings from earlier studies [28,29,30], a $BER$ that closes at or below 0.01 is indicative of potentially high-quality data transmission within an optical chaos communication framework. As depicted in Figure 2c, when demodulated through correlation and refined by various digital signal processing methods, four distinct baseband signal sets (or bitstreams) encapsulated within the decoded messages ${S}_{1}^{\prime}$(t), ${S}_{2}^{\prime}$(t), ${S}_{3}^{\prime}$(t) and ${S}_{4}^{\prime}$(t) are successfully reconstructed. Consequently, the $BER$ ranges for these retrieved baseband signals (${b}_{1}^{\prime}$, ${b}_{2}^{\prime}$, ${b}_{3}^{\prime}$ and ${b}_{4}^{\prime}$) remain constant and effectively zero, irrespective of ${k}_{inj}$ and ${k}_{f}$ variations. Figure 10 delves into the performance of the four retrieved baseband signals by presenting their temporal trajectories and eye-diagrams alongside those of the original baseband signals ${b}_{1}$–${b}_{4}$. An inspection of Figure 10 reveals a striking similarity between the temporal profiles of the original signals ${b}_{1}$, ${b}_{2}$, ${b}_{3}$ and ${b}_{4}$ and their retrieved counterparts ${b}_{1}^{\prime}$, ${b}_{2}^{\prime}$, ${b}_{3}^{\prime}$ and ${b}_{4}^{\prime}$, respectively. The eye-diagrams corresponding to the original and retrieved baseband signals also correspond closely, with ${b}_{1}$, ${b}_{2}$, ${b}_{3}$ and ${b}_{4}$, showing a remarkable resemblance to ${b}_{1}^{\prime}$, ${b}_{2}^{\prime}$, ${b}_{3}^{\prime}$ and ${b}_{4}^{\prime}$. Notably, the eye openings in the eye-diagrams for ${b}_{1}^{\prime}$, ${b}_{2}^{\prime}$, ${b}_{3}^{\prime}$ and ${b}_{4}^{\prime}$ are sufficiently large, which is an important indicator of signal integrity. The insights gathered from Figure 9 and Figure 10 strongly support the effectiveness of our proposed coherent optical chaotic communication system in delivering secure and high-quality data transmission.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Principle block diagram of four-channel coherent optical chaotic secure communication based on four parallel photonic reservoir computers. Here, TM: transmitter module; RCM: reservoir computing module; CDM: coherent demodulation module; CDU: coherent demodulation unit; ${b}^{1}$–${b}^{4}$: baseband signals (bit sequence signals); ${b}^{\prime 1}$–${b}^{\prime 4}$: demodulation baseband signals; GS-PC

_{x}and GS-PC

_{y}: X-PC and Y-PC from the ground state emission of the QD-spin-VCSEL, respectively; WDM Mux: wavelength division multiplexer; WDM DeMux: wavelength division demultiplexer; and ES-PC

_{x}and ES-PC

_{y}: X-PC and Y-PC from the excited state emission of the QD-spin-VCSEL, respectively.

**Figure 2.**Simulation experiment setup of a four-channel COCSC system, founded on four parallel reservoirs. Here, (

**a**) Transmitter; (

**b**) Chaos-synchronization prediction and demodulation using reservoirs; (

**c**) Coherent demodulation and DSP processing; PL: pumped light; PCL: polarization controller; IS: isolator; FPBS: fiber-optic polarization splitter; QPSKT: QPSK transmitter; 4QAMT: 4QAM transmitter; 8QAMT: 8QAM transmitter; 16QAMT: 16QAM transmitter; NS: empty source; BP: bidirectional ports; PC: power combiner; FPC: fiber polarization coupler; WDM Mux: wavelength division multiplexer; WDM DeMux: wavelength division demultiplexer; CW: continuous wave laser; NDF: the neutral density filter; PD: photodetector; AM: amplitude modulator; DL: delay line; FC: fiber coupler; OL: output layer; CSM: co-simulation module; EA: Electrical amplifier; DM: discrete module; SC: proportional operation circuit; Mask: masked signal; PDDCR: polarization-diversity digital coherent receiver; DSP: digital signal processor; SF: submatrix finder; BEREM: bit error rate estimation module; and NA: numerical analyzer.

**Figure 3.**Power spectral density (PSD) distributions of the four polarization components GS-PC

_{x}, GS-PC

_{y}, ES-PC

_{x}and ES-PC

_{y}from the driving QD Spin-VCSEL output. Here, (

**a**) the PSD of GS-PC

_{x}(${\mathrm{PSD}}_{Gx}$); (

**b**) the PSD of the GS-PC

_{y}(${\mathrm{PSD}}_{Gy}$); (

**c**) the PSD of ES-PC

_{x}(${\mathrm{PSD}}_{Ex}$); and (

**d**) the PSD of the ES-PC

_{y}(${\mathrm{PSD}}_{Ey}$).

**Figure 4.**Samples of four delayed polarization components emitted by the driving QD spin-VCSEL (blue solid line) and the outputs of four parallel reservoir based on the reservoir QD Spin-VCSEL (red dashed line). Here, (

**a**) ${C}_{Gx}$(n−${L}_{Gx}$) and ${y}_{Gx}^{\prime}(n)$; (

**b**) ${C}_{Gy}$(n−${L}_{Gy}$) and ${y}_{Gy}^{\prime}(n)$; (

**c**) ${C}_{Ex}$(n−${L}_{Ex}$) and ${y}_{Ex}^{\prime}(n)$; and (

**d**) ${C}_{Ey}$(n−${L}_{Ey}$) and ${y}_{Ey}^{\prime}(n)$.

**Figure 5.**Dependence of the prediction errors ($NMS{E}_{Gx}$, $NMS{E}_{Gy}$, $NMS{E}_{Ex}$, and $NMS{E}_{Ey}$) on the sampling period T and the virtual node interval $\theta $. Here, (

**a**) $NMS{E}_{Gx}$, $NMS{E}_{Gy}$, $NMS{E}_{Ex}$, and $NMS{E}_{Ey}$ via T, when $\theta $ = 40 fs. (

**b**) $NMS{E}_{Gx}$, $NMS{E}_{Gy}$, $NMS{E}_{Ex}$, and $NMS{E}_{Ey}$ via $\theta $, while T = 10 ps.

**Figure 6.**Dependences of the correlation coefficients (${\rho}_{Gx}$, ${\rho}_{Gy}$, ${\rho}_{Ex}$, ${\rho}_{Ey}$) on the parameters ${k}_{inj}$ and ${k}_{f}$ when T = 10 ps and $\theta $ = 40 fs. Here, (

**a**) ${\rho}_{Gx}$, ${\rho}_{Gy}$, ${\rho}_{Ex}$, ${\rho}_{Ey}$∝${k}_{inj}$; (

**b**) ${\rho}_{Gx}$, ${\rho}_{Gy}$, ${\rho}_{Ex}$, ${\rho}_{Ey}$∝${k}_{f}$.

**Figure 7.**Temporal trajectories of the delayed encoding messages, the delayed chaos masked messages, and the decoding messages in the reservoir computing system. Here, (

**a**) the delayed encoding message ${S}_{1}$(n−${L}_{Gx}$) via time step n; (

_{1}**a**) the delayed chaos masked message ${U}_{Gx}$(n−${L}_{Gx}$) via time step n; (

_{2}**a**) the decoding message ${S}_{1}^{\prime}$(n) via time step n; (

_{3}**a**) ${S}_{2}$(n−${L}_{Gy}$) via time step n; (

_{4}**a**) ${U}_{Gy}$(n−${L}_{Gy}$) via time step n; (

_{5}**a**) ${S}_{2}^{\prime}$(n) via time step n; (

_{6}**a**) ${S}_{3}$(n−${L}_{Ex}$) via time step n; (

_{7}**a**) ${U}_{Ex}$(n−${L}_{Ex}$) via time step n; (

_{8}**a**) ${S}_{3}^{\prime}$(n) via time step n; (

_{9}**a**) ${S}_{4}$(n−${L}_{Ey}$) via time step n; (

_{10}**a**) ${U}_{Ey}$(n−${L}_{Ey}$) via time step n; and (

_{11}**a**) ${S}_{4}^{\prime}$(n) via time step n.

_{12}**Figure 8.**Eye-diagrams of the decoded messages (${S}_{1}^{\prime}$(n), ${S}_{2}^{\prime}$(n), ${S}_{3}^{\prime}$(n) and ${S}_{4}^{\prime}$(n)). Here, (

**a**) the eye-diagram of ${S}_{1}^{\prime}$(n); (

**b**) the eye-diagram of ${S}_{2}^{\prime}$(n); (

**c**) the eye-diagram of ${S}_{3}^{\prime}$(n); and (

**d**) the eye-diagram of ${S}_{4}^{\prime}$(n).

**Figure 9.**The dependences of the $BER$s for the decoding messages (${S}_{1}^{\prime}$(t), ${S}_{2}^{\prime}$(t), ${S}_{3}^{\prime}$(t) and ${S}_{4}^{\prime}$(t)) and their corresponding baseband signals (${b}_{1}^{\prime}$, ${b}_{2}^{\prime}$, ${b}_{3}^{\prime}$ and ${b}_{4}^{\prime}$) on two key parameters ${k}_{inj}$ and ${k}_{f}$. Here, (

**a**) ${S}_{1}^{\prime}$(t), ${S}_{2}^{\prime}$(t), ${b}_{1}^{\prime}$, ${b}_{2}^{\prime}$ via ${k}_{inj}$; (

_{1}**a**) ${S}_{3}^{\prime}$(t), ${S}_{4}^{\prime}$(t), ${b}_{3}^{\prime}$, ${b}_{4}^{\prime}$ via ${k}_{inj}$; (

_{2}**a**) ${S}_{1}^{\prime}$(t), ${S}_{2}^{\prime}$(t), ${b}_{1}^{\prime}$, ${b}_{2}^{\prime}$ via ${k}_{f}$; and (

_{3}**a**) ${S}_{3}^{\prime}$(t), ${S}_{4}^{\prime}$(t), ${b}_{3}^{\prime}$, ${b}_{4}^{\prime}$ via ${k}_{f}$.

_{4}**Figure 10.**Temporal trajectories and eye-diagrams of the original baseband signals ${b}_{1}$–${b}_{4}$ and their respectively retrieved baseband signals ${b}_{1}^{\prime}$–${b}_{4}^{\prime}$. Here, (

**a**,

_{1}**a**) the temporal trajectories of the ${b}_{1}$ and ${b}_{1}^{\prime}$, respectively, and (

_{2}**b**,

_{1}**b**) their respectively eye-diagrams. (

_{2}**a**,

_{3}**a**) the temporal trajectories of the ${b}_{2}$ and ${b}_{2}^{\prime}$, respectively, and (

_{4}**b**,

_{3}**b**) their corresponding eye-diagrams; (

_{4}**a**,

_{5}**a**) the temporal trajectories of the ${b}_{3}$ and ${b}_{3}^{\prime}$, respectively, and (

_{6}**b**,

_{5}**b**) their respectively eye-diagrams; and (

_{6}**a**,

_{7}**a**) the temporal trajectories of the ${b}_{4}$ and ${b}_{4}^{\prime}$, respectively, and (

_{8}**b**,

_{7}**b**) their corresponding eye-diagrams.

_{8}The Parameter and Symbol | Value | The Parameter and Symbol | Value |
---|---|---|---|

The photon decay rate ${\kappa}_{D}$ | 250 ${\mathrm{ns}}^{-1}$ | The capture rate ${\gamma}_{D,0}$ | 400 ${\mathrm{ns}}^{-1}$ |

Linewidth enhancement factor ${\alpha}_{D}$ | 3 | Intradot relaxation rate ${\gamma}_{D,21}$ | 50 ${\mathrm{ns}}^{-1}$ |

Total pump intensity $\eta $ | 4 | Spin relaxation rate ${\gamma}_{D,s}$ | 10 ${\mathrm{ns}}^{-1}$ |

Dichroism ${\gamma}_{D,a}$ | 0 ${\mathrm{ns}}^{-1}$ | Carrier recombination rate ${\gamma}_{D,n}$ | 1 ${\mathrm{ns}}^{-1}$ |

Birefringence ${\gamma}_{D,p}$ | 30 ${\mathrm{ns}}^{-1}$ | Electron charge e | 1.6 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-19}$ C |

Quantum dot density ${N}_{QD}$ | 1.5 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{17}$ ${\mathrm{m}}^{-2}$ | The gain coefficient ${h}_{D,1}$ | 1.1995 |

The Parameter and Symbol | Value | The Parameter and Symbol | Value |
---|---|---|---|

The field decay rate $\kappa $ | 300 ${\mathrm{ns}}^{-1}$ | Central frequency detuning $\Delta {\omega}_{E}$ | −20 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{9}\phantom{\rule{3.33333pt}{0ex}}\mathrm{rad}/\mathrm{s}$ |

Line-width enhancement factor $\alpha $ | 3 | The capture rate ${\gamma}_{0}$ | 600 ${\mathrm{ns}}^{-1}$ |

Total pump intensity $\eta $ | 4 | Intradot relaxation rate ${\gamma}_{21}$ | 40 ${\mathrm{ns}}^{-1}$ |

Dichroism ${\gamma}_{a}$ | 0.1 ${\mathrm{ns}}^{-1}$ | Spin relaxation rate ${\gamma}_{s}$ | 20 ${\mathrm{ns}}^{-1}$ |

Birefringence ${\gamma}_{p}$ | 20 ${\mathrm{ns}}^{-1}$ | Carrier recombination rate ${\gamma}_{n}$ | 1 ${\mathrm{ns}}^{-1}$ |

Center frequency ${\omega}_{G}$ | 2 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{14}\phantom{\rule{3.33333pt}{0ex}}\mathrm{rad}/\mathrm{s}$ | Injection strength ${k}_{inj}$ | 35 ${\mathrm{ns}}^{-1}$ |

Center frequency ${\omega}_{E}$ | ${10}^{14}\phantom{\rule{3.33333pt}{0ex}}\mathrm{rad}/\mathrm{s}$ | Feedback strength ${k}_{f}$ | 30 ${\mathrm{ns}}^{-1}$ |

Central frequency detuning $\Delta {\omega}_{G}$ | 0 $\mathrm{rad}/\mathrm{s}$ | The gain coefficient ${h}_{1}$ | 1.1665 |

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## Share and Cite

**MDPI and ACS Style**

Zhong, D.; Wang, T.; Chen, Y.; Wu, Q.; Qiu, C.; Zeng, H.; Wang, Y.; Xi, J.
Exploration of Four-Channel Coherent Optical Chaotic Secure Communication with the Rate of 400 Gb/s Using Photonic Reservoir Computing Based on Quantum Dot Spin-VCSELs. *Photonics* **2024**, *11*, 309.
https://doi.org/10.3390/photonics11040309

**AMA Style**

Zhong D, Wang T, Chen Y, Wu Q, Qiu C, Zeng H, Wang Y, Xi J.
Exploration of Four-Channel Coherent Optical Chaotic Secure Communication with the Rate of 400 Gb/s Using Photonic Reservoir Computing Based on Quantum Dot Spin-VCSELs. *Photonics*. 2024; 11(4):309.
https://doi.org/10.3390/photonics11040309

**Chicago/Turabian Style**

Zhong, Dongzhou, Tiankai Wang, Yujun Chen, Qingfan Wu, Chenghao Qiu, Hongen Zeng, Youmeng Wang, and Jiangtao Xi.
2024. "Exploration of Four-Channel Coherent Optical Chaotic Secure Communication with the Rate of 400 Gb/s Using Photonic Reservoir Computing Based on Quantum Dot Spin-VCSELs" *Photonics* 11, no. 4: 309.
https://doi.org/10.3390/photonics11040309