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Abstract
In this work, with the mixing fractions being known in advance or unknown, the schemes and theories for the separations of two groups of the mixed optical chaotic signals are proposed in detail, using the VCSEL-based reservoir computing (RC) systems. Here, two groups of the mixed optical chaotic signals are linearly combined with many beams of the chaotic x-polarization components (X-PCs) and Y-PCs emitted by the optically pumped spin-VCSELs operation alone. Two parallel reservoirs are performed by using the chaotic X-PC and Y-PC output by the optically pumped spin-VCSEL with both optical feedback and optical injection. Moreover, we further demonstrate the separation performances of the mixed chaotic signal linearly combined with no more than three beams of the chaotic X-PC or Y-PC. We find that two groups of the mixed optical chaos signals can be effectively separated by using two reservoirs in single RC system based on optically pumped Spin-VCSEL and their corresponding separated errors characterized by the training errors are no more than 0.093, when the mixing fractions are known as a certain value in advance. If the mixing fractions are unknown, we utilize two cascaded RC systems based on optically pumped Spin-VCSELs to separate each group of the mixed optical signals. The mixing fractions can be accurate predicted by using two parallel reservoirs in the first RC system. Based on the values of the predictive mixing fractions, two groups of the mixed optical chaos signals can be effectively separated by utilizing two parallel reservoirs in the second RC system, and their separated errors also are no more than 0.093. In the same way, the mixed optical chaos signal linearly superimposed with more than three beams of optical chaotic signals can be effectively separated. The method and idea for separation of complex optical chaos signals proposed by this paper may provide an impact to development of novel principles of multiple access and demultiplexing in multi-channel chaotic cryptography communication.
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1. Introduction
The optical chaotic signals are of interest in many different fields such as security optical communication [1–5], optical chaotic computing [6,7], lidar ranging [8–10], high-speed random code generation [11,12] and so on. The problem of extracting an optical chaotic signal from ambient noise or from mixed optical chaotic signals have potential applications in various fields such as optical signal processing, the ranging to multi-target using synchronized chaotic lidars [13,14], optical imaging [15,16], and multi-channel optical cryptography communication [1,17]. The problem of separation of optical chaotic signals is of great interest in analysis of nature or artificial optical chaotic signals. In particular, its efficient solution can give an impact to development of novel principles of multiple access and demultiplexing in multi-channel chaotic cryptography communication.
In the above-mentioned applications of optical chaotic signal separation, it is needed to extract the original optical chaotic signal from a group of combined optical chaotic signals. As we all know, the separation of optical chaotic signals is rarely reported. Many previous works mainly focused on the separation of electric chaotic signals [18–22]. In these works, various methods were applied in the separation of electric chaotic signals, such as blind source separation [23–25], chaos synchronization [19,26–28], neural network [29,30]. The problem of bind source separation typically assumes that multiple linear combinations of signals are being measured, with the number of independent measurements at least as large as the number of signals. If the equations governing chaotic process are known, chaotic signals can be separated by using chaotic synchronization [19]. The neural network applied for signal separation need accurate information about the structure and parameters of the chaotic system. Compared with the separation of electric chaotic signals, the separation of optical chaotic signals is faced with more difficulty, owing to that they have faster and more complex dynamical behaviors, and higher dimension than electric chaotic signals.
In recent years, reservoir computing has been proven to be an effective approach in the prediction of chaotic systems from data. The delay-based RC firstly proposed by Appeltant $et$ $al$ [31], composed of a nonlinear physical node and a delay feedback-loop, is proved to be an effective and simple implementation for neural network computing in hardware. Many hardware implementations for delayed-based RC have been experimentally and theoretically demonstrated, such as electronic system [21], opto-electronic system [32], all-optical system [33], and laser dynamical system [34]. It is of special interest that the delay-based RC using nonlinear semiconductor lasers is promising for high-speed computing, due to high relaxation oscillation frequency and the ability of transmitting low dimensional data into high state space. A growing number of studies have shown that this technique has the advantages of fast-speed, high efficiency, parallel computing [35–37]. This technique has been applied to several real-world problems, such as the prediction for fast time series [38], the prediction for optical chaotic synchronization [39], optical chaotic secure communication [40], optical packet header recognition [41] and so on. Our recent works proved that the delay-based RC using chaotic semiconductor lasers can well predict the dynamical behaviors of optical chaotic signals [40,42]. Therefore, an effective separation of optical chaotic signals with high-speed and high-dimension is expected to be realized by using optical delay-based RC.
Optically pumped spin-polarized vertical-cavity surface-emitting lasers (OP-Spin-VCSELs) proposed in [43] might provide properties superior to those of their conventional counterparts. For instance, they promise to offer reduced lasing threshold, spin amplification, the ability to effectively control their emitted light polarization, ultrafast dynamics, and extra-large bandwidth (about 200GHz) [44,45]. Therefore, the chaotic signal emitted by the OP-Spin-VCSEL without self-feedback or light injection is of interest for a wide range of applications including optical chaotic time division multiplexing communication, optical chaotic wavelength division multiplexing communications with multi-channel, multiple access in optical chaotic communication, polarization multiplexing optical chaotic communication and photonic neural network [46,47]. In these applications, extracting an optical chaotic signal from ambient noise or mixed optical chaotic signals plays an important role. However, how to separate subsequently mixed optical chaotic signals into their components still faces new challenges. The mechanism of the separation of optical chaotic signals needs to be further explored. For these purposes, in this paper, we consider optical RC approach instead of the above-mentioned methods, to separate mixed optical chaotic signals into their components. Here, the OP-Spin-VCSEL with both optical injection and optical feedback is considered as the reservoir laser, which is also named as the R-Spin-VCSEL. Two chaotic polarization components (X-PC and Y-PC) emitted by it are utilized as two nonlinear nodes to form two parallel reservoirs. The chaotic signals to be separated include the linearly superimposed chaotic X-PC signals and the linearly combined Y-PC signals, which are from a group of the OP-Spin-VCSELs operating alone. Moreover, using the reservoirs based on the R-Spin-VCSEL, we explore the separations of the linearly mixed chaotic X-PC signals and the linearly mixed Y-PC signals when their mixing fraction is known or unknown. We further discuss the influences of the mixing fraction, sampling period and virtual node interval on the separation errors characterized by the training error.
2. Theory and model
Figure 1 depicts a scheme diagram of the separations of the linearly mixed chaotic X-PC signals and the linearly mixed Y-PC signals from a group of the OP-Spin-VCSELs operating alone, using two parallel reservoirs based on the R-Spin-VCSEL with both optical feedback and optical injection. Here, the VCSELs (the subscripts of 1-$n$) all are OP-Spin-VCSEL, and operate alone. The OP-Spin-VCSEL with both optical feedback and optical injection is considered as the reservoir based on R-Spin-VCSEL. The chaotic X-PC and Y-PC emitted by R-Spin-VCSEL are utilized as two nonlinear nodes to form two parallel reservoirs. The CW$_1$ and CW$_2$ are continuous wave laser. Optical isolators (ISs) (the subscripts of 1-($n$+3)) are used to optical feedback. The fiber polarization beam splitter (FPBS) (the subscripts of 1-($n$+1)) are utilized for the separation of the light into two polarization components. The FC is fiber coupler. The FPC, as fiber polarization coupler, is used for the coupling of two chaotic PCs. The polarization controllers (PCS) (the subscripts of 1-4) are utilized to control the polarization of light. The EA$_1$ and EA$_2$ both are electrical amplifier. The DM$_1$ and DM$_2$ both are discrete module. The Mask$_1$ and Mask$_2$ both denote masked signal. The SC$_1$ and SC$_2$ both represent scaling operation circuit. The PM$_1$ and PM$_2$ both are phase modulator. The neutral density filters (NDFs) (with the subscripts of 1-(2$n$+1)) are utilized for control of the light intensity. The VOA$_1$ and VOA$_2$ both denote variable optical attenuators, which are used for the control of the feedback light-intensity. The DL$_1$ and DL$_2$ both represent delay line. The photoelectric detectors (PDs) with the subscripts of 1-(2$n$+2) are utilized to convert optical signals into current signals.
Fig. 1. Schematic diagram of the separations of the linearly mixed chaotic X-PC signals and the linearly mixed Y-PC signals from a group of optically pumped spin-VCSEL operation alone, using two parallel reservoirs based on the optically pumped spin-VCSEL with both optical feedback and optical injection (see texts for the detailed descriptions). Here the mixing fractions of $U_{x}(n)$ and $U_{y}(n)$ are known in advance.
The system presented by Fig. 1 mainly consists of the chaotic laser sources, two input layers, two parallel reservoirs, two output layers. Here, the chaotic laser sources are composed of the OP-Spin-VCSELs with number of $N$ that all operate alone. These laser sources generate $N$ beams of the chaotic signals. Each beam of chaotic signal is further divided into the X-PC and Y-PC by using the FPBS. The intensities of the X-PC and Y-PC from the $i$th beam of the chaotic signal are defined as $C_{xi}$($t$) and $C_{yi}$($t$) ($i$=1, ${\ldots }$, $n$), which are reduced to $\beta _{i}C_{xi}$ and $\beta _{i}C_{yi}$ by using the NDF, respectively. The term $\beta _i$ denotes the scale factor. The $N$ beams of chaotic X-PC signals are converted into $N$ beams of chaotic electrical signals by the PDs (s=2$n$-1), linearly combined into a beam of the mixed chaotic signals by using the combiner1 (CB$_1$). The mixed X-PC signals are defined as $U_x$($t$), and $U_x$($t$)= $\beta _1C_{x1}$($t$)+…+$\beta _nC_{xn}$($t$), where $\beta _{1}^{2}+\beta _{2}^{2}+{\ldots },+\beta _{n}^{2}=1$,(The same below). The $N$ beams of chaotic Y-PC signals are converted into $N$ beams of chaotic electrical signals by the PDs (s=2$n$), linearly superimposed into a beam of the mixed chaotic signals by using the CB$_2$. The mixed Y-PC signals are called as $U_y$($t$), and $U_y$($t$)= $\beta _1C_{y1}$($t$)+…+$\beta _nC_{yn}$($t$).
The input layers provide the input connections with reservoirs. In the input layers 1 and 2, the mixed signals $U_x$($t$) and $U_y$($t$) re amplified by the EA$_1$ and EA$_2$, then sampled as two groups of input data by the DM$_1$ and DM$_2$, respectively. These sampled input data are defined as $U_x$($n$) and $U_y$($n$), respectively, where $n$ is the discrete time index, $U_x$($n$)= $\beta _1C_{x1}$($n$)+…+$\beta _nC_{xn}$($n$) and $U_y$($n$)= $\beta _1C_{y1}$($n$)+…+$\beta _nC_{yn}$($n$). The sampled chaotic signals $C_{xi}$($n$) and $C_{yi}$($n$) are considered as two independent prediction targets. The sampled data $U_x$($n$) and $U_y$($n$) holding for a period $T$ are multiplied by a mask signal (named as Mask) with the periodicity of $T$. Here, the Mask is a chaotic signal emitted by two mutually coupled semiconductors, which is presented in [36,48]. After being scaled by a scaling factor $\gamma$ by the scaling operation circuits (SC$_1$ and SC$_2$), these two masked input signals are denoted as $S_x$($n$) and $S_y$($n$), respectively. They are utilized to modulate the phases of the optical fields output by the CW$_1$ and CW$_2$ by using the PM$_1$ and PM$_2$, respectively, where two CW lasers are used to convert the masked input signals into optical injection signals.
Within the reservoirs, the X-PC and Y-PC of the R-Spin-VCSEL with double optical feedbacks provided by the feedback Loop 1 and Loop 2 are named as the X-PC$_R$ and Y-PC$_R$, which will exhibit chaotic dynamic. Hence, they are utilized as nonlinear nodes to implement two parallel reservoirs. The X-PC${\rm _R}$ and Y-PC${\rm _R}$ are fed back into the R-Spin-VCSEL by the Loops 1 and 2. The feedback time along any of the delay lines (DL$_1$ and DL$_2$) is set as $\tau$. In the output layers, the X-PC${\rm _R}$ and Y-PC${\rm _R}$ are divided by using the FPBS with the subscript of n+1. After being extracted at time intervals $\theta$, their intensities ($I_{Rx}$=|$E_{Rx}$|$^2$ and $I_{Ry}$=|$E_{Ry}$|$^2$) are considered virtual node states. The number $N$ of the virtual nodes $N$=$T$/$\theta$, and $\tau$= $T$. For two independent prediction targets $C_{xi}$($n$) and $C_{yi}$($n$), the states of $N$ virtual nodes along these two delay lines are first converted into current signals by the PD$_{2n+1}$ and PD$_{2n+2}$, respectively. These current signals that are weighted and linearly summed up are denoted as $C^{'}_{xi}(n)$ and $C^{'}_{yi}(n)$, respectively. Here, the weights need to be trained using linear least-squares method that minimize the mean-square error between each input signal and the corresponding reservoir output [48]. In such a system, by training the output weights corresponding to $C_{xi}$($n$) and $C_{yi}$($n$), $C^{'}_{xi}(n)$ and $C^{'}_{yi}(n)$ can well reproduce $C_{xi}(n)$ and $C_{yi}(n)$, respectively.
Based on the modified spin-dependent model developed by San Miguel $et$ $al$ [49]. The four coupled rate-equations for each OP-Spin-VCSEL operation alone are described by
The mixed chaotic signals $U_{x}(t)$ and $U_{y}(t)$ are written as
In some cases, such as the multiple access in optical chaotic communication and so on, the ratio of amplitudes in a mixed optical chaos signal may not be known. Based on two cascaded RC systems, we propose a scheme for the separations of the mixed optical chaos signals ($U_{x}(t)$ and $U_{y}(t)$) without knowledge of their mixing fractions ($F_{1}$ and $F_{2}$), as displayed in Fig. 2. Here, the optical structure and parameters of each RC is the same with those of the RC based on the R-spin-VCSEL presented in Fig. 1. The input layers and output layers are identical to those given in Fig. 1. We use two cascaded RCs in sequence as follows. The first RC (RC$_1$) is trained as described in the previous paragraph, except that the original outputs are $F_{1}$ and $F_{2}$ instead of $C_{xi}^{'}(n)$ and $C_{yi}^{'}(n)$. Once trained, we use the RC$_1$ to estimate the values of $F_{1}$ and $F_{2}$ for the given mixed optical chaos signals ( $U_{x}(n)$ and $U_{y}(n)$). Then we input the same mixed optical chaos signals to a second RC (RC$_2$), which is trained to output $C_{xi}^{'}(n)$ and $C_{yi}^{'}(n)$. The RC$_2$ can be trained after the RC$_1$ has run, using the estimated values of $F_{1}$ and $F_{2}$. Namely, the output weight matrix for the RC$_2$ can be interpolated from output weight matrices that have been pretrained on discrete values of $F_{1}$ and $F_{2}$.
Fig. 2. Schematic diagram of the separations of the linearly mixed chaotic X-PC signals and the linearly mixed Y-PC signals from a group of optically pumped spin-VCSEL operation alone, using two cascaded reservoirs based on the optically pumped spin-VCSEL with both optical feedback and optical injection. Here, the mixing fractions of $U_{x}(n)$ and $U_{y}(n)$ are unknown.
3. Results and discussions
We take the mixed signal composed of two or three beams of the optical chaos signals as an example to explore its separation into components, due to limitation of the length of this paper. Namely, $U_{x}(n)=\beta _{1}C_{x1}(n)+\beta _{2}C_{x2}(n)$ or $\beta _{1}C_{x1}(n)+\beta _{2}C_{x2}(n)+\beta _{3}C_{x3}(n)$, $U_{y}(n)=\beta _{1}C_{y1}(n)+\beta _{2}C_{y2}(n)$ or $\beta _{1}C_{y1}(n)+\beta _{2}C_{y2}(n)+\beta _{3}C_{y3}(n)$. Here, $C_{x1}(n)$, $C_{x2}(n)$, and $C_{x3}(n)$ are from the X-PC$_1$, X-PC$_2$ and X-PC$_3$, respectively. $C_{y1}(n)$, $C_{y2}(n)$, and $C_{y3}(n)$ are from the Y-PC$_1$, Y-PC$_2$ and Y-PC$_3$, respectively. In the following calculations, for the separation of the mixed signal composed of two beams of the optical chaos signals, the parameters of the spin-VCSELs (the subscripts of 1-2) operation alone and the reservoir based on R-spin-VCSEL are presented in Table 1. For the separation of the mixed signal composed of three beams of the scaled optical chaos signals, the parameters of the spin-VCSELs (the subscripts of 1-3) operation alone and the reservoir based on the R-spin-VCSEL are presented in Table 2.
Table 1. The parameter values used for calculations in the separation of two chaotic X-PCs or Y-PCs signals
Table 2. The parameter values used for calculations in the separation of three chaotic X-PCs or Y-PCs signals
To obtain the chaotic dynamics of $\beta _{1}C_{x1}(n)$-$\beta _{3}C_{x3}(n)$ and $\beta _{1}C_{y1}(n)$-$\beta _{3}C_{y3}(n)$, we first slove numerically Eqs. (1)–(8) by using fourth-order Runge-Kutta method, with a step of 1 ps. Then, we calculate the dynamical evolutions of the OP-Spin-VCSELs (the subscripts of 1-3) operation alone in different parameter spaces, as displayed in Fig. 3. Here, $\eta _{1}, \eta _{2}, \eta _{3} =\eta$; $\gamma _{1a}, \gamma _{2a}, \gamma _{3a} =\gamma _{a}$; $\gamma _{1p}, \gamma _{2p}, \gamma _{3p} =\gamma _{p}$; $p_{1}, p_{2}, p_{3}=p$; $\alpha _{1}, \alpha _{2}, \alpha _{3}=4$; $k$= 300ns$^{-1}$; $\gamma _{1s}, \gamma _{2s}, \gamma _{3s}$ = 50ns$^{-1}$; $\gamma$=1ns$^{-1}$. As seen from Fig. 3, the X-PCs (X-PC$_1$, X-PC$_2$ and X-PC$_3$) emitted by the OP-Spin-VCSELs with the subscripts of 1-3 under operation alone exhibit chaotic state in large parameter spaces of $\gamma _{a}$ and $p$, $\gamma _{p}$ and $p$, and $\eta$ and $p$. By the observation of Fig. 3, it is found that, with the parameters given in Table 2, the OP-Spin-VCSELs with the subscripts of 1-3 all generate chaotic X-PC and Y-PC. Therefore, the signals ($\beta _{1}C_{x1}(n)$-$\beta _{3}C_{x3}(n)$ and $\beta _{1}C_{y1}(n)$-$\beta _{3}C_{y3}(n)$) to be separated appear chaotic state. In order for the separations of these optical chaotic signals, they need to be predicted by using two parallel reservoirs system based on R-spin-VCSEL. For the predictions of these chaotic signals, we introduce the normalized mean square errors ($NMSE_{s}$) to describe their training errors. Here, the $NMSE_{xi}$ ($i$=1, 2, 3) is utilized to describe the training error between the trained reservoir output $C_{xi}^{'}(n)$ and the original chaotic signal $C_{xi}(n)$. The $NMSE_{yi}$ ($i$=1,2,3) is used to measure the training error between the trained reservoir output $C_{yi}^{'}(n)$ and $C_{yi}(n)$. The $NMSE_{xi}$ and $NMSE_{yi}$ are expressed as follows
Fig. 3. Maps of the dynamical evolutions of the optically pumped spin-VCSELs (the subscripts of 1-3) operation alone in different parameter spaces. Here, $\eta _{1}, \eta _{2}, \eta _{3} =\eta$; $\gamma _{1a}, \gamma _{2a}, \gamma _{3a} =\gamma _{a}$; $\gamma _{1p}, \gamma _{2p}, \gamma _{3p} =\gamma _{p}$; $p_{1}, p_{2}, p_{3}=p$; $\alpha _{1}, \alpha _{2}, \alpha _{3}=4$; $k$= 300ns$^{-1}$; $\gamma _{1s}, \gamma _{2s}, \gamma _{3s}$ = 50ns$^{-1}$; $\gamma$=1ns$^{-1}$. CO: Chaotic state; QP: quasi-periodic oscillation; P2: period-two oscillation; P1: period-one oscillation; CW: stable operation. Moreover, (a$_1$)-(a$_2$): the parameter space of $\gamma _{a}$ and $p$ where $\gamma _{p}$=20ns$^{-1}$ and $\eta$=3.5; (b$_1$)-(b$_2$): that of $\gamma _{p}$ and $p$ where $\gamma _{a}$=2ns$^{-1}$ and $\eta$=3.5; (c$_1$)-(c$_2$): that of $\eta$ and $p$ where $\gamma _{a}$=2ns$^{-1}$ and $\gamma _{p}$=20ns$^{-1}$
3.1 Separation of the mixed signal combined with two beams of optical chaotic signals
3.1.1 Mixing fractions $F_{1}$ and $F_{2}$ are known in advance
Two beams of chaotic X-PCs and Y-PCs emitted by the OP-Spin-VCSELs with the subscripts of 1-2 are linearly combined into the mixed signals $U_{x}(n)$ and $U_{y}(n)$, respectively, which are expressed as
where $F_{1}$, $F_{2}$=$F$, $\beta _{1}=\sqrt {F}$, $\beta _{2}=\sqrt {1-F}$ and $\beta _{1}^{2}+\beta _{2}^{2}=1$. For the numerically solving Eqs. (1)–(4), 6000 samples of the input data ($U_{x}(n)$ and $U_{y}(n)$) are recorded under the sampling interval of 10 ps. After discarding the first 1000 samples (to eliminate transient states), we use 3000 samples for training two parallel reservoirs performed by the X-PC and Y-PC emitted by the reservoir based on the R-spin-VCSEL, and take their remaining 2000 samples to test these reservoirs. Moreover, two mask signals (Mask$_1$ and Mask$_2$) both are chaotic signals by two mutually-coupled semiconductors as presented in [46]. The amplitudes of these mask signals are adjusted, making their standard deviations to be 1 and mean values of 0. In each reservoir, the interval between two adjacent virtual nodes is denoted by $\theta$ and set as 20 ps. The sampling period $T$ of input data is set as 5 ns. The number of virtual nodes $N$ is obtained as 250 under $N=\tau /\theta$ and $\tau$ = $T$. The scale factor is set as 1.To further explore the predictive performances of the two parallel trained reservoirs to the trajectories of the chaotic signals ($\beta _{1}C_{x1}(n)$- $\beta _{2}C_{x2}(n)$ and $\beta _{1}C_{y1}(n)$- $\beta _{2}C_{y2}(n)$). Figure 4 shows the training errors (${NMSE}_{x1}$-${NMSE}_{x2}$ and ${NMSE}_{y1}$-${NMSE}_{y2}$) as a function of the period $T$. when the mixing fraction $F$ is known as 0.5 in advance, $\theta$ is set as 20 ps, $L$ is fixed at 2000. One sees from Fig. 4 that with the increase of $T$ from 1 ns to 10 ns, the ${NMSE}_{x1}$ appears a decrease in oscillation from 0.093 to 0.065, the ${NMSE}_{x2}$ exhibits an oscillatory decrease from 0.093 to 0.073. The ${NMSE}_{y1}$ shows an oscillatory decrease from 0.076 to 0.057. The ${NMSE}_{y2}$ appears a decrease oscillation from 0.082 to 0.061. The reason that a longer sampling period $T$ results in an oscillation decreasing training error may be explained as follows. In this work, $\theta =T/N$ is fixed at 20 ps, a smaller $N$ is accompanied by a smaller $T$. This means a lower dimension of state space, making the training of the system be implemented more difficult and therefore results in a larger $NMSE$.
Fig. 4. Training errors (${NMSE}_{x1}$-${NMSE}_{x2}$ and ${NMSE}_{y1}$-${NMSE}_{y2}$) as a function of the period $T$ when the mixing fraction $F$ is known as 0.5 in advance and $\theta$=20 ps.
The dependences of these training errors on the virtual node interval $\theta$ are presented in Fig. 5 under $T$=5 ns and $F$=0.5. From this figure, it is found that when $\theta$ increases from 3 ps to 50 ps, the ${NMSE}_{x1}$ and ${NMSE}_{x2}$ rapidly increase to 0.06 and 0.105, respectively. The ${NMSE}_{y1}$ and ${NMSE}_{y2}$ fast increase to 0.05 and 0.08. With the increase of $\theta$ further increases from 3 ps to 50 ps, the ${NMSE}_{x1}$ and ${NMSE}_{x2}$ gradually stabilize at 0.087 and 0.098, respectively. The ${NMSE}_{y1}$ and ${NMSE}_{y2}$ gradually stabilize at 0.074 and 0.08, respectively. This can be explained as follows: in the two ultrashort feedback loops presented in Fig. 1, when $T=N\theta$ is fixed at 5 ns, the larger $\theta$ is accompanied by the smaller $N$, indicating that the reservoir has a lower space dimension. In such a case, the prediction of the trained reservoir to the original chaotic signal becomes unstable and more difficult. In addition, a large $\theta$ can lead to insufficient coupling between two adjacent nodes. As a result, the joint action of above two factors may cause a relatively bad performance. Figure 6 presents the dependences of the training errors on the mixing fraction $F$. As seen from this figure, all these training errors show oscillatory changes in a small range of the mixing fraction $F$, denoting the $F$ has a slight influence on these training errors. In addition, as displayed in Figs. 4–6, the ${NMSE}_{x2}$ and ${NMSE}_{y2}$ are obviously bigger than the ${NMSE}_{x1}$ and ${NMSE}_{y1}$, respectively, when $F$ is mixed at a certain value. The reason is that $\beta _{2}C_{x2}(n)$ and $\beta _{2}C_{y2}(n)$ (the original chaotic signals) show more violent and irregular oscillation than $\beta _{1}C_{x1}(n)$ and $\beta _{1}C_{y1}(n)$ (the original chaotic signals), respectively, as observed from Fig. 7. This case leads to the prediction of the trained reservoir to $\beta _{2}C_{x2}(n)$ and $\beta _{2}C_{y2}(n)$ becoming more difficult. By observations of Figs. 4–6, we take the values $T$, $\theta$ and $F$ as 5 ns, 20 ps and 0.5, respectively, to obtain relatively small training errors for the accurate separation of the mixed signals ($U_{x}(n)$ and $U_{y}(n)$) into their components.
Fig. 5. Training errors (${NMSE}_{x1}$-${NMSE}_{x2}$ and ${NMSE}_{y1}$-${NMSE}_{y2}$) as a function of the virtual node interval $\theta$ when the mixing fraction $F$ is known as 0.5 in advance and $T$=5 ns.
Fig. 6. Training errors (Training errors (${NMSE}_{x1}$-${NMSE}_{x2}$ and ${NMSE}_{y1}$-${NMSE}_{y2}$) as a function of the mixing fraction $F$.
Fig. 7. (a$_1$)-(a$_2$) The samples of the time series data of the mixed optical chaos signals $U_{x}(n)$ and $U_{y}(n)$; (b$_1$) The samples of the time series data of $\beta _{1}C_{x1}(n)$ and $\beta _{1}C_{x1}^{'}(n)$; (b$_2$) The samples of the time series data of $\beta _{1}C_{y1}(n)$ and $\beta _{1}C_{y1}^{'}(n)$; (c$_1$) The samples of the time series data of $\beta _{2}C_{x2}(n)$ and $\beta _{2}C_{x2}^{'}(n)$; (c$_2$) The samples of the time series data of $\beta _{2}C_{y2}(n)$ and $\beta _{2}C_{y2}^{'}(n)$. $\beta _{1}=\sqrt {F}$, $\beta _{2}=\sqrt {1-F}$.
Under $F$=0.5, $T$=5 ns, $\theta$=20 ps, $N$=250 and $L$=2000, Figs. 7(a$_1$)–7(c$_2$) displays the samples of time series data of the mixed chaotic signals ($U_{x}(n)$ and $U_{y}(n)$), the original chaotic signals ($\beta _{1}C_{x1}(n)$- $\beta _{2}C_{x2}(n)$ and $\beta _{1}C_{y1}(n)$- $\beta _{2}C_{y2}(n)$), and the trained reservoir outputs ( $\beta _{1}C_{x1}^{'}(n)$- $\beta _{2}C_{x2}^{'}(n)$ and $\beta _{1}C_{y1}^{'}(n)$- $\beta _{2}C_{y2}^{'}(n)$ ), respectively. One sees from Figs. 7(a$_1$)-(a$_2$) that the mixed signals $U_{x}(n)$ and $U_{y}(n)$ show chaotic sate, and their components are indistinguishable. As shown in Figs. 7(b$_1$)-(b$_2$), the values of $\beta _{1}C_{x1}(n)$ and $\beta _{1}C_{y1}(n)$ are highly similar to those of $\beta _{1}C_{x1}^{'}(n)$ and $\beta _{1}C_{y1}^{'}(n)$, respectively. Their corresponding training errors are 0.065 and 0.079, respectively (sees Fig. 4). From Figs. 7(c$_1$) and 7(c$_2$), the values of $\beta _{2}C_{x2}(n)$ and $\beta _{2}C_{y2}(n)$ are highly identical to those of $\beta _{2}C_{x2}^{'}(n)$ and $\beta _{2}C_{y2}^{'}(n)$, respectively. Their corresponding training errors becomes 0.06 and 0.08(sees Fig. 4), respectively. These results indicate that the mixed optical chaos signals $U_{x}(n)$ and $U_{y}(n)$ are effectively separated into their components. Their separation errors characterized by the training errors are very small and less than 0.08.
3.1.2 Mixing fractions $F_{1}$ and $F_{2}$ are unknown
Often, the mixing fractions $F_{1}$ and $F_{2}$ may be unknown in some applications such as multiple access in optical chaotic communication, optical chaotic time division multiplexing communication and so on. The methodology to separate signals without knowledge of the mixing fractions is presented in Fig. 2. We suppose that the samples of time series data of the mixed optical signals $U_{x}(n)$ and $U_{y}(n)$ are known in advance, but the mixing fractions $F_{1}$ and $F_{2}$ are unknown. When the two parallel reservoirs in the RC$_1$ system are trained by using $C_{x1}(n)$- $C_{x2}(n)$ and $C_{y1}(n)$- $C_{y2}(n)$, respectively, their outputs are obtained as $C_{x1}^{'}(n)$- $C_{x2}^{'}(n)$ and $C_{y1}^{'}(n)$- $C_{y2}^{'}(n)$. As a result, we rewrite two groups of mixed optical signals as
where $\beta _{11}=\sqrt {F_{1}^{'}}$, $\beta _{12}=\sqrt {1-F_{1}^{'}}$, $\beta _{21}=\sqrt {F_{2}^{'}}$ and $\beta _{22}=\sqrt {1-F_{2}^{'}}$. $F_{1}^{'}$ and $F_{2}^{'}$ are the mixing fractions to be estimated. In order to verify the effectiveness of the method for estimating these two mixing fractions, as shown in Fig. 2, in training, two parallel reservoirs in the RC$_{1}$ system are respectively given mixed optical signals $U_{x}(n)$ and $U_{y}(n)$, with a range of discrete actual values of $F_{1}$ and $F_{2}$ from 0.1 to 0.9 in intervals of 0.1. Namely, We using Eqs. (17)–(20) to construct two mappings, which are respectively from the predicted $F_{1}^{'}$ and $F_{2}^{'}$ to the original $F_{1}$ and $F_{2}$ via a fit a third-order polynomial function. These two mappings are shown in Fig. 8. One sees from this figure that the reservoir-predicted $F_{1}^{'}$ and $F_{2}^{'}$ almost appear linear change with the actual $F_{1}$ and $F_{2}$, respectively, denoting our method can accurately predict the mixing fractions.Fig. 8. Plots of the actual $F_{1}$ and $F_{2}$ used in the mixed optical chaotic signals versus the estimated $F_{1}^{'}$ and $F_{2}^{'}$ by using two parallel reservoirs in the RC$_1$ system, respectively. Here, each mixed chaotic signal is combined with two groups of optical chaotic signals.
As we mentioned above, having estimated the values of $F_{1}$ and $F_{2}$, we subsequently train two parallel reservoirs in the RC$_2$ system using the estimated $F_{1}^{'}$ and $F_{2}^{'}$, to further separate two groups of the mixed optical chaos signals. The actual $F_{1}$ and $F_{2}$ are assumed as 0.3 and 0.6, respectively. Correspondingly, the estimated $F_{1}^{'}$ and $F_{2}^{'}$ are respectively obtained as 0.32 and 0.618 by observation of Fig. 8. In such a case, the mixed chaotic signals $U_{x}(n)$ and $U_{y}(n)$ are further separated by using two parallel reservoirs in the RC$_2$ system (sees Fig. 2). Figure 9 gives the samples of time series data of the mixed optical chaos signals, their original components and their separated components output from two parallel reservoirs in the RC$_2$ system, where $F_{1}^{'}$=0.32, $F_{2}^{'}$=0.618. As shown in Figs. 9(a$_1$)–9(a$_2$), the mixed chaotic signals $U_{x}(n)$ and $U_{y}(n)$ with the estimated mixing fractions exhibit chaotic sate and their components are indistinguishable. One sees from Figs. 9(b$_1$)–9(b$_2$) that the trajectories of the original components $\beta _{11}C_{x1}(n)$ and $\beta _{21}C_{y1}(n)$ are highly similar to those of the separated components $\beta _{11}C_{x1}^{'}(n)$ and $\beta _{21}C_{y1}^{'}(n)$, respectively. As further seen from Figs. 9(c$_1$)–9(c$_2$), the traces of $\beta _{12}C_{x2}(n)$ and $\beta _{22}C_{y2}(n)$ are highly identical to those of $\beta _{12}C_{x2}^{'}(n)$ and $\beta _{22}C_{y2}^{'}(n)$, respectively. These results obtained from Figs. 8 and 9 indicate that the mixed optical chaos signals with unknown mixing fractions can effectively separated into their components by using two cascaded RCs based on the reservoir VCSELs.
Fig. 9. (a$_1$)-(a$_2$) The samples of the time series data of the mixed optical chaos signals $U_{x}(n)$ and $U_{y}(n)$; (b$_1$) The samples of the time series data of $\beta _{11}C_{x1}(n)$ and $\beta _{11}C_{x1}^{'}(n)$ ; (b$_2$) The samples of the time series data of $\beta _{21}C_{y1}(n)$ and the $\beta _{21}C_{y1}^{'}(n)$ ; (c$_1$) The samples of the time series data of $\beta _{12}C_{x2}(n)$ and $\beta _{12}C_{x2}^{'}(n)$ ; (c$_2$) The samples of the time series data of $\beta _{22}C_{y2}(n)$ and $\beta _{22}C_{y2}^{'}(n)$. Moreover, in (a$_1$)-(c$_2$), the estimated mixing fractions $F_{1}^{'}$ and $F_{2}^{'}$ are obtained as 0.32 and 0.618, respectively. $T$=5 ns and $\theta$=20 ps.
3.2 Separation of the mixed signal combined with three beams of optical chaotic signals
3.2.1 Mixing fractions $F_{1}$ and $F_{2}$ are known in advance
Three beams of the X-PCs emitted by the OP-Spin-VCSELs with the subscripts of 1-3 are linearly combined into the mixed signal $U_{x}(n)$. Three beams of the Y-PCs output by OP-Spin-VCSELs with the subscripts of 1-3 are linearly superimposed into the mixed signal $U_{y}(n)$. Therefore, $U_{x}(n)$ and $U_{y}(n)$ are expressed as
where $F_{1}$, $F_{2}$=$F$, $\beta _{1}=\sqrt {2F}$, $\beta _{2},\beta _{3}=\sqrt {0.5-F}$ and $\beta _{1}^{2}+\beta _{2}^{2}+\beta _{3}^{2}=1$. In the scheme proposed in Fig. 1, for each group of the mixed signals combined with three beams of optical chaotic signals, we first observe the predictive performances of two parallel reservoirs to the trajectories of the original components ( $\beta _{1}C_{x1}^{'}(n)$- $\beta _{3}C_{x3}^{'}(n)$ and $\beta _{1}C_{y1}^{'}(n)$- $\beta _{3}C_{y3}^{'}(n)$). The predictive performances are described by the corresponding training errors and displayed in Figs. 10–11, when the mixing fraction $F$ is known as 0.25 in advance and $L$ is fixed at 2000. Figure 10 presents the dependences of the training errors (${NMSE}_{x1}$-${NMSE}_{x3}$ and ${NMSE}_{y1}$-${NMSE}_{y3}$) on the period $T$ under $\theta$=20 ps. It is found from Fig. 10 that these training errors show a slightly oscillatory decrease with the increase of $T$ from 1 ns to 10 ns. For example, ${NMSE}_{x1}$ and ${NMSE}_{y1}$ decrease from 0.09 to 0.068 and from 0.078 to 0.059, respectively. ${NMSE}_{x2}$ and ${NMSE}_{y2}$ vary from 0.101 to 0.071 and from 0.083 to 0.062, respectively. ${NMSE}_{x3}$ and ${NMSE}_{y3}$ change from 0.095 to 0.069 and from 0.088 to 0.064, respectively. Figure 11 gives the dependences of these training errors on the virtual interval $\theta$ when $T$=5 ns. One sees from Fig. 11 that all training errors first show rapid linear-increase, then exhibit slow linear-increase with the change of $\theta$ from 3 ps to 50 ps. These training errors are no more than 0.093. Moreover, Fig. 12 shows the dependences of these training errors on the mixing fraction $F$, where $\theta$=20 ps and $T$=5 ns. From this figure, it is seen that these training errors appear slightly oscillatory change with the increase of $F$, indicating that the mixing fraction F has slight effect on these training errors. Notably, as shown in Figs. 10–12, the value of the ${NMSE}_{x3}$ is obviously bigger than that of the ${NMSE}_{x1}$ or ${NMSE}_{x2}$, and the ${NMSE}_{y3}$ also becomes obviously bigger than the ${NMSE}_{y1}$ or ${NMSE}_{y2}$. The reason is given as follows: as observed from Fig. 13, the original signal $\beta _{3}C_{x3}(n)$ has more complex and irregular chaotic dynamical behaviors than the original signal $\beta _{2}C_{x2}(n)$ or $\beta _{1}C_{x1}(n)$. Similarly, the chaotic dynamical behaviors of the original signal $\beta _{3}C_{y3}(n)$ appears more complex and irregular than those of the original signal $\beta _{2}C_{y2}(n)$ or $\beta _{1}C_{y1}(n)$. These result in the case that the predictions of the trained reservoir to $\beta _{3}C_{x3}(n)$ and $\beta _{3}C_{y3}(n)$ become more difficult. In the following, from Figs. 10–12, we take $T$, $\theta$ and $F$ as 5 ns, 20 ps, and 0.25, respectively, to illustrate the separations of the mixed signals $U_{x}(n)$ and $U_{y}(n)$ into their components.Fig. 10. Dependences of the training errors (${NMSE}_{x1}$-${NMSE}_{x3}$ and ${NMSE}_{y1}$-${NMSE}_{y3}$) on the period $T$, where the mixing fraction $F$ is known as 0.25 in advance and $\theta$=20 ps.
Fig. 11. Dependences of the training errors (${NMSE}_{x1}$-${NMSE}_{x3}$ and ${NMSE}_{y1}$-${NMSE}_{y3}$) on the virtual node interval $\theta$, the mixing fraction $F$ is known as 0.25 in advance and $T$=5 ns.
Fig. 12. Dependences of the training errors (${NMSE}_{x1}$-${NMSE}_{x3}$ and ${NMSE}_{y1}$-${NMSE}_{y3}$) on the mixing fraction $F$ when $\theta$=20 ps and $T$=5 ns.
Fig. 13. (a$_1$)-(a$_2$) The samples of time series data of the mixed signals $U_{x}(n)$ and $U_{y}(n)$, respectively; (b$_1$) The samples of time series data of $\beta _{1}C_{x1}(n)$ and $\beta _{1}C_{x1}^{'}(n)$; (b$_2$) The samples of time series data of $\beta _{1}C_{y1}(n)$ and $\beta _{1}C_{y1}^{'}(n)$; (c$_1$) The samples of time series data of $\beta _{2}C_{x2}(n)$ and $\beta _{2}C_{x2}^{'}(n)$; (c$_2$) The samples of time series data of $\beta _{2}C_{y2}(n)$ and $\beta _{2}C_{y2}^{'}(n)$; (d$_1$) The samples of time series data of $\beta _{3}C_{x3}(n)$ and $\beta _{3}C_{x3}^{'}(n)$; (d$_2$) The samples of time series data of $\beta _{3}C_{y3}(n)$ and $\beta _{3}C_{y3}^{'}(n)$. In (a$_1$)-(d$_2$), the mixing fraction $F$ is known as 0.25 in advance, $T$=5 ns and $\theta$=20 ps.
Figure 13 displays the samples of time series data of the mixed signals ($U_{x}(n)$ and $U_{y}(n)$), their original components ($\beta _{1}C_{x1}(n)$- $\beta _{3}C_{x3}(n)$ and $\beta _{1}C_{y1}(n)$- $\beta _{3}C_{y3}(n)$) and separated components($\beta _{1}C_{x1}^{'}(n)$- $\beta _{3}C_{x3}^{'}(n)$ and $\beta _{1}C_{y1}^{'}(n)$- $\beta _{3}C_{y3}^{'}(n)$) output from two parallel reservoirs presented in Fig. 1. As observed from Figs. 13(a$_1$) and 13(a$_2$), the mixed signals $U_{x}(n)$ and $U_{y}(n)$ with the known mixing $F$ as 0.25 exhibits chaotic sate and their components are indistinguishable. One sees from Figs. 13(b$_1$) – 13(d$_2$) that the trajectories of $\beta _{1}C_{x1}(n)$, $\beta _{2}C_{x2}(n)$ and $\beta _{3}C_{x3}(n)$ are highly similar to those of $\beta _{1}C_{x1}^{'}(n)$, $\beta _{2}C_{x2}^{'}(n)$ and $\beta _{3}C_{x3}^{'}(n)$, respectively. The traces of $\beta _{1}C_{y1}(n)$, $\beta _{2}C_{y2}(n)$ and $\beta _{3}C_{y3}(n)$ are highly identical to those of $\beta _{1}C_{y1}^{'}(n)$, $\beta _{2}C_{y2}^{'}(n)$ and $\beta _{3}C_{y3}^{'}(n)$, respectively. These results denote the mixed each group of the mixed chaotic signals combined with three beams of optical chaotic signals can be effectively separated by using two parallel reservoirs in single RC system given in Fig. 1, when the mixing fractions is known in advance.
3.2.2 Mixing fractions $F_{1}$ and $F_{2}$ are unknown
When the mixed chaotic signals $U_{x}(n)$ and $U_{y}(n)$ are linearly combined with three groups of optical chaotic signals emitted by the OP-Spin-VCSELs with the subscripts of 1-3, the scheme of their separations is presented in Fig. 2. As displayed in Fig. 2, we suppose that $U_{x}(n)$ and $U_{y}(n)$ are known in advance, but their mixing fractions are unknown. When the two parallel reservoirs in the RC$_1$ system are trained by using $C_{x1}(n)$- $C_{x3}(n)$ and $C_{y1}(n)$- $C_{y3}(n)$, respectively, the outputs of the reservoirs are obtained as $C_{x1}^{'}(n)$-$C_{x3}^{'}(n)$ and $C_{y1}^{'}(n)$-$C_{y3}^{'}(n)$, respectively. Under these conditions, $U_{x}(n)$ and $U_{y}(n)$ are described as
where $\beta _{11}=\sqrt {2F_{1}^{'}}$ and $\beta _{12}$, $\beta _{13}$=$\sqrt {0.5-F_{1}^{'}}$ under $\beta _{11}^{2}+\beta _{12}^{2}+\beta _{13}^{2}=1$; $\beta _{21}=\sqrt {2F_{2}^{'}}$ and $\beta _{22}$, $\beta _{23}$=$\sqrt {0.5-F_{2}^{'}}$ under $\beta _{21}^{2}+\beta _{22}^{2}+\beta _{23}^{2}=1$. $F_{1}^{'}$ and $F_{2}^{'}$ are the mixing fractions to be estimated by using two parallel reservoirs in the RC$_1$ system. In order to prove the effectiveness of the method for estimating these two mixing fractions, as shown in Fig. 2, in training, two parallel reservoirs in the RC$_1$ system are respectively given the mixed optical signals $U_{x}(n)$ and $U_{y}(n)$, with a range of discrete actual values of $F_{1}$ and $F_{2}$ from 0.1 to 0.4 in intervals of 0.05. In such a case, $U_{x}(n)$ and $U_{y}(n)$ are rewritten as We using Eqs. (23)–(26) to construct two mappings, which are respectively from the estimated $F_{1}^{'}$ and $F_{2}^{'}$ to the actual $F_{1}$ and $F_{2}$ via a fit a third-order polynomial function. These two mappings are displayed in Fig. 14. It is found from Fig. 14 that the reservoir-estimated mixing fractions $F_{1}^{'}$ and $F_{2}^{'}$ almost linearly change with the actual mixing fractions $F_{1}$ and $F_{2}$, respectively. Namely, $F_{1}^{'}$ and $F_{2}^{'}$ are very close to $F_{1}$ and $F_{2}$, respectively. These results indicate that our method can accurately predict the mixing fractions of the mixed chaotic signal combined with three groups of optical chaotic signals.Fig. 14. Plots of the actual $F_{1}$ and $F_{2}$ used in the mixed optical chaotic signals versus the estimated $F_{1}^{'}$ and $F_{2}^{'}$ by using two parallel reservoirs in the RC$_1$ system, respectively. Here, each mixed chaotic signal is combined with three groups of optical chaotic signals.
As shown in Fig. 2, using the estimated values of $F_{1}^{'}$ and $F_{2}^{'}$, we subsequently train two parallel reservoirs in the RC$_2$ system to further separate the mixed chaotic signals into their components. The actual $F_{1}$ and $F_{2}$ are assumed as 0.2 and 0.25, respectively. Correspondingly, the estimated $F_{1}^{'}$ and $F_{2}^{'}$ are respectively obtained as 0.217 and 0.257 by observation of Fig. 14. Under these conditions, the mixed chaotic signals $U_{x}(n)$ and $U_{y}(n)$ are further separated into their components by using two parallel reservoirs in the RC$_2$ system (sees Fig. 2). Under $F_{1}^{'}$=0.217 and $F_{2}^{'}$=0.257, Fig. 15 shows the samples of time series data of the mixed chaotic signals $U_{x}(n)$ and $U_{y}(n)$, their original components ($\beta _{11}C_{x1}(n)$- $\beta _{13}C_{x3}(n)$ and $\beta _{21}C_{y1}(n)$- $\beta _{23}C_{y3}(n)$) and their separated components ( $\beta _{11}C_{x1}^{'}(n)$- $\beta _{13}C_{x3}^{'}(n)$ and $\beta _{21}C_{y1}^{'}(n)$- $\beta _{23}C_{y3}^{'}(n)$) output by two parallel reservoirs in the RC$_2$ system. One sees from Figs. 15(a$_1$)–15(a$_2$) that $U_{x}(n)$ and $U_{y}(n)$ appear chaotic state and their components are nearly indistinguishable. As observed from Figs. 15(b$_1$) and 15(b$_2$), the trajectories of $\beta _{11}C_{x1}(n)$ and $\beta _{21}C_{y1}(n)$ are nearly similar to those of $\beta _{11}C_{x1}^{'}(n)$ and $\beta _{21}C_{y1}^{'}(n)$, respectively. One sees from Figs. 15(c$_1$) and 15(c$_2$) that the traces of $\beta _{12}C_{x2}(n)$ and $\beta _{22}C_{y2}(n)$ are almost identical to those of $\beta _{12}C_{x2}^{'}(n)$ and $\beta _{22}C_{y2}^{'}(n)$, respectively. As displayed in Figs. 15(d$_1$) and 15(d$_2$), the trajectories of $\beta _{13}C_{x3}(n)$ and $\beta _{23}C_{y3}(n)$ are highly similar to those of $\beta _{13}C_{x3}^{'}(n)$ and $\beta _{23}C_{y3}^{'}(n)$, respectively. These results presented in Fig. 15 denotes that each mixed chaotic signal combined with three beams of optical chaotic signals can be effectively separated into their components under the case that the mixing fractions are unknown.
Fig. 15. (a$_1$)-(a$_2$) The samples of the time series data of the mixed signals $U_{x}(n)$ and $U_{y}(n)$; (b$_1$) The samples of the time series data of $\beta _{11}C_{x1}(n)$ and $\beta _{11}C_{x1}^{'}(n)$; (b$_2$) The samples of the time series data of $\beta _{21}C_{y1}(n)$ and the $\beta _{21}C_{y1}^{'}(n)$; (c$_1$) The samples of the time series data of $\beta _{12}C_{x2}(n)$ and $\beta _{12}C_{x2}^{'}(n)$; (c$_2$) The samples of the time series data of $\beta _{22}C_{y2}(n)$ and $\beta _{22}C_{y2}^{'}(n)$.(d$_1$) The samples of time series data of $\beta _{13}C_{x3}(n)$ and $\beta _{13}C_{x3}^{'}(n)$; (d$_2$) The samples of time series data of $\beta _{23}C_{y3}(n)$ and $\beta _{23}C_{y3}^{'}(n)$. Moreover, in (a$_1$)-(d$_2$), the estimated mixing fractions $F_{1}^{'}$ and $F_{2}^{'}$ are obtained as 0.217 and 0.257, respectively. $T$=5 ns and $\theta$=20 ps.
Generally, under the case the mixing fraction is known in advance, each mixed chaotic signal linearly combined with two or three groups of optical chaotic signals can be effectively separated into their components, using two parallel reservoirs system based on the R-Spin-VCSEL presented in Fig. 1. As shown in Fig. 2, if the mixing fraction is unknown, the mixing fraction in each mixed chaotic signal can be estimated by using first reservoir based on the R-Spin-VCSEL. Using the value of the estimated mixing fraction, each mixed chaotic signal linearly combined with two or three groups of optical chaotic signals can be effectively separated by using second reservoir based on the R-Spin-VCSEL. In particular, whether the mixing fraction is unknown or known, in the same way, the mixed chaotic signals linearly combined with more than three beams of optical chaotic signals also can be effectively separated.
4. Conclusion
In this article, we propose the schemes and theories for the separation of the mixed chaotic signal linearly combined with many beams of optical chaotic signals under the case that the mixing fraction is known in advance or unknown, using the RC systems based on R-Spin-VCSEL. Here, each group of the mixed optical chaotic signals originate from the chaotic X-PC or Y-PC output from the optically pumped spin-VCSELs operation alone. Two parallel optical reservoirs are formed by the chaotic X-PC and Y-PC emitted by the optically pumped spin-VCSEL with both optical feedback and optical injection. We take the mixed chaotic signal linearly combined with no more than three beams of the chaotic X-PCs or Y-PCs as an example to explore its separation. When the mixing fractions are known as a certain value in advance, two groups of the mixed chaotic signals can be effectively separated by using two parallel reservoirs in the single RC system based on the R-Spin-VCSEL, and their separated errors characterized by the training errors are no more than 0.093. In some cases, when the mixing fractions are unknown, we use two cascaded RCs based on the R-Spin-VCSEL to separate each group of the mixed chaotic signals. The mixing fractions can be accurately estimated by using two parallel reservoirs in the first RC system based on the R-Spin-VCSEL. Using values of the estimated mixing fractions, two groups of the mixed chaotic signals can be effectively separated by two parallel reservoirs in the second RC system based on R-Spin-VCSEL, and their errors also are no more than 0.093. whether the mixing fraction is unknown or known, in the same way, the mixed chaotic signals linearly combined with more than three beams of optical chaotic signals also can be effectively separated. Therefore, optical RC has ability to act an efficient and robust method for separation of complex and high-dimensional optical chaos signals. The research results presented by our scheme can give an impact to development of novel principles of multiple access and demultiplexing in multi-channel chaotic cryptography communication.
In general, compared to the previously reported optical reservoir computing schemes, the advantages of our scheme presented by this paper are given as follows: accurate prediction for dual-channel high-dimensional and complex optical chaotic dynamics; the ability to act an efficient and robust method for separation of the mixed high-dimensional optical chaos signals when the mixing fraction is known or unknown. Our scheme also can give an impact to development of novel principles of multiple access and demultiplexing in multi-channel chaotic cryptography communication.
Funding
National Natural Science Foundation of China (62075168); Special project in key fields of Guangdong Provincial Department of Education of China the new generation of communication technology (2020ZDZX3052); Basic and Applied Basic Research Foundation of Guangdong Province (2020A1515011088); Innovation team project of colleges and universities in Guangdong Province (2021KCXTD051); Major Project of Basic Research and Applied Research for Natural Science in GuangDong Province (2017KZDXM086).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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