1. Introduction

As we all know, so far, the methods for optical communication multiplexing mainly include range from wavelength division multiplexing (WDM), optical time division multiplexing (OTDM) to polarization division multiplexing (PDM). In these methods, the rate of single-channel communication provided by OTDM [1–6] technology is far higher than that of any single-channel communication system based on electronic devices in the same period. Since Tucker et al. [5] took the lead in realizing 4-channel OTDM with 4Gbit/s in 1987, OTDM technology has attracted great attention [7–9]. By using different multiplexing and demultiplexing technology, the researchers have greatly improved the rate of OTDM. For example, in 2005, the single-channel rate of OTDM had reached 160 Gbit/s [1,2,10–13]. In 2014, Hu et al. achieved a rate of 1.2 Tbit/s for OTDM by the combination of time division multiplexing and polarization multiplexing [14]. In 2018, Hirooka et al. realized an OTDM system with a single channel rate of 10.2Tbit/s by using optical Nyquist pulse [15]. In 2020, Watanabe et al. performed a 10-channel optical multiplexing communication system with 12.8Tbit/s by the combination of OTDM and WDM [16]. OTDM provides a rate unmatched by the other multiplexing methods. In the OTDM system, the most critical technology is time division demultiplexing technology implemented by clock extraction. Previous works have reported many traditional clock extraction schemes. For example, high Q value and high gain loop based on photoelectric phase-locked loop [17] or electro-absorption modulator (EAM) [18], and frame clock extraction based on unequal amplitude encoding [19]. Many devices have been proposed to realize optical time division demultiplexing [20,21], such as EAM-based demultiplexer, nonlinear optical fiber demultiplexer, semiconductor optical amplifier demultiplexer, four-wave mixing demultiplexer using optical fiber or semiconductor optical amplifier, and all-optical demultiplexer utilizing cross-phase modulation effect, and so on. However, these reported demultiplexing technology in OTDM system were implemented by clock extraction, where complex optical hardware system with high cost was needed to be used for optical clock extraction and optical sampling.

In addition, optically pumped spin-VCSEL (OP-Spin-VCSEL) has many advantages over traditional edge-emitting lasers. For example, smaller threshold current, larger modulation bandwidth (about 200GHz) [22,23], independently controlled output polarization and intensity, and ultra-fast dynamic behavior. Utilizing these characteristics, OP-spin-VCSEL can be widely used in optical communication, optical information processing, data storage, quantum computing and biosensing. OP-Spin-VCSEL can generate different forms of ultrafast instabilities, such as periodic oscillation, polarization conversion and chaotic dynamics. The potential applications of chaotic dynamics of OP-spin-VCSEL in ultrafast chaotic computation, ultrafast random code generation and ultrafast chaotic secure communication have attracted great attention. In recent decades, optical chaotic secure communications have been widely studied [24]. In order to promote the development of optical chaotic secure communication toward high capacity and high speed, future works will focus on multi-channel optical chaotic secure communications, including WDM and OTDM chaotic secure communications. It is expected that the OTDM chaotic secure communication with high-speed can be realized by using OP-spin-VCSELs, due to their femtosecond dynamic characteristics, ultra-large bandwidth and independent control of output polarization. However, in such a system, how to realize high-quality chaotic synchronization and time division demultiplexing will be faced challenge. Traditional chaotic synchronizations (such as leading synchronization and lagging synchronization, and so on) is limited by the symmetry between driving laser and response one, and the perfect match of their parameters. In addition, most of the traditional time division demultiplexing schemes using clock extraction possessed complex structures and high costs. The recently developed photon reservoir computing (RC) systems [25] have showed good performances in chaotic synchronization prediction and chaotic signal separation. These RC systems are expected to solve the faced challenges in high-speed OTDM secure communication.

In recent years, photonic artificial intelligence has greatly developed [26–28]. Significantly, deep learning and delay-based reservoir computing (RC) technology have been applied in synchronous prediction of optical chaos [25], nonlinear equalization and optical communication performance monitoring [29]. Deep learning technology applied for the prediction of optical chaos synchronization needs accurate information about structure and parameters of the optical system. In addition, when dealing with complex tasks such as face or speech recognition, the deep-learning with too many neurons or layers are usually blamed for time-consuming. In comparison, delay-based RC typically uses a single delay loop configuration and time-multiplexing of the input data to emulate a ring topology. The introduction of this concept led to a better understanding of RC, its minimal requirements, and suitable parameter conditions. Moreover, it facilitated their implementation on various nonlinear systems [30–34], such as photoelectric feedback system [35], active nonlinear device [36], optical feedback semiconductor laser [37] and so on. In fact, the delay-based RC concept inspired successful implementations in terms of hardware efficiency [30], processing speed and task performance [31,38]. Most notably, the delay-based RC using nonlinear semiconductor lasers has the advantages of fast-speed, high efficiency and parallel computing capability for processing time-dependent signals. A growing number of studies have demonstrated that this effective technique can well forecast trajectories of optical chaotic systems [25,39], distinct chaotic signals [40] and synchronize with chaotic system [39]. Recently, our work further shows that high-quality chaotic synchronization between the driving laser and the response laser can be achieved by using the delay-based photon RC method [41].

Recently, a few works proposed a delay-based RC system based on electronically pumped spin-VCSELs [25]. In such a RC system, the nonlinear dynamical x-PC and y-PC from the VCSEL output can be utilized to perform two parallel reservoir computers, which can well predict two independent chaotic time-series in parallel. Our recent work further demonstrates the effective separation of linearly superimposed optical chaos by using a delay-based RC based on OP-Spin-VCSEL [42]. In addition, OP-Spin-VCSEL offers the flexible spin control of the lasing output, as well as offering more control parameters. Based on this, QP-Spin-VCSEL has better controllability for polarization switch [22,23], which is conducive to the realization of two parallel RCs. Moreover, it can produce ultra-fast chaotic response with short delay feedback or without feedback, thus forming very short spacing between two virtual nodes under sufficient nodes. These show that two RC using two chaotic polarization components emitted by an optically pumped VCSEL can deal with high-speed chaotic time series in parallel. Therefore, the delay-based RC system based on QP-Spin-VCSEL can be potentially applied in the demultiplexing in a dual-channel OTDM system.

Based on the above considerations, in this paper, using two cascaded reservoir computing systems based on the multi beams of chaotic polarization components emitted by four optically pumped VCSELs, we propose a novel OTDM chaotic secure communication system. In such a system, there are four parallel reservoirs in each reservoir layer. We further explore the training errors for these four reservoirs in each reservoir layer in some key parameter spaces, and discuss the evolutions of chaotic synchronization between any one reservoir in last-level reservoir layer and the corresponding prediction target in different parameter spaces. Finally, we discuss the performances of the OTDM chaotic secure communication system, such as bit error rates and eye diagrams.

2. Theory and model

Figure 1 depicts a scheme diagram for dual-channel OTDM chaotic secure communication system with high-speed using two cascaded RCs based optically pumped VCSELs. Here, QP-Spin-VCSEL operating along, as a light source, is used to generate ultrafast chaotic carrier. Response Spin-VCSELs (R-Spin-VCSELs) (with subscript 1-8) represent reservoir lasers. The reservoir computing formed by the R-Spin-VCSEL with the subscript of $j$ is defined as RC$_j$, where $j$=1, 2, 3, $\ldots$, 8. The chaotic X polarization component X-PC and Y-PC emitted by the QP-Spin-VCSEL with the subscript of $j$, as two nonlinear nodes, are used to form two parallel sub-reservoirs, respectively. The four sub-reservoirs formed by four beams of the chaotic X-PCs from the R-Spin-VCSELs with the subscripts of 1-4 are respectively utilized for separating four chaotic masked signals ([$C_{x}$($t$-$dt_{1}$)+$m_{x1}$($t$-$dt_{1}$)]-[$C_{x}$($t$-$dt_{4}$)+$m_{x4}$($t$-$dt_{4}$)]). These separated signals are defined as $C^{'}_{x}$($t$-$dt_{1}$), $C^{'}_{x}$($t$-$dt_{2}$), $C^{'}_{x}$($t$-$dt_{3}$) and $C^{'}_{x}$($t$-$dt_{4}$), respectively. Here, $C_{x}$ is the light intensity of the X-PC. $dt_{1}$-$dt_{4}$ indicate four different delay times. The four sub-reservoirs using four beams of the chaotic Y-PCs from the R-Spin-VCSELs with subscript 1-4 are respectively used for separating four chaotic masked signals ([$C_{y}$($t$-$dt_{1}$)+$m_{y1}$($t$-$dt_{1}$)]-[$C_{y}$($t$-$dt_{4}$)+$m_{y4}$($t$-$dt_{4}$)]). These separated signals are defined as $C^{'}_{y}$($t$-$dt_{1}$), $C^{'}_{y}$($t$-$dt_{2}$), $C^{'}_{y}$($t$-$dt_{3}$) and $C^{'}_{y}$($t$-$dt_{4}$), respectively. Here, $C_{y}$ is the light intensity of the Y-PC. The four sub-reservoirs formed by four beams of the chaotic X-PCs from the R-Spin-VCSELs with the subscripts of 5-8 are used for the predictions of the original chaotic X-PCs ($C_{x}$($t$-$dt_{1}$)-$C_{x}$($t$-$dt_{4}$)), respectively. Their outputs are respectively defined as $C^{''}_{x1}$($t$)-$C^{''}_{x4}$($t$). The four sub-reservoirs based on the four beams of the chaotic Y-PCs from the R-Spin-VCSELs with subscripts of 5-8 are utilized for the predictions of the original chaotic Y-PCs ($C_{y}$($t$-$dt_{1}$)-$C_{y}$($t$-$dt_{4}$)), respectively. Their outputs are respectively defined as $C^{''}_{y1}$($t$)-$C^{''}_{y4}$($t$). The delay multiplexing module for the X-PC (X-DMM) are used for the delay of modulation signals $m_{x1}$-$m_{x4}$. The delay multiplexing module for the Y-PC (Y-DMM) are utilized for the delay of the multiplex modulation signals $m_{y1}$-$m_{y4}$. Each delay multiplexing module consists of four modulators (MD), four delay lines (DL) and optical time multiplexer (OTM). The fiber polarization couplers (FPCs) (the subscript 1-6) are used to couple X-PC and Y-PC into a beam of light. The fiber polarization splitters (FPBSs) (the subscripts of 1-10) are applied to separate a beam of light into the X-PC and Y-PC. The continuous-wave (CW) lasers (the subscripts of 1-4) are utilized to modulate the masked input signal. The optical isolators (ISs) (the subscripts of 1-24) are used to avoid optical feedback. The polarization controllers (PCs) with subscript 1-34 are used to control the polarization of light. The photodetectors (PDs) with the subscripts of 1-26 are utilized to convert optical signals into electrical signals. The phase modulators (PMs) (the subscripts of 1-10) are applied for the phase modulation of the mask input signals. The neutral density filters (NDFs) (the subscripts of 1-5) are used to control the light intensity. The optical circulator is named as OC for short, which subscripts are 1-8. The variable attenuators (VOAs) (the subscripts of 1-16) are used to control the intensity of feedback light. The delay line is named as DL for short. The EA is the electric amplifier. The DM is a discrete module. The SC is scaling operation circuits. The OSC is oscilloscope. Any one the feedback loops (Loop$_1$-Loop$_{16}$) consists of the PC, DL and VOA. The $Mask$ is a mask signal.

 

Fig. 1. Schematic diagram of the dual-channel OTDM chaotic secure communication with high-speed using two cascaded RCs based on optically pumped VCSELs (The detailed descriptions see the above texts).

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The system shown in Fig. 1 is mainly composed of the time division multiplexing modules, the input layers, cascaded reservoir layers and output layers. In the X-DMM, the chaotic carrier-wave emitted by the spin-VCSEL operating alone is separated into X-PC and Y-PC by the FPBS$_1$. Their light intensities are defined as $C_{x}$($t$) and $C_{y}$($t$) respectively. The encoded signals $m_{x1}$($t$)- $m_{x4}$($t$) are modulated to $C_{x}$($t$) by using the modulators (MDs) to $C_{x}$($t$), then respectively are delayed by delay lines (DLs). Here, their delay times are $dt_{1}$-$dt_{4}$, respectively. These delayed signals are multiplexed in time by the OTM. The multiplexed signal $U_{x}$($t$) is expressed as follows: $U_{x}(t)$=$\sum _{i=1}^{4}[c_{x}(t-dt_{i})+m_{xi}(t-dt_{i})]$ ($i$=1,2,3,4, the same below). In the Y-DMM, the encoded signals $m_{y1}$($t$)-$m_{y4}$($t$) are delayed and multiplexed in the same way. The multiplexed signal $U_{y}$($t$) for them are described as follows: $U_{y}(t)$=$\sum _{i=1}^{4}[c_{y}(t-dt_{i})+m_{yi}(t-dt_{i})]$.

The input layers provide the connections with reservoirs. In the input layers 1 and 2, the multiplexed signals $U_{x}$($t$) and $U_{y}$($t$) are first converted into current signals by the PDs, then amplified by the EAs, and finally sampled into two groups of input data $U_{x}$($n$) and $U_{y}$($n$) by the discrete modules (DMs). The sampled data $U_{x}$($n$) and $U_{y}$($n$) with period $T$ are multiplied by the Mask signals with period $T$. The Mask signals both are the chaotic signal generated by the mutually-coupled semiconductor lasers [43,44]. After scaled with a scaling factor $\gamma$ by the scaling operation circuits (SCs), the input layers 1 and 2 yield output signals, denoted as $S_{1x}$($n$) and $S_{1y}$($n$), respectively. They are modulated to the optical-field phases of the CW$_1$- CW$_2$ by using phase modulators PM$_1$ and PM$_2$, respectively. The modulated $S_{1x}$($n$) and $S_{1y}$($n$) are first coupled into one beam by the FPC$_2$, and then divided into four beams of lights by the FBS$_2$. These four beams of lights are respectively injected into the R-Spin-VCSELs with the subscripts of 1-4.

For simplicity, within any one of the RC$_1$-RC$_4$, two polarization components emitted by the R-Spin-VCSEL with double feedbacks are defined as the X-PC${\rm _R}$ and Y-PC${\rm _R}$. These two PCs will show chaotic state under a certain condition, and be used as two nonlinear nodes to form two sub-reservoirs. The X-PC${\rm _R}$ and Y-PC${\rm _R}$ in each RC are fed back to the R-Spin-VCSEL itself by using the Loop$_1$ and Loop$_2$. The feedback time along any delay line (DL) is set to $\tau$. The light intensities of the X-PC${\rm _R}$ and Y-PC${\rm _R}$ emitted by each R-Spin-VCSEL are respectively extracted at an interval of $\theta$, and interpreted as the states of virtual nodes. Here, the number of virtual node along each delay line (DL) satisfies $N=\tau _{R}/\theta$, where $\tau$ = $T$. Two outputs of each RC are first converted into current signals by the PDs, then injected into the corresponding output layers. In the output layers of 1-4, for the $i$th prediction target ($C_{x}$($n$-$L_{i}$)+$m_{xi}$($n$-$L_{i}$))($L_{i}$ is the length of the discrete delay time), the corresponding current signal is defined as $C^{'}_{x}$($n$-$L_{i}$) after being weighted and linearly summed in the $i$th output layer. In the output layers of 5-8, for the $i$th prediction target ($C_{y}$($n$-$L_{i}$)+$m_{yi}$($n$-$L_{i}$)), after being weighted and linearly summed in the output layer of (i+4), the corresponding current signal is defined as $C^{'}_{y}$($n$-$L_{i}$). Here, the weights need to be trained using linear least-squares method that minimize the mean-square error between each target signal and the corresponding reservoir output. In such a reservoir system, by training the weights, $C^{'}_{x}$($n$-$L_{i}$) and $C^{'}_{y}$($n$-$L_{i}$) can well replicate ($C_{x}$($n$-$L_{i}$)+$m_{xi}$($n$-$L_{i}$)) and ($C_{y}$($n$-$L_{i}$)+$m_{yi}$($n$-$L_{i}$)), respectively, indicating that the reservoirs RC$_1$-RC$_4$ can effectively separate the chaotic masking signals, which are multiplexed in time. The chaotic masking signals ($C^{'}_{x}$($n$-$L_{1}$)-$C^{'}_{x}$($n$-$L_{4}$)) output from the output layers of 1-4 are first combined into one beam by using the combiner (CB$_1$), then injected into the input layer 3. Here, the combined signal $U^{'}_{x}$($n$)=$\sum _{i=1}^{4}C^{'}_{x}$($n$-$L_{i}$). The chaotic masking signals ($C^{'}_{y}$($n$-$L_{1}$)-$C^{'}_{y}$($n$-$L_{4}$)) from the output layers of 5-8 are first combined into one beam by using the combiner (CB$_2$), then injected into the input layer 4, where the combined signal $U^{'}_{y}$($n$)=$\sum _{i=1}^{4}C^{'}_{y}$($n$-$L_{i}$). After being experienced the similar processing with those in the input layers 1 and 2, these chaotic masking signals $U^{'}_{x}$($n$)and $U^{'}_{y}$($n$) from the input layers of 3-4 are injected into the R-Spin-VCSEL with the subscripts of 5-8. The outputs from the R-Spin-VCSELs (the subscripts of 5-8) are first injected to the output layers of 9-16, respectively, and then weighted and linearly summed. Under this condition, the outputs layers of 9-12 ($C^{''}_{x1}$($n$)-$C^{''}_{x4}$($n$)) can well reproduce ($C_{x}$($n$-$L_{1}$)-$C_{x}$($n$-$L_{4}$)), respectively. The outputs layers of 13-16 ($C^{''}_{y1}$($n$)-$C^{''}_{y4}$($n$)) can well modeled ($C_{y}$($n$-$L_{1}$)-$C_{y}$($n$-$L_{4}$)), respectively. Therefore, the reservoirs RC$_5$-RC$_8$ can effectively separate the chaotic carrier-waves from the above-mentioned chaotic making signals. When ($C^{''}_{x1}$($n$)-$C^{''}_{x4}$($n$)) from the output layer 9-12 are synchronously subtracted from ($C_{x}$($n$-$L_{1}$)+$m_{x1}$($n$-$L_{1}$))-($C_{x}$($n$-$L_{4}$)+$m_{x4}$($n$-$L_{4}$)) from the output layers of 1-4, respectively, the decoded signals ($m^{'}_{x1}$-$m^{'}_{x4}$) can be obtained. Moreover, when ($C^{''}_{y1}$($n$)-$C^{''}_{y4}$($n$)) from the output layer 13-16 are synchronously subtracted from ($C_{y}$($n$-$L_{1}$)+$m_{y1}$($n$-$L_{1}$))-($C_{y}$($n$-$L_{4}$)+$m_{y4}$($n$-$L_{4}$)) from the output layer 5-8, respectively, the decoded signals ($m^{'}_{y1}($n$)$-$m^{'}_{y4}($n$)$) can be achieved.

Based on the modified spin-dependent model developed by San Miguel et al. [45], the four coupled rate equations of the optically pumped spin-VCSEL with optical feedback can be described as follows

Suppose that the continuous-wave lasers (CW$_1$-CW$_4$) have the same center frequency, we obtain the nonlinear dynamic equations of R-Spin-VCSEL (subscript 1-8) with optical feedback and optical injection as follows

In Eqs. (1)–(8), the subscript $R$ represents the R-spin-VCSEL. The subscript $j$ ($j$=1-8, the same below) represents the $j$th R-spin-VCSEL. $E_{x}$ and $E_{Rx}$ are the slowly varying amplitudes of the X-PC and X-PC${\rm _R}$, respectively. $E_{y}$ and $E_{Ry}$ are the slowly varying amplitudes of the Y-PC and Y-PC${\rm _R}$, respectively. $E_{xinj,1}$ and $E_{yinj,2}$ are the optical-field amplitudes of the CW$_1$ and CW$_2$, respectively. $E_{xinj,3}$ and $E_{yinj,4}$ are the optical-field amplitudes of the CW$_3$ and CW$_4$, respectively. Here, $E_{xinj,1}$ is injected into the X-PCs emitted by the R-Spin-VCSELs with the subscripts of 1-4. $E_{xinj,3}$ is injected into the X-PCs from the R-Spin-VCSELs with the subscripts of 5-8. $E_{yinj,2}$ is injected into the Y-PCs emitted by the R-Spin-VCSELs with the subscripts 1-4. $E_{yinj,4}$ is injected the Y-PCs of the R-Spin-VCSELs with the subscripts of 5-8. In these lasers, the circularly polarized electric field components are coupled by crystal birefringence, and characterized by birefringence $\gamma _{p}$ and dichroic $\gamma _{a}$. The normalized carrier variable $M$ and $n$ appearing in Eqs. (1)–(8) are defined by $M=(n^{+}+n^{-})$/2 and $n=(n^{+}-n^{-})$/2, where $n^{+}$ and $n^{-}$ are the corresponding normalized densities of electrons with spin-up and spin-down, respectively. $k$ and $k_{R}$ are the cavity decay rates. $a$ and $a_{R}$ are the linewidth enhancement factor. $\gamma$ and $\gamma _{R}$ are the electron density decay rate. $\gamma _{s}$ and $\gamma _{Rs}$ are the spin relaxation rates. $\eta$ and $\eta _{R}$ are the total normalized pump energy. $p$ and $p_{R}$ are the pump polarization ellipticity. $k_{f}$ and $k_{Rf}$ are the feedback intensity. $k_{x}$ and $k_{y}$ are the injection intensities of the X-PC${\rm _R}$ and Y-PC${\rm _R}$ of each R-Spin-VCSEL, respectively. $\Delta \omega _{j}$ is the center frequency detuning between the R-Spin-VCSEL with the subscript of $j$ and the CW with the subscript of $j$. $\omega _{R}$ is the center frequency of each R-Spin-VCSEL; $\beta _{sp}$ is the spontaneous emission coefficient, which can also be considered as the noise intensity. $\xi _{x}$, $\xi _{y}$ are independent Gaussian white noise with a mean of 0 and a variance of 1.

The time-division multiplexing signals $U_{x}(t)$ and $U_{y}(t)$ from the x-DMM and the y-DMM are respectively expressed as

Here, $i$=1, 2, 3, 4. The encoded signals $m_{x1}$-$m_{x4}$ and $m_{y1}$-$m_{y4}$ are considered as aperiodic rectangular pulses, which are expressed as

In the system shown in Fig.1, chaotic synchronization plays a key role in security and encrypted message recovery. In the following, we use the reservoirs (RC$_1$- RC$_8$) to address the chaotic synchronizations. According to the theory of complete chaotic synchronization, we get the synchronization solution as follows

3. Results and discussions

In the following calculation, the parameter values for the Spin-VCSEL and any one of the R-Spin-VCSELs are presented in Table 1. The center wavelengths of the Spin-VCSEL and any one of the R-Spin-VCSELs are both 1550 nm. The duration ($T_{m}$) of the periodic rectangular pulse signals $m_{xi}(n)$ or $m_{yi}(n)$ is 16ps. The sampling frequency of $U_{x}(n)$ or $U_{y}(n)$ is 1/$T$, and $T$=20ps. The delay times are given as follows: $dt_{1}$= 0 ns, $dt_{2}$= 0.03 ns, $dt_{3}$= 0.06 ns, $dt_{4}$= 0.09 ns. The delay lengths are presented as follows: $L_{1}$=15000, $L_{2}$=15030, $L_{3}$=15060 and $L_{4}$=15090. We first numerically solve Eqs. (1)–(8) by the fourth-order Runge-Kutta method, where Eqs. (1)–(4) are solved with a step of 1ps, and Eqs. (5)–(8) are solved with a step of 0.01ps. Here, the four sub-reservoirs formed by the X-PCs of the R-Spin-VCSELs with the subscripts of 1-4 are used to predict the targets ([$C_{x}(n-L_{1})$+$m_{x1}(n-L_{1})$]-[$C_{x}(n-L_{4})$+$m_{x4}(n-L_{4})$]), respectively. The four sub-reservoirs formed by the Y-PC from the R-Spin-VCSELs with the subscripts of 1-4 are utilized to predict the target ([$C_{y}(n-L_{1})$+$m_{y1}(n-L_{1})$]-[$C_{y}(n-L_{4})$+$m_{y4}(n-L_{4})$]), respectively. 7,000 sampling data of the prediction targets ($C_{x}(n-L_{i})$+$m_{xi}(n-L_{i})$) and ($C_{y}(n-L_{i})$+$m_{yi}(n-L_{i})$), as the input data, are recorded with the sampling interval of 1ps. After discarding the first 1000 points (to eliminate transients), we use the 3000 points for training one of the sub-reservoirs in the reservoir layers RC$_1$-R$_4$, then take the remain 3000 points to test the corresponding sub-reservoir. The four sub-reservoirs formed by the X-PCs from the R-Spin-VCSELs with the subscript of 5-8 are applied to predict the targets ($C^{'}_{x}(n-L_{1})$-$C^{'}_{x}(n-L_{4})$), respectively. The four sub-reservoirs formed by the Y-PC from the R-Spin-VCSELs with the subscripts of 5-8 are employed to predict the targets ($C^{'}_{y}(n-L_{1})$-$C^{'}_{y}(n-L_{4})$). For these sub-reservoirs, 3000 sampled data of the prediction targets ($C^{'}_{x}(n-L_{i})$ and $C^{'}_{y}(n-L_{i})$), as the input data, are recorded with the sampling interval of 1 ps. After discarding the first 1000 points (to eliminate transients), we utilize 1000 points to train one of the sub-reservoirs in the reservoir layers RC$_5$-RC$_8$, then take the remain 1000 points to test the corresponding sub-reservoir.

Table 1. The parameter values for the Spin-VCSEL and the R-Spin-VCSELs with the subscripts of 1-8

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Figure 2 shows the samples of the predicted targets ($C_{x}(n-L_{i})$+$m_{xi}(n-L_{i})$) and ($C_{y}(n-L_{i})$+$m_{yi}(n-L_{i})$) ($i$=1-4) and the outputs ($C^{'}_{x}(n-L_{i})$ and $C^{'}_{y}(n-L_{i})$) the trained sub-reservoirs in the reservoir-layers RC$_1$-RC$_4$. Here, $T$=20ps, $\theta$=0.1ps and $N$=200. As seen from Fig. 2, the trajectories of the prediction targets and the outputs of these sub-reservoirs all are chaotic. In addition, the trajectories of ($C^{'}_{x}(n-L_{1})$-$C^{'}_{x}(n-L_{4})$) is almost the same as those of ([$C_{x}(n-L_{1})$+$m_{x1}(n-L_{1})$]-[$C_{x}(n-L_{4})$+$m_{x4}(n-L_{4})$]), respectively. The trajectories of ($C^{'}_{y}(n-L_{1})$-$C^{'}_{y}(n-L_{4})$) are almost identical to those of ([$C_{y}(n-L_{1})$+$m_{y1}(n-L_{1})$]-[$C_{y}(n-L_{4})$+$m_{y4}(n-L_{4})$]), respectively. These results show that the four sub-reservoirs formed by the X-PCs from the R-Spin-VCSELs with the subscripts of 1-4 can effectively predict the trajectories of the targets ([$C_{x}(n-L_{1})$+$m_{x1}(n-L_{1})$]-[$C_{x}(n-L_{4})$+$m_{x4}(n-L_{4})$]), respectively. The four sub-reservoirs formed by the Y-PCs from the R-Spin-VCSELs with the subscripts of 1-4 can effectively model the trajectories of the targets ([$C_{y}(n-L_{1})$+$m_{y1}(n-L_{1})$]-[$C_{y}(n-L_{4})$+$m_{y4}(n-L_{4})$]), respectively. In other words, each sub-reservoir in the reservoir-layers RC$_1$-RC$_4$ can effectively separate each group of chaotic masking signals multiplexed in time.

 

Fig. 2. The samples of the predictive targets ($C_{x}(n-L_{i})$+$m_{xi}(n-L_{i})$) and ($C_{y}(n-L_{i})$+$m_{yi}(n-L_{i})$) ($i$=1-4) and the outputs ($C^{'}_{x}(n-L_{i})$ and $C^{'}_{y}(n-L_{i})$) of the trained sub-reservoirs in the reservoir-layers RC$_1$-RC$_4$. Here, $T$=20ps; $\theta$=0.1ps; $N$=200. (a$_1$): $C_{x}(n-L_{1})$+$m_{x1}(n-L_{1})$ and $C^{'}_{x}(n-L_{1})$ ; (a$_2$): $C_{y}(n-L_{1})$+$m_{y1}(n-L_{1})$ and $C^{'}_{y}(n-L_{1})$ ; (b$_1$): $C_{x}(n-L_{2})$+$m_{x2}(n-L_{2})$ and $C^{'}_{x}(n-L_{2})$ ; (b$_2$): $C_{y}(n-L_{2})$+$m_{y2}(n-L_{2})$ and $C^{'}_{y}(n-L_{2})$ ; (c$_1$): $C_{x}(n-L_{3})$+$m_{x3}(n-L_{3})$ and $C^{'}_{x}(n-L_{3})$ ; (c$_2$): $C_{y}(n-L_{3})$+$m_{y3}(n-L_{3})$ and $C^{'}_{y}(n-L_{3})$ ; (d$_1$): $C_{x}(n-L_{4})$+$m_{x4}(n-L_{4})$ and $C^{'}_{x}(n-L_{4})$ ; (d$_2$): $C_{y}(n-L_{4})$+$m_{y4}(n-L_{4})$ and $C^{'}_{y}(n-L_{4})$ .

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Figure 3 further presents the trajectories of the predictive targets $C_{x}(n-L_{i})$ and $C_{y}(n-L_{i})$ ($i$=1-4) and the outputs ($C^{''}_{xi}(n)$ and $C^{''}_{yi}(n)$) of the trained sub-reservoirs in the reservoir-layers RC$_5$-RC$_8$, where $T$=20ps, $\theta$=0.1ps, $N$=200. As displayed in Fig. 3, the trajectories of these predictive targets and the outputs by these sub-reservoirs all are chaotic. The trajectories of ($C^{''}_{x1}(n)$-$C^{''}_{x4}(n)$) are almost the same with those of the predictive targets ($C_{x}(n-L_{1})$-$C_{x}(n-L_{4})$), respectively. Moreover, the trajectories of ($C^{''}_{y1}(n)$-$C^{''}_{y4}(n)$) are consistent with the those of the predictive targets ($C_{y}(n-L_{1})$-$C_{y}(n-L_{4})$), respectively. These results show that the four sub- reservoirs formed by the X-PCs from the R-Spin-VCSELs with the subscripts of 5-8 can effectively predict the trajectories of the targets ($C_{x}(n-L_{1})$-$C_{x}(n-L_{4})$), respectively. The four sub-reservoirs formed by the Y-PCs from the R-Spin-VCSELs with the subscripts of 5-8 can effectively model the trajectories of the targets ($C_{y}(n-L_{1})$-$C_{y}(n-L_{4})$), respectively.

 

Fig. 3. The samples of the predictive targets ($C_{x}(n-L_{i})$ and $C_{y}(n-L_{i})$) ($i$=1-4) and the outputs ($C^{''}_{xi}(n)$ and $C^{''}_{yi}(n)$) of the trained sub-reservoirs in the reservoir-layers RC$_5$-RC$_8$. Here, $T$=20ps; $\theta$=0.1ps; $N$=200. (a$_1$): $C_{x}(n-L_{1})$ and $C^{''}_{x1}(n)$; (a$_2$): $C_{y}(n-L_{1})$ and $C^{''}_{y1}(n)$ ; (b$_1$): $C_{x}(n-L_{2})$ and $C^{''}_{x2}(n)$; (b$_2$): $C_{y}(n-L_{2})$ and $C^{''}_{y2}(n)$ ; (c$_1$): $C_{x}(n-L_{3})$ and $C^{''}_{x3}(n)$; (c$_2$): $C_{y}(n-L_{3})$ and $C^{''}_{y3}(n)$ ; (d$_1$): $C_{x}(n-L_{4})$ and $C^{''}_{x4}(n)$; (d$_2$): $C_{y}(n-L_{4})$ and $C^{''}_{y4}(n)$.

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To quantitatively observe the training errors of the prediction targets, we introduce the expressions of the normalized mean square errors as follows

Figure 4 gives the dependences of the training errors ($\it NMSE_{x1}^{(1)}$-$\it NMSE_{x4}^{(1)}$ and $\it NMSE_{y1}^{(1)}$-$\it NMSE_{y4}^{(1)}$) on the period $T$. Here, the interval $\theta$ between virtual nodes is fixed at 0.1ps. As observed from Fig.4, when $T$ is between 10ps and 50ps, these training errors decrease almost linearly with the increase of $T$. The reason is given as follows. When $\theta$=$T/N$ is fixed at 0.1ps, smaller $N$ is accompanied by smaller $T$, indicating that that the reservoirs have a lower space dimension. In such a case, the predictions of the trained reservoirs to the original target signals become unstable and more difficult. When $T$ is fixed at a certain value, these training errors have the following relations: $\it NMSE_{x4}^{(1)}$>$\it NMSE_{x3}^{(1)}$>$\it NMSE_{x2}^{(1)}$>$\it NMSE_{x1}^{(1)}$; $\it NMSE_{y4}^{(1)}$>$\it NMSE_{y3}^{(1)}$>$\it NMSE_{y2}^{(1)}$>$\it NMSE_{y1}^{(1)}$. The reasons can be expressed as follows. As shown in Fig.2, compared with the other predictive targets, $C_{x}(n-L_{4})$+$m_{x}(n-L_{4})$ and $C_{y}(n-L_{4})$+$m_{y}(n-L_{4})$ appear the most complicated chaotic trajectories, which makes the predictions of the reservoirs to be more difficult. Besides, when $T$ ranges from 10ps to 50ps, $\it NMSE_{x1}^{(1)}$-$\it NMSE_{x4}^{(1)}$ are less than 0.059, 0.075, 0.0945 and 0.097, respectively, and $\it NMSE_{y1}^{(1)}$-$\it NMSE_{y4}^{(1)}$ are less than 0.052, 0.061, 0.086 and 0.091, respectively. These show that the reservoirs RC$_1$-RC$_4$ can effectively separate each group of chaotic masking signals multiplexed in time.

 

Fig. 4. Training errors ($\it NMSE_{x1}^{(1)}$-$\it NMSE_{x4}^{(1)}$ and $\it NMSE_{y1}^{(1)}$-$\it NMSE_{y4}^{(1)}$) as a function of the period $T$ when $\theta$=0.1ps.

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Figure 5 further offers the dependences of the training errors ($\it NMSE_{x1}^{(2)}$-$\it NMSE_{x4}^{(2)}$ and $\it NMSE_{y1}^{(2)}$-$\it NMSE_{y4}^{(2)}$) on the period $T$. With the further increase of $T$, $\it NMSE_{x1}^{(2)}$-$\it NMSE_{x4}^{(2)}$ appear an oscillatory decrease from 0.054 to 0.042, and $\it NMSE_{y1}^{(2)}$-$\it NMSE_{y4}^{(2)}$ exhibit an oscillatory decrease from 0.045 to 0.037. When $T$ is fixed at 20ps, Fig. 6 shows the dependences of the training errors ($\it NMSE_{x1}^{(1)}$-$\it NMSE_{x4}^{(1)}$ and $\it NMSE_{y1}^{(1)}$-$\it NMSE_{y4}^{(1)}$) on the interval $\theta$ between the virtual nodes. It is found from Fig. 6 that when $\theta$ increases from 0.01ps to 0.1ps, $\it NMSE_{x1}^{(1)}$ and $\it NMSE_{y1}^{(1)}$ rapidly increase to 0.04 and 0.038, respectively. When $\theta$ further increases from 0.1ps to 0.5ps, $\it NMSE_{x1}^{(1)}$ and $\it NMSE_{y1}^{(1)}$ radually stabilize at 0.051 and 0.048 respectively. Moreover, any one of $\it NMSE_{x2}^{(1)}$-$\it NMSE_{x4}^{(1)}$ and $\it NMSE_{y2}^{(1)}$-$\it NMSE_{y4}^{(1)}$ remains a unchanged value with the increase of $\theta$ from 0.01ps to 0.5ps. In other words, $\it NMSE_{x2}^{(1)}$- $\it NMSE_{x4}^{(1)}$ keep at 0.068, 0.089 and 0.092, respectively $\it NMSE_{y2}^{(1)}$-$\it NMSE_{y4}^{(1)}$ keep at 0.063, 0.078 and 0.089, respectively. Figure 7 displays the dependence of training errors ($\it NMSE_{x1}^{(2)}$-$\it NMSE_{x4}^{(2)}$ and $\it NMSE_{y1}^{(2)}$-$\it NMSE_{y4}^{(2)}$) on $\theta$, where $T$=20ps, As seen from Fig. 7, these training errors have the same change trends. When $\theta$ increases from 0.01ps to 0.5ps, $\it NMSE_{x1}^{(2)}$-$\it NMSE_{x4}^{(2)}$ first increase quickly to 0.051, and then gradually stabilize at 0.059. $\it NMSE_{y1}^{(2)}$-$\it NMSE_{y4}^{(2)}$ first increase rapidly to 0.045, and then gradually keep at 0.0526.

 

Fig. 5. Training errors ($\it NMSE_{x1}^{(2)}$-$\it NMSE_{x4}^{(2)}$ and $\it NMSE_{y1}^{(2)}$-$\it NMSE_{y4}^{(2)}$) as a function of the period $T$ when $\theta$=0.1ps.

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Fig. 6. Training errors ($\it NMSE_{x1}^{(1)}$-$\it NMSE_{x4}^{(1)}$ and $\it NMSE_{y1}^{(1)}$-$\it NMSE_{y4}^{(1)}$) as a function of the nterval $\theta$ between the virtual nodes when $T$=20ps.

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Fig. 7. Training errors ($\it NMSE_{x1}^{(2)}$-$\it NMSE_{x4}^{(2)}$ and $\it NMSE_{y1}^{(2)}$-$\it NMSE_{y4}^{(2)}$) as a function of the nterval $\theta$ between the virtual nodes when $T$=20ps.

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The results obtained from Figs. 4-7 show that the reservoirs RC$_1$-RC$_4$ in the first-level reservoir layers can effectively separate each group of chaotic masking signals multiplexed in time. The reservoirs RC$_5$-RC$_8$ in the second-level reservoir layers can accurately predict the original chaotic carriers ($C_{x}(n-L_{1})$-$C_{x}(n-L_{4})$) and ($C_{y}(n-L_{1})$-$C_{y}(n-L_{4})$).

In the system presented in Fig.1, the synchronization between the output from one of the reservoirs RC$_5$-RC$_8$ and the corresponding original chaotic carrier plays a key role in extracting the time division multiplexed signals. The correlation coefficients used to quantitatively describe the synchronization qualities are introduced and expressed as

As observed from Figs. 8(a$_1$)-8(b$_4$), when $\eta$ is between 1 and 5, and $\Delta \omega$ varies from $-80{\times }10^{9}$ rad/s to $80{\times }10^{9}$ rad/s, $\rho _{x1}$-$\rho _{x4}$ all are more than 0.972, and $\rho _{y1}$-$\rho _{y4}$ all are more than 0.97. As shown in Figs. 8(c$_1$)-8(d$_4$), when $\Delta \omega$ varies from $-80{\times }10^{9}$ rad/s to $80{\times }10^{9}$rad/s and $\gamma$ varies from 0.1 to 2, the values of $\rho _{x1}$-$\rho _{x4}$ all exceed 0.973. The values of $\rho _{y1}$-$\rho _{y4}$ all are greater than 0.97. As displayed in Figs. 8(e$_1$)-8(f$_4$), when $\eta$ varies from 1 to 5 and $\gamma$ changes from 0.1 to 2, $\rho _{x1}$-$\rho _{x4}$ all are more than 0.974, and $\rho _{y1}$-$\rho _{y4}$ all are more than 0.97. These results show that the outputs ($C^{''}_{x1}(n)$-$C^{''}_{x4}(n)$) rom the reservoirs RC$_5$-RC$_8$ can be well synchronized with the original chaotic carriers ($C_{x}(n-L_{1})$-$C_{x}(n-L_{4})$), respectively. Furthermore, ($C^{''}_{y1}(n)$-$C^{''}_{y4}(n)$) can be well synchronized with ($C_{y}(n-L_{1})$-$C_{y}(n-L_{4})$), respectively. To clearly observe the dependences of the correlation coefficients ($\rho _{x1}$-$\rho _{x4}$ and $\rho _{y1}$-$\rho _{y4}$) on different parameters, Fig. 9 presents the plots of $\rho _{x1}$-$\rho _{x4}$ and $\rho _{y1}$-$\rho _{y4}$ via different parameters when $T$ = 20ps, $\theta$=0.1ps, $N$=20 and $\tau _{x}$=$\tau _{y}$=15ns. As seen from this figure, under different parameters, $\rho _{x1}$-$\rho _{x4}$ and $\rho _{y1}$-$\rho _{y4}$ all appear an oscillatory change in small range. For example, when $\Delta \omega$ is between −80rad/ns and 80rad/ns, $\rho _{x1}$-$\rho _{x4}$ show an oscillatory change from 0.938 to 0.961(sees Fig. 9(a)), and $\rho _{y1}$-$\rho _{y4}$ vary in oscillation from 0.9708 to 0.9746 (sees Fig. 9(b)). In other parameter spaces, $\rho _{x1}$-$\rho _{x4}$ and $\rho _{y1}$-$\rho _{y4}$ have similar oscillatory change. In the following, to obtain high-quality synchronization ($\rho _{x1}$-$\rho _{x4}$=0.972 and $\rho _{y1}$-$\rho _{y4}$=0.97), some key parameters of the system are taken values as follows: $T$=20ps, $\theta$=0.1ps, $\gamma$=1, $\rho _{x}$=$\rho _{y}$=15ns, $\eta$=2, $\Delta \omega ={-80}{\times }10^{9}$rad/s.

 

Fig. 8. Evolutions of correlation coefficients $\rho _{x1}$-$\rho _{x4}$ and $\rho _{y1}$-$\rho _{y4}$ in different parameter spaces. Where $T$ = 20ps, $\theta$=0.1ps, $N$=20 and $\tau _{x}$=$\tau _{y}$=15ns. (a$_1$)-(b$_4$): The evolutions of the correlation coefficients in the parameter space of $\eta$ and $\Delta \omega$ when $\gamma$ =1${\rm ns}^{-1}$. (c$_1$)-(d$_4$): The evolutions of the correlation coefficients in the parameter space of $\Delta \omega$ and $\gamma$ under $\eta$=2. (e$_1$)-(f$_4$): The evolutions of the correlation coefficients in the parameter space of $\eta$ and $\gamma$ under $\Delta \omega ={-80}{\times }10^{9}$ rad/s.

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Fig. 9. Dependences of correlation coefficients $\rho _{x1}$-$\rho _{x4}$ and $\rho _{y1}$-$\rho _{y4}$ on different parameters, Where $T$ = 20ps, $\theta$=0.1ps, $N$=20 and $\tau _{x}$=$\tau _{y}$=15ns. Here, (a):$\rho _{x1}$-$\rho _{x4}$ versus $\Delta \omega$ when $\gamma$ =1${\rm ns}^{-1}$ and $\eta$=2; (b):$\rho _{y1}$-$\rho _{y4}$ versus $\Delta \omega$ when $\gamma$ =1${\rm ns}^{-1}$ and $\eta$=2; (c):$\rho _{x1}$-$\rho _{x4}$ versus $\eta$ when $\gamma$ =1${\rm ns}^{-1}$ and $\Delta \omega ={-80}{\times }10^{9}$ rad/s; (d):$\rho _{y1}$-$\rho _{y4}$ versus $\eta$ when $\gamma$ =1${\rm ns}^{-1}$ and $\Delta \omega ={-80}{\times }10^{9}$ rad/s; (e):$\rho _{x1}$-$\rho _{x4}$ versus $\gamma$ when $\eta$=2 and $\Delta \omega = {-80}{\times }10^{9}$ rad/s; (f):$\rho _{y1}$-$\rho _{y4}$ versus $\gamma$ when $\eta$=2 and $\Delta \omega ={-80}{\times }10^{9}$ rad/s;

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Under $\rho _{x1}$-$\rho _{x4}$=0.972 and $\rho _{y1}$-$\rho _{y4}$=0.97, each encoded message is decoded by the synchronous subtraction between the output of one sub-reservoir in the second-level reservoir layer and the output of the corresponding sub-reservoir in first-level reservoir layer. The eye-diagrams of the decoded messages ($m^{'}_{x1}(n)$-$m^{'}_{x4}(n)$) and ($m^{'}_{y1}(n)$-$m^{'}_{y4}(n)$) are presented in Fig. 10. As seen from this figure, 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.

 

Fig. 10. Eye-diagrams of the decoded messages $m^{'}_{x1}(n)$-$m^{'}_{x4}(n)$ and $m^{'}_{y1}(n)$-$m^{'}_{y4}(n)$. (a$_1$): The eye- diagram for $m^{'}_{x1}(n)$; (a$_2$): The eye- diagram for $m^{'}_{y1}(n)$; (b$_1$): The eye- diagram for $m^{'}_{x2}(n)$; (b$_2$): The eye- diagram for $m^{'}_{y2}(n)$; (c$_1$): The eye- diagram for $m^{'}_{x3}(n)$; (c$_2$): The eye- diagram for $m^{'}_{y3}(n)$; (d$_1$): The eye- diagram for $m^{'}_{x4}(n)$; (d$_2$): The eye- diagram for $m^{'}_{y4}(n)$;

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Bit error rate ($BER$) is usually used to measure the data transmission quality of optical chaotic secure communication system, and defined as

The variable $erfc$ is a complementary error function. Figure 11 presents the dependences of the bit error rates ($BER_{x}$ and $BER_{y}$) on the parameters $\gamma$, $K_{x}$, $K_{f}$ and $\eta$. As displayed in Fig.11, the bit error rates of $m^{'}_{x2}(n)$-$m^{'}_{x4}(n)$ and $m^{'}_{y2}(n)$-$m^{'}_{y4}(n)$ are almost equal to 0 under different parameters. However, for $m^{'}_{x1}(n)$ and $m^{'}_{y1}(n)$, their bit error rates show an irregular oscillation in a small range, and are less than $7{\times }10^{-3}$. The reason is given as follows: the amplitudes of the encoded messages of $m_{x1}(n)$ and $m_{y1}(n)$ are smallest among these encoded messages, which are close to the synchronization error between $C_{x}(n-L_{1})$ and $C^{''}_{x1}(n)$. According to Eq. (31), this synchronization error can lead to the case that the bit error rates for $m^{'}_{x1}(n)$ and $m^{'}_{y1}(n)$ are obviously bigger than those of the other decoded messages. If $m^{'}_{x1}(n)$ and $m^{'}_{y1}(n)$ are filtered, shaped and regenerated, their bit error rates may be less than $10^{-10}$. The above results show that the system presented in Fig.1 can effectively demultiplex the optical time-division multiplexing signals.

To further observe the quality of the decoded messages, Figs. 12(a$_1$) and 12(a$_2$) display the time-division multiplexing trajectories of the original delay encoded messages ($m_{x1}(n-L_{1})$-$m_{x4}(n-L_{4})$) and ($m_{y1}(n-L_{1})$-$m_{y4}(n-L_{4})$), respectively. As observed from these figures, according to Eq. (10), the original delay encoded messages can be effectively multiplexed in time-division by adjusting the delays of the pulses. Figures 12(b$_1$)-12(c$_2$) further give the temporal traces of the decoded messages $m^{'}_{x1}(n)$-$m^{'}_{x4}(n)$ and $m^{'}_{y1}(n)$-$m^{'}_{y4}(n)$. It can be further seen from Figs. 12(b$_1$)-12(c$_2$) that the temporal traces of the original delay encoded messages $m_{x1}(n-L_{1})$-$m_{x4}(n-L_{4})$ are almost the same with those of the decoded messages $m^{'}_{x1}(n)$-$m^{'}_{x4}(n)$, respectively. The temporal traces of the original delay encoded messages $m_{y1}(n-L_{1})$-$m_{y4}(n-L_{4})$ are almost identical to $m^{'}_{y1}(n)$-$m^{'}_{y4}(n)$, respectively. However, compared with the original encoded pulses, there appear a certain degree of jitters at the center of the decoded pulses, and there exist low-amplitude noises in the interval between two adjacent pulses can be weaken by using the electric filters. Moreover, compared with decoded messages $m^{'}_{x3}(n)$, $m^{'}_{x4}(n)$, $m^{'}_{y3}(n)$ and $m^{'}_{y4}(n)$, there are much more jitters that can be observed for the decoded messages $m^{'}_{x1}(n)$, $m^{'}_{x2}(n)$, $m^{'}_{y1}(n)$ and $m^{'}_{y2}(n)$. The reason for this case is given as follows: compared with the original encoded messages $m_{x3}(n)$, $m_{x4}(n)$, $m_{y3}(n)$ and $m_{y4}(n)$, the original encoded messages $m_{x1}(n)$, $m_{x2}(n)$, $m_{y1}(n)$ and $m_{y2}(n)$ have smaller amplitude. Their amplitudes approach to the synchronization error between $C_{x}(n-L_{1})$ and $C^{''}_{x1}(n)$ and that between $C_{x}(n-L_{2})$ and $C^{''}_{x2}(n)$, respectively. Therefore, much more jitters can be observed for the decoded messages $m^{'}_{x1}(n)$, $m^{'}_{x2}(n)$, $m^{'}_{y1}(n)$ and $m^{'}_{y2}(n)$.

In a word, by observations of the eye-diagram qualities, the change curves of the bit error rates and the temporal traces of the decoded messages from Figs. 10–12, it is concluded that the high-quality encoded messages can be decoded synchronously by training the sub-reservoirs in the second-level reservoir layer presented in Fig. 1.

 

Fig. 11. Dependences of the bit error rates ($BER_{x}$ and $BER_{y}$) on different parameters. Here, (a$_1$): $BER_{x}$ versus $\gamma$; (b$_1$): $BER_{x}$ versus $K_{x}$; (c$_1$): $BER_{x}$ versus $K_{f}$; (d$_1$): $BER_{x}$ versus $\eta$; (a$_2$): $BER_{y}$ versus $\gamma$; (b$_2$): $BER_{y}$ versus $K_{x}$; (c$_2$): $BER_{y}$ versus $K_{f}$; (d$_2$): $BER_{y}$ versus $\eta$.

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Fig. 12. (a$_1$) The time-division multiplexing trajectories of the original delay encoded messages ($m_{x1}(n-L_{1})$-$m_{x4}(n-L_{4})$). (a$_2$) The time-division multiplexing trajectories of the original delay encoded messages ($m_{y1}(n-L_{1})$-$m_{y4}(n-L_{4})$). (b$_1$) The temporal traces of $m_{x1}(n-L_{1})$ or $m_{y1}(n-L_{1})$ and $m^{'}_{x1}(n)$ or $m^{'}_{y1}(n)$ ; (b$_2$) Those of $m_{x2}(n-L_{2})$ or $m_{y2}(n-L_{2})$ and $m^{'}_{x2}(n)$ or $m^{'}_{y2}(n)$ ; (c$_1$) Those of $m_{x3}(n-L_{3})$ or $m_{y3}(n-L_{3})$ and $m^{'}_{x3}(n)$ or $m^{'}_{y3}(n)$ ; (c$_2$) Those of $m_{x4}(n-L_{4})$ or $m_{y4}(n-L_{4})$ and $m^{'}_{x4}(n)$ or $m^{'}_{y4}(n)$.

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4. Conclusion

In this paper, we propose a novel dual-channel OTDM chaotic secure communication system, using two cascaded reservoir computing systems based on many beams of chaotic polarization components emitted by four optically pumped Spin-VCSELs. Here, there are four parallel reservoirs in each level reservoir layer, and each parallel reservoir contains two sub-reservoirs. Each group of chaotic masking messages multiplexed in time-division can be effectively separated by training the sub-reservoirs in the first-level reservoir layer, where the training errors are far less than 0.1. In addition, the high-quality synchronization between the output of each sub-reservoir in the second-level reservoir layers and its corresponding delayed chaotic carrier can be achieved by training sub-reservoirs in the second-level reservoir layer, where the training errors are far less than 0.1 and the correlation coefficients used to characterize the synchronization qualities are more than 0.97. Under these high-quality synchronizations, we further explore the performances of dual-channel OTDM chaotic secure communication with a rate of 4$\times$60Gb/s. By the observations of the eye diagram, bit error rate and temporal trajectory of each decoded message, it is found that there is a large eye opening in the eye diagram, and there appears a low bit error rate for each decoded message. Except that the bit error rate of one decoded message is lower than $7{\times }10^{-3}$ in different parameter spaces, and those of the other decoded messages are close to 0, showing that high-quality data transmissions are expected to be performed in the system. These results show that our OTDM system based on reservoir computing technology can recover each encoded message with high quality. Therefore, the multi-cascaded reservoir computing systems based on multiple optically pumped VCSELs open a new direction for the realization of multi-channel OTDM chaotic secure communications with high-speed. However, our proposed scheme can face some potential challenges in practical application. For example, for the encoded messages with large amplitude, there can exit large training errors in the multi-cascaded reservoir computing system. Moreover, since the rate of each encoded message is limited to the feedback delay time of each reservoir laser, the realization of the OTDM chaotic secure communication with higher speed can be more difficult. Finally, to meet communication standards, the decoded messages need to be further filtered, shaped and regenerated, and their bit-error rates should less than $10^{-10}$.