High-Speed Optical Chaotic Data Selection Logic Operations with the Performance of Error Detection and Correction

Abstract

Based on the chaotic polarization system of optically injected cascaded vertical-cavity surface-emitting lasers (VCSELs), we propose a novel implementation scheme for high-speed optical chaotic data selection logic operations. Under the condition where the slave VCSEL (S-VCSEL) outputs a chaotic laser signal, we calculate the range of the applied electric field and the optical injection amplitude. We also investigate the evolution of the correlation characteristics between the polarized light output from the periodic poled LiNbO3 (PPLN) and the S-VCSEL as a function of the optical injection amplitude under different applied electric fields. Furthermore, we analyze the polarization bistability of the polarized light from the PPLN and S-VCSEL. Based on these results, we modulate the optical injection amplitude as the logic input and the applied electric field as the control logic signal. Using a mean comparison mechanism, we demodulate the polarized light from the PPLN and S-VCSEL to obtain two identical logic outputs, achieving optical chaotic data selection logic operations with an operation speed of approximately 114 Gb/s. Finally, we investigate the influence of noise on the logic outputs and find that both logic outputs do not show any error symbols under the noise strength as high as 180 dBw. The anti-noise performance of logic output O₁ is superior to that of optical chaotic logic output O₂. For noise strengths up to 185 dBw, error symbols in O₂ can be detected and corrected by comparison with O₁.

1. Introduction

As is well known, chaotic laser signals are highly sensitive to system initial conditions and external disturbances, exhibiting a high degree of randomness. Currently, they are widely applied in various fields such as chaotic laser radar, physical random numbers, chaotic neural networks, image recognition and encryption and optical reservoir computing [1,2,3,4,5,6,7,8]. Moreover, they also have potential applications in chaos computing.

Compared with the transmission capacity of optical chaotic networks, the switching capacity of them is still relatively weak. At present, dynamic packet switching technology for optical chaotic signals is an effective measure to enhance the switching capability of optical chaotic networks. The foundation for developing dynamic packet switching technology is to achieve optical chaotic signal processing such as multiplexing, demultiplexing, switching, regeneration and storage involved in optical chaotic signal packet switching nodes. However, the prerequisite for achieving the above chaotic signal processing is to develop low-power, low-loss and high-speed optical chaos computing. Therefore, chaos computing has attracted the great interest of many scholars.

Vertical-cavity surface-emitting lasers (VCSELs), as a microchip semiconductor laser, exhibit many advantages over edge-emitting lasers, in some areas such as low threshold current, single longitudinal mode operation, high modulation frequency, low cost and large-scale integration into two-dimensional arrays [9,10,11]. When VCSELs are subjected to optical injection or optical feedback, they easily generate chaotic x-PC and chaotic y-PC which are orthogonal to each other [12,13,14,15,16,17,18,19,20,21]. The dynamic behaviors such as polarization switching and polarization bistability in VCSELs can be induced by changing the pump current, the strength of the injected light or the detuning of the injected light [22,23,24,25,26]. Therefore, based on the high-dimensional chaotic system of VCSELs, optical chaotic logic operations with different functions can be designed and implemented by using its rich polarization modes.

In recent years, research in chaos computing has primarily focused on nanosecond-scale basic computations, with very few studies at the picosecond scale. For example, based on a VCSEL with tunable optical injection and polarization bistability, C Masoller et al. implemented an all-optical stochastic logic gate [9,10,11]. Based on electro-optic (EO) modulation theory and the VCSEL subjected to external optical injection, Zhong et al. achieved optoelectronic composite logic gates such as AND, NAND, OR, NOR, XOR and XNOR [25]. Based on chaos synchronization theory, Yan implemented all-optical logic gates [12]. Our previous works focused on reconfigurable chaotic logic operations [27,28]. The above implementation of chaotic logic gates almost lacks error detection and correction capabilities. However, the development of more complex combinational chaotic logic operations, such as optical chaotic data selection, is still relatively lagging behind, and few people pay attention to it. In 2019, using the polarization bistability in a VCSEL injected by a sampled grating distributed Bragg reflector laser, Zhong et al. first implemented picosecond-level optical chaotic data selection logic operations, and the operation speed was up to 10 Gb/s [29]. In Zhong’s scheme, the main reasons limiting the operational speed and noise resistance are twofold: insufficient mutual inhibition between x-PC and y-PC and the poor capability of the threshold demodulation mechanism for short bit durations. In the presence of significant noise, logic outputs may produce errors, but the scheme does not provide appropriate error detection and correction methods. Motivated by these issues, we propose a novel approach for achieving high-speed optical chaotic data selection logic operations with error detection and correction capabilities. In our approach, based on EO modulation theory, we achieve complete mutual inhibition between x-PC and y-PC. By employing an outstanding mean comparison demodulation mechanism, we enable the optical chaotic data selection logic operations to achieve a maximum speed of up to 114 Gb/s. Additionally, leveraging the generalized chaos synchronization theory, the system is capable of accurately detecting and correcting errors in chaotic logic outputs, with a noise strength limit as high as 185 dBw, exceeding the noise strength limit reported in reference [28] by approximately 100 dBw. These research results have great applications and reference values for the construction of high-speed optical chaos computing architecture and the development of optical chaotic network communication systems.

2. Theory and Model

Figure 1 depicts a detailed optical path diagram for implementing high-speed optical data selection logic operations. The optical isolator (IS) prevents feedback from the fiber polarization beam splitter (FPBS) into the laser. The fiber beam splitter (FBS) divides light into multiple beams. Meanwhile, the neutral density filter (NDF) controls light intensity. The Faraday rotator (FR) and half-wave plate (HWP) change the polarization direction of light. The optical amplifier (OA) boosts light power. Since the distributed feedback (DFB) laser can emit polarized light in any direction, to ensure that arbitrary polarized light from IS₁ can be parallelly injected into the x-PC and y-PC of the master VCSEL (M-VCSEL), it is necessary to convert the arbitrary polarized light into linearly polarized light by some optical passive devices placed between the IS₁ and the fiber coupler (FC₁). Here, we consider that the converted linear polarization light is the x-PC. Arbitrarily polarized light from the DFB laser is split into x-PC and y-PC by the FPBS₁. The y-PC is converted into x-PC by FR₁ and HWP₁, and then these two x-PCs are coupled into a single x-PC beam by the FC₁. Arbitrarily polarized light emitted from the DFB laser is initially split into x-PC and y-PC, using the FPBS₁. The y-PC is converted into x-PC through the use of FR₁ and HWP₁. Subsequently, these two x-PC beams are combined into a unified x-PC beam using FC₁. The unified x-PC beam is divided into three separate beams using a 1 × 3 FBS₁. The amplitudes of these beams are individually denoted as E_inj1, E_inj2 and E_inj3. Each beam’s intensity is controlled by NDF₁, NDF₂ and NDF₃, respectively. Here, the amplitudes E_inj1 and E_inj2 are modulated as optical logic inputs I₁ and I₂, respectively, while the amplitude E_inj3 is modulated as the optical clock signal I₃. The three beams of x-PC are combined into a single beam using FC₂. This unified x-PC beam is then split into two separate x-PC beams using a 1 × 2 FBS₂. One of these x-PC beams is directed into the M-VCSEL after passing through NDF₄. The second x-PC beam is converted into y-PC using FR₂ and HWP₂, before being injected into the M-VCSEL following passage through NDF₅. The polarized light emitted by the M-VCSEL is separated into x-PC and y-PC by the FPBS₂ after transmission through the IS2. The x-PC, designated as o-light, is directly introduced into the periodic poled LiNbO3 (PPLN). Conversely, the y-PC, referred to as e-light, is directed into the PPLN after polarization conversion facilitated by FR₃ and HWP₃. Under the influence of the applied electric field E_A, the PPLN exhibits the electro-optic effect. The o-light (x-PC) generated by the PPLN is directed into the S-VCSEL following amplification by OA₁. Conversely, the e-light output is converted into y-PC using FR₄ and HWP₄, before being introduced into the slave VCSEL (S-VCSEL) subsequent to amplification by OA₂. The time delay between the light emission from the PPLN to its arrival at the S-VCSEL is denoted as τ. Subsequently, the light emitted by the S-VCSEL undergoes polarization separation into x-PC and y-PC using FPBS₃, following its transmission through the IS₃. Here, the logic outputs O₁ and O₂ are derived by demodulating the x_P-PC and y_P-PC outputs from the PPLN and the x_S-PC and y_S-PC outputs from the S-VCSEL, respectively. The subscripts P and S denote the origins from the PPLN and S-VCSEL, respectively.

In our experimental setup, the applied electric field E_A used to modulate the control logic signal C is a square wave signal characterized by distinct low and high levels, denoted as E_A1 and E_A2, respectively, i.e., C = 0 when E_A = E_A1, and C = 1 if E_A = E_A2. Under the appropriate conditions of E_A and utilizing a suitable logic output demodulation mechanism, data selection logic operations can be effectively implemented, i.e., $O_1\left(t\right)=I_1\left(t\right)\overline{I_3}\left(t\right)+I_2\left(t\right)I_3\left(t\right)$. According to the theory of generalized chaos synchronization, when the S-VCSEL undergoes strong light injection, its output polarization light exhibits high correlation with the polarization light emitted from the PPLN. Therefore, it is feasible to establish $O_2\left(t+\tau \right)=O_1\left(t\right)=I_1\left(t\right)\overline{I_3}\left(t\right)+I_2\left(t\right)I_3\left(t\right)$. This expression signifies the successful implementation of delay storage in data selection logic operations.

Under the influence of spontaneous emission noise, based on the SFM of VCSEL, the rate equation of M-VCSEL subjected to external optical injection can be expressed as follows:

$$\begin{array}{cc}\hfill \frac{d{E}_{Mx}(t)}{dt}=& k(1+ia)[N_M{E}_{Mx}(t)+i{n}_M{E}_{My}(t)-E_{Mx}(t)]-i({\gamma }_p+\Delta \omega )E_{Mx}(t)\hfill \\ & -{\gamma }_a{E}_{Mx}(t)+\sqrt{{\beta }_{sp}{\gamma }_e{N}_M}{\xi }_x+k_x{E}_{inj}\hfill \end{array}$$

(1)

$$\begin{array}{cc}\frac{d{E}_{My}(t)}{dt}=& k(1+ia)[N_M{E}_{My}(t)-i{n}_M{E}_{Mx}(t)-E_{My}(t)]+i({\gamma }_p-\Delta \omega )E_{My}(t)\hfill \\ & +{\gamma }_a{E}_{My}(t)+\sqrt{{\beta }_{sp}{\gamma }_e{N}_M}{\xi }_y+k_y{E}_{inj}\hfill \end{array}$$

(2)

$$\begin{array}{cc}\frac{d{N}_M(t)}{dt}=& -{\gamma }_e[N_M(t)(1+{\left|E_{Mx}(t)\right|}^2+{\left|E_{My}(t)\right|}^2)]+{\gamma }_e{\mu }_M\hfill \\ & -i{\gamma }_e{n}_M[E_{My}(t)E_{Mx}^{*}-E_{Mx}{E}_{My}^{*}(t)]\hfill \end{array}$$

(3)

$$\begin{array}{cc}\frac{d{n}_M(t)}{dt}=& -{\gamma }_s{n}_M(t)-{\gamma }_e{n}_M(t)({\left|E_{Mx}(t)\right|}^2+{\left|E_{My}(t)\right|}^2)\hfill \\ & -i{\gamma }_e{N}_M(t)[E_{My}(t)E_{Mx}^{*}(t)-E_{Mx}(t)E_{My}^{*}(t)]\hfill \end{array}$$

(4)

In the above formulas, the subscript M represents M-VCSEL; the subscripts x and y represent x-PC and y-PC, respectively; N is the total carrier concentration, while n is the difference in concentration between carriers with spin-up and spin-down; E is the normalized amplitude, and $E=\sqrt{\mathrm{g}/{\gamma }_e}A$, where g is the differential material gain; A is the slowly varying amplitude; γ_e is the nonradiative carrier relaxation rate; a is the linewidth enhancement factor; k is the field decay rate; γ_s is the spin-flip relaxation rate; γ_a is the linear dichroism; γ_p is the linear birefringence; μ_M is the normalized bias current of the M-VCSEL; $\sqrt{{\gamma }_e{N}_M{\beta }_{sp}}$ denotes the noise term, with β_sp being the spontaneous emission factor, and noise strength N_oi = 10lg($\sqrt{{\gamma }_e{N}_M{\beta }_{sp}}$)². ξ_x and ξ_y are two independent Gaussian white noises with a mean value of 0 and a variance of 1; k_x and k_y are the injection strength for x-PC and y-PC, respectively; $E_{inj}=\sqrt{\mathrm{g}/{\gamma }_e}{A}_{inj}$ is the sum of E_inj1, E_inj2 and E_inj3, where A_inj is the slowly varying amplitude of the injected field; Δω = ω_inj − ω_ref is the detuning between the frequency of the injected field and the reference frequency of the M-VCSEL; ω_inj is the angular optical frequency of the injected field; the reference optical frequency ω_ref is defined as (ω_x + ω_y)/2, where ω_x = −γ_p + aγ_a and ω_y = γ_p − aγ_a are the optical frequency of the x-PC and the y-PC of the free-running VCSEL.

The x-PC and y-PC emitted by the M-VCSEL are injected into the PPLN as the initial input of the o-light and the e-light, respectively, so we have

$$U_{o,e}(0,t)=\sqrt{\frac{\hslash {\omega }_{0}V}{S_A{T}_L{v}_c{n}_1{n}_2}}{E}_{Mx,My}(t)$$

(5)

where U_o and U_e are the amplitudes of the o-light and the e light, respectively; ℏ is the Planck constant; S_A is the effective area of the light spot; V is the volume of the active layer of the VCSEL; v_c is the light velocity in a vacuum; T_L = 2n_gυ_c/L_v refers to the round trip time in the laser cavity; L_v is the length of the laser cavity; n_g is the effective refractive index of the laser active layer; ω₀ is the central frequency of the M-VCSEL; n₁ and n₂ are the undisturbed refractive indices of the o-light and the e-light, respectively. With the phase mismatch and the weak second-order nonlinear effect, the analytical solutions of the wave coupling equations of the linear EO effect for the two PCs in the PPLN are written as follows:

$$U_{o,e}(L,t)={\rho }_{x,y}(L,t)\exp (i{\beta }_{0}L)\exp [i{\varphi }_{x,y}(L,t)].$$

(6)

where L is the length of the PPLN; ρ_x,y, β₀ and ϕ_x,y are presented in Ref. [30]. The amplitudes of x_P-PC and y_P-PC after undergoing electro-optic modulation in PPLN can be represented as follows:

$$E_{Px,Py}(t-\tau )=\sqrt{\frac{S_A{T}_L{\nu }_c{n}_{1,2}}{\hslash {\omega }_{0}V}}{U}_{o,e}(L,t-\tau )$$

(7)

The rate equations for the S-VCSEL subjected to strong optical injection with a delay of τ can be represented as follows:

$$\begin{array}{cc}\frac{d}{dt}\left(\begin{array}{c}{E}_{Sx}(t)\\ E_{Sy}(t)\end{array}\right)=& k(1+ia)[N_S(t)-1]\left(\begin{array}{c}{E}_{Sx}(t)\\ E_{Sy}(t)\end{array}\right)\pm ik(1+ia)n_S(t)\left(\begin{array}{c}{E}_{Sy}(t)\\ E_{Sx}(t)\end{array}\right)\mp ({\gamma }_a+i{\gamma }_p)\left(\begin{array}{c}{E}_{Sx}(t)\\ E_{Sy}(t)\end{array}\right)\hfill \\ & +\sqrt{{\beta }_{sp}{\gamma }_e{N}_S}\left(\begin{array}{c}{\xi }_x\\ {\xi }_y\end{array}\right)+\left(\begin{array}{c}{k}_{injx}\\ k_{injy}\end{array}\right)\left(\begin{array}{c}{E}_{Px}(t-\tau )\\ E_{Py}(t-\tau )\end{array}\right)\exp (-i{\omega }_0\tau +i\Delta {\omega }_{S}t)\hfill \end{array}$$

(8)

$$\begin{array}{cc}\frac{d{N}_S(t)}{dt}=& -{\gamma }_e\{N_S(t)-{\mu }_S+N_S(t)({\left|E_{Sx}(t)\right|}^2+{\left|E_{Sy}(t)\right|}^2)\hfill \\ & +{in}_S(t)[E_{Sy}(t)E_{Sx}^{*}(t)-E_{Sx}(t)E_{Sy}^{*}(t)]\}\hfill \end{array}$$

(9)

$$\begin{array}{ll}\frac{d{n}_S(t)}{dt}=& -{\gamma }_s{n}_S(t)-{\gamma }_e\{n_S(t)({\left|E_{Sx}(t)\right|}^2+{\left|E_{Sy}(t)\right|}^2)\hfill \\ & +{iN}_S(t)[E_{Sy}(t)E_{Sx}^{*}(t)-E_{Sx}(t)E_{Sy}^{*}(t)]\}\hfill \end{array}$$

(10)

where the subscript S refers to the S-VCSEL; ∆ω_S is the center frequency detuning between the M-VCSEL and the S-VCSEL; k_injx and k_injy are the injection strength of the x_P-PC and y_P-PC, respectively; μ_S is the normalized bias current of the S-VCSEL.

3. Results and Discussions

We first calculate Equations (1)–(10) using the fourth-order Runge–Kutta method, employing the parameters listed in

Table 1. The main system parameters.

Parameters	Value	Parameters	Value
Linewidth enhancement factor a	3	Noise strength Noi	100 dBw
Field decay rate k	300 ns−1	Polar angle θ	π/2
Spin relaxation rate γs	50 ns−1	Azimuth φ	π/2
Nonradiative carrier relaxation γe	1 ns−1	Crystal temperature F	293 K
Dichroism γa	2 ns−1	Poled period of crystal Λ	5.8 × 105 m−1
Birefringence γp	60 ns−1	Duty ratio D	0.5
Delay time τ	5 ns	Crystal length L	15 mm
Effective area of light spot SA	7.0686 μm2	Refractive index of o-light n1	2.24
Length of laser cavity Lv	3 μm	Refractive index of e-light n2	2.17
Effective refractive index of active layer ng	3.6	Frequency detuning Δωs	0 GHz
Volume of active layer V	21.206 μm3	Frequency detuning Δω	150 GHz
Normalized bias current μM = μS	1.2	Wavelength of VCSEL λ	1550 nm
Optical injection strength kx = ky	10 ns−1	Optical injection strength kinjx = kinjy	500 ns−1
Threshold current of VCSEL Ith	6.8 mA	Bit duration time T	10 ps

. Since the PPLN cannot generate chaotic polarized light, we investigate the influence of the applied electric field E_A and the optical injection amplitude E_inj on the dynamics of x_S-PC and y_S-PC outputs from the S-VCSEL, illustrated in Figure 2a,b. The figures demonstrate that within the parameter space defined by E_A and E_inj, x_S-PC and y_S-PC exhibit various typical dynamic states such as single-period oscillation (P1), two-period oscillation (P2), quasi-periodic oscillation (QP) and chaotic behavior (CO). Our focus here is specifically on the evolution of the chaotic state. From Figure 2a, it can be observed that x_S-PC remains in a chaotic state within the regions corresponding to 0–42 kV/mm (E_A) and 0.1–10 (E_inj). As the E_A varies from 42 kV/mm to 100 kV/mm, the chaotic state of x_S-PC shows quasi-periodic variations with E_inj. Specifically, x_S-PC exhibits chaotic states when E_inj falls within the ranges such as 0.18–0.23, 0.89–0.93, 1.59–1.64, 2.3–2.35 and 3.01–3.06, with a period of approximately 0.7. Figure 2b indicates that y_S-PC is in a chaotic state across most regions corresponding to 0–42 kV/mm (E_A) and 0.07–10 (E_inj). As the E_A varies from 42 kV/mm to 58 kV/mm, the chaotic state of y_S-PC also exhibits quasi-periodic variations with E_inj.

Based on the theory of generalized chaos synchronization, when the S-VCSEL experiences intense optical injection from the PPLN, significant correlations may arise between x_P-PC and x_S-PC, as well as between y_P-PC and y_S-PC. Here, the correlation coefficient ρ_x,y is introduced to quantify the magnitude of the correlation between these pairs:

$${\rho }_{x,y}=\frac{\langle [I_{Px,Py}(t-\tau )-\langle I_{Px,Py}(t-\tau )\rangle ][I_{Sx,Sy}(t)-\langle I_{Sx,Sy}(t)\rangle ]\rangle }{{\left\{\langle {[I_{Px,Py}(t-\tau )-\langle I_{Px,Py}(t-\tau )\rangle ]}^2\rangle \langle [{[I_{Sx,Sy}(t)-\langle I_{Sx,Sy}(t)\rangle ]}^2]\rangle \right\}}^{1/2}}$$

(11)

where I_Px,Py(t − τ) = |E_Px,Py((t − τ))|², I_Sx,Sy(t) = |E_Sx,Sy(t)|² and the symbol < > denotes the time average. The correlation coefficient ρ_x,y ranges from 0 to 1, with higher values indicating a stronger correlation between the respective pairs.

Here, we calculate the dependence of the correlation coefficient ρ_x,y on the optical injection amplitude E_inj, as shown in Figure 3. Under the condition of E_A = 11 kV/mm as depicted in Figure 3a, as E_inj increases from 0 to 2.8, ρ_x gradually rises to 0.73. As E_inj further increases to 4.2, ρ_x shows a declining trend, dropping from 0.73 to 0.41. With E_inj increasing to 4.4, ρ_x rapidly rises from 0.41 to a peak of 0.97. As E_inj progresses from 4.4 to 10, ρ_x slowly decreases to 0.84. For ρ_y, as E_inj increases from 0 to 3.6, it swiftly rises from 0 to 0.82; with a further increase in E_inj to 10, ρ_y decreases slowly to 0.45. From Figure 3b, it is observed that when E_A = 25 kV/mm, the evolution curve of ρ_x with E_inj closely matches that in Figure 3a. However, the evolution of ρ_y differs: as E_inj increases from 0 to 3.6, ρ_y increases from 0 to 0.82; as E_inj continues to 4.2, ρ_y decreases slowly to 0.77; then, as E_inj rises from 4.2 to 4.4, ρ_y swiftly increases to a maximum of 0.98. Finally, as E_inj progresses to 10, ρ_y decreases gradually from 0.98 to 0.9. In our approach, a stronger correlation is advantageous for enhancing the storage performance of data selection logic operations. Therefore, it is concluded that under the condition where the E_A equals 11 kV/mm, ρ_x reaches a relatively high value around E_inj = 4.4, peaking at 0.97. Similarly, ρ_y achieves a relatively high value around E_inj = 3.6, reaching a maximum of 0.82. When the E_A is 25 kV/mm and E_inj approaches 4.4, both ρ_x and ρ_y reach relatively high values, with ρ_x peaking at 0.96 and ρ_y at 0.98.

To further determine the value range of E_inj for modulating logic inputs, we calculate the evolutions of the polarization bistability of x_P-PC, y_P-PC, x_S-PC and y_S-PC under applied electric fields E_A of 11 kV/mm and 25 kV/mm, respectively, as illustrated in Figure 4. From Figure 4a, it is evident that under an applied electric field E_A of 11 kV/mm, when E_inj ranges from 0 to 3.9, x_P-PC predominates, while y_P-PC is suppressed. However, the extent of mutual suppression between them is not pronounced. When E_inj varies between 3.9 and 4.5, both x_P-PC and y_P-PC display bistable loops, accompanied by an increased degree of mutual suppression. When E_inj ranges from 4.5 to 10, x_P-PC dominates completely, while y_P-PC is nearly entirely suppressed. In Figure 4b, when E_inj is below 3.9, the evolution of the polarization bistability for x_S-PC and y_S-PC is similar to that observed in Figure 4a. When E_inj ranges from 3.9 to 4.7, both x_S-PC and y_S-PC demonstrate bistable loops. As E_inj exceeds 4.7, the mutual suppression between x_S-PC and y_S-PC intensifies, leading to the complete suppression of y_S-PC by x_S-PC. From Figure 4c, it is evident that under an applied electric field E_A of 25 kV/mm, when E_inj ranges from 0 to 3.9, y_P-PC dominates, while x_P-PC is suppressed. When E_inj ranges from 3.9 to 4.7, both x_P-PC and y_P-PC exhibit bistable loops, with an increased degree of mutual suppression observed between them. When E_inj ranges from 4.7 to 10, y_P-PC dominates completely, while x_P-PC is entirely suppressed. The polarization bistability evolutions of x_S-PC and y_S-PC depicted in Figure 4d closely resemble those observed in Figure 4c. In our scheme, the mean comparison mechanism is employed to demodulate x_P-PC and y_P-PC and x_S-PC and y_S-PC to obtain logic outputs O₁ and O₂ (as described below). The bit error rate of O₁ and O₂ is influenced by the extent of mutual suppression between x_P-PC and y_P-PC, as well as between x_S-PC and y_S-PC. A higher degree of mutual suppression correlates with a lower bit error rate. Therefore, based on the above analysis, it can be inferred that the value of E_inj used for modulating the logic input should be greater than 4.7. Considering that maximizing the correlation coefficient ρ_x,y is desirable and ensuring that x_S-PC and y_S-PC maintain a chaotic state, we opt to choose appropriate values of E_inj within the range of 5.35 to 5.46 for modulating the logic inputs.

Assume that the optical injection amplitude E_inj is equal to the sum of three square wave signals, i.e., E_inj = E_inj1 + E_inj2 + E_inj3, where E_inj1 and E_inj2 are modulated as optical logic inputs I₁ and I₂, and E_inj3 is modulated as an optical clock signal I₃. Since the logic input can be either 0 or 1, there are eight possible input combinations: (0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0) and (1, 1, 1). Here, we modulate these eight input combinations using four-level signals E_injI, E_injII, E_injIII and E_injIV (where E_injI, E_injII, E_injIII and E_injIV are the four possible values of E_inj). Specifically, E_injI denotes (0, 0, 0), E_injII represents (0, 0, 1), (0, 1, 0) and (1, 0, 0), E_injIII for (0, 1, 1), (1, 0, 1) and (1, 1, 0) and E_injIV for (1, 1, 1). In this case, the four-level signals remain constant during a bit duration T. Here, T is equal to 10 ps, which means the data selection logic operates at a speed of 100 Gb/s. We take E_injI = 5.37, E_injII = 5.4, E_injIII = 5.43 and E_injIV = 5.46, which means when E_inj1, E_inj2 and E_inj3 are all equal to 1.79, I₁, I₂ and I₃ are all equal to 0, and if E_inj1, E_inj2 and E_inj3 are all equal to 1.82, I₁, I₂ and I₃ are all equal to 1. Additionally, when E_A = E_A1 = 11 kV/mm, C = 1, and if E_A = E_A2 = 25 kV/mm, C = 0. The demodulation mechanism for the logic output is described as follows: Suppose that the duration of C being equal to 0 or 1, denoted as T₁, is n’ times the bit duration T, i.e., T₁ = n’T. During the time duration T₁, the amplitude of x_P-PC, y_P-PC, x_S-PC and y_S-PC is sampled M times, where M is equal to T_1/h, and h represents the sampling interval. Within T₁, the average amplitudes of x_P-PC, y_P-PC, x_S-PC and y_S-PC are respectively denoted by A_Px, A_Py, A_Sx and A_Sy:

$$A_{\mathrm{P}\mathrm{x}}=\sum _{j=1}^M\frac{E_{P_{x(j)}}}{M}$$

(12)

$$A_{\mathrm{P}\mathrm{y}}=\sum _{j=1}^M\frac{E_{P_{y(j)}}}{M}$$

(13)

$$A_{\mathrm{S}\mathrm{x}}=\sum _{j=1}^M\frac{E_{S_{x(j)}}}{M}$$

(14)

$$A_{\mathrm{S}\mathrm{y}}=\sum _{j=1}^M\frac{E_{S_{y(j)}}}{M}$$

(15)

where E_Px(j), E_Py(j), E_Sx(j) and E_Sy(j) respectively represent the jth amplitude sampling value of x_P-PC, y_P-PC, x_S-PC and y_S-PC within T₁. By comparing A_Px and A_Py, as well as A_Sx and A_Sy, we can determine the logic outputs O₁ and O₂ within T₁. The demodulation rules are as follows: when A_Px > A_Py and A_Sx > A_Sy, O₁ = O₂ = 1; if A_Px ≤ A_Py and A_Sx ≤ A_Sy, then O₁ = O₂ = 0.

We calculate the numerical values of A_Px, A_Py, A_Sx and A_Sy across different time intervals, as presented in

Table 2. The numerical values of A_Px, A_Py, A_Sx and A_Sy in different time periods.

Time (ns)	APx	APy	Time (ns)	ASx	ASy
3–3.04	8.74 · 10−5	0.24	8–8.04	0.0078	0.92
3.04–3.05	0.25	5.83 · 10−4	8.04–8.05	0.56	0.35
3.05–3.06	2.53 · 10−5	0.25	8.05–8.06	0.31	0.75
3.06–3.1	0.25	5.64 · 10−4	8.06–8.1	0.5	0.19
3.1–3.11	4.01 · 10−5	0.24	8.1–8.11	0.19	0.79
3.11–3.16	0.25	5.66 · 10−4	8.11–8.16	0.48	0.14
3.16–3.18	5.21 · 10−5	0.24	8.16–8.18	0.11	1
3.18–3.19	0.24	3.98 · 10−4	8.18–8.19	0.5	0.33
3.19–3.2	1.02 · 10−4	0.25	8.19–8.2	0.3	0.82
3.2–3.24	0.24	5.75 · 10−4	8.2–8.24	0.51	0.2
3.24–3.29	7.08 · 10−5	0.25	8.24–8.29	0.04	0.9
3.29–3.3	0.25	5.83 · 10−4	8.29–8.3	0.56	0.35

. Figure 5 illustrates the optical data selection logic operations involving two identical logic outputs. Figure 5a depicts the relationship between the applied electric field E_A and the optical injection amplitude E_inj. Based on Figure 5b–e, and Table 2, it is evident that during the time interval from 3 ns to 3.04 ns, where C = 0, the logic inputs consist of (0, 0, 0), (0, 1, 0) and (0, 0, 1). In this scenario, A_Px = 8.74 · 10⁻⁵ < A_Py = 0.24, leading to the conclusion that O₁ = 0 (as shown in Figure 5f,g); after a delay of τ (τ = 5 ns), specifically within the time interval from 8 ns to 8.04 ns, it is observed that A_Sx = 0.0078 < A_Sy = 0.92. Consequently, we conclude that O₂ = 0 (as shown in Figure 5h,i). Therefore, it follows that $O_2(t+\tau )=O_1\left(t\right)=I_1\left(t\right)\overline{I_3}\left(t\right)+I_2\left(t\right)I_3\left(t\right)$. Between the time interval of 3.04 ns and 3.05 ns, the logic input corresponds to (1, 0, 0) with C = 1. Here, A_Px = 0.25 > A_Py = 5.83 · 10⁻⁴, resulting in O₁ = 1. After a delay of τ, specifically within the time interval from 8.04 ns to 8.05 ns, it is observed that A_Sx = 0.56 > A_Sy = 0.35. Consequently, O₂ = 1. Thus, we establish that $O_2(t+\tau )=O_1\left(t\right)=I_1\left(t\right)\overline{I_3}\left(t\right)+I_2\left(t\right)I_3\left(t\right)$. Between 3.05 ns and 3.06 ns, the logic input transitions to (1, 0, 1) with C = 0. Here, A_Px = 2.53 · 10⁻⁵ < A_Py = 0.25, leading to O₁ = 0. Between 8.05 ns and 8.06 ns, it is observed that A_Sx = 0.31 < A_Sy = 0.75. Consequently, O₂ = 0. Thus, we conclude that $O_2(t+\tau )=O_1\left(t\right)=I_1\left(t\right)\overline{I_3}\left(t\right)+I_2\left(t\right)I_3\left(t\right)$. The analysis principle for logic operations in other time intervals follows the same methodology; hence, it is not reiterated here. Based on the aforementioned analysis, it is concluded that $O_2(t+\tau )=O_1\left(t\right)=I_1\left(t\right)\overline{I_3}\left(t\right)+I_2\left(t\right)I_3\left(t\right)$. This signifies the successful implementation of data selection logic operations and their delay storage.

It is crucial to acknowledge that the fidelity of logic outputs is influenced by various system parameters, such as the bit duration time T and the noise strength N_oi. These factors can introduce bit errors, potentially compromising the integrity of logic operations. Therefore, the success probability (SP) is employed as a metric to quantify the reliability of data selection logic operations. The SP is calculated as the ratio of the number of correctly identified symbols to the total number of symbols present in the output of the logic operations. The evolution of success probability (SP) for the logic outputs O₁ and O₂ is analyzed across the parameter space defined by noise strength N_oi and bit duration time T, as depicted in Figure 6. Figure 6a illustrates that for N_oi values below 185 dBw, the SP of O₁ consistently remains at unity across a T range of 0.25 ps to 50 ps. Beyond 185 dBw, the SP of O₁ falls below unity; however, its anti-noise performance can be significantly improved by increasing T. Figure 6b reveals that for T values below 8.75 ps, the SP of O₂ is less than unity. In the scenario where N_oi is less than 180 dBw and T exceeds 8.75 ps, the SP of O₂ consistently achieves unity. Nevertheless, as N_oi surpasses 180 dBw, the SP of O₂ exhibits a gradual decline. From the analysis presented, it can be inferred that to ensure error-free operation for both O₁ and O₂, the bit duration time T must be set to at least 8.75 ps, and the noise strength N_oi should be maintained below 180 dBw. This configuration suggests that the maximum operational speed for optical chaotic data selection is approximately 114 Gb/s. Furthermore, the superior anti-noise performance of O₁ compared to O₂ is attributed primarily to the more pronounced mutual suppression observed between x_P-PC and y_P-PC as opposed to that between x_S-PC and y_S-PC, which is illustrated in Figure 5f–i.

In summary, it is evident that when the bit duration time T exceeds 8.75 ps and the noise strength N_oi falls within the range of 180 dBw to 185 dBw, the logic output O₂ may encounter errors, whereas O₁ remains error-free. Consequently, by comparing O₂ with O₁, it is feasible to detect erroneous symbols within O₂ and subsequently implement error correction procedures. In this section, we exemplify the system’s error detection and correction capabilities by examining noise strengths of 179 dBw, 185 dBw and 190 dBw. The remaining system parameters are consistent with those depicted in Figure 5. At a noise strength of 179 dBw, Figure 7a illustrates that the noise exerts a pronounced effect on the amplitudes of x_P-PC, y_P-PC, x_S-PC and y_S-PC. Nonetheless, leveraging the superior mean comparison demodulation mechanism, the logic outputs O₁ and O₂ exhibit no errors. At a noise strength of 185 dBw, Figure 7b indicates that the logic output O₁ remains error-free. Consequently, by comparing it with O₁, it is observed that the logic output O₂ incurs errors within the time frame of 8.11 ps to 8.16 ps. These errors are indicated with a black solid line and marked with the symbol ‘x’. The corrected logic outputs are depicted by a pink dotted line. When the noise strength is elevated to 190 dBw, as shown in Figure 7c, the logic output O₁ encounters errors within the interval of 3.18 ps to 3.19 ps. Similarly, logic output O₂ experiences errors in two distinct intervals: 8.1 ps to 8.11 ps and 8.18 ps to 8.19 ps. Thus, under these conditions, both logic outputs are susceptible to errors, leading to the failure of the logic operations.

4. Conclusions

Based on the chaotic polarization system of optically injected cascaded VCSELs, we propose a scheme for implementing high-speed optical chaotic data selection logic operations with the performance of error detection and correction. It has been observed that the S-VCSEL can emit chaotic laser output within a broader parameter space encompassing an applied electric field and optical injection amplitude. When the applied electric field is set to 11 kV/mm or 25 kV/mm and the optical injection amplitude is around 4.4, the polarized light emitted by the PPLN exhibits strong correlation with that emitted by the S-VCSEL. Moreover, both the PPLN and S-VCSEL demonstrate identical polarization bistability characteristics: under an applied electric field of 11 kV/mm, x-PC predominates, while y-PC is suppressed. Additionally, y-PC is entirely suppressed by x-PC when the optical injection amplitude exceeds 4.7. Under an applied electric field of 25 kV/mm, the predominant polarization is y-PC, with x-PC being suppressed. Similarly, when the optical injection amplitude exceeds 4.7, x-PC is completely suppressed by y-PC. Based on the aforementioned findings, we utilize the modulation of optical injection amplitude and an applied electric field as the logic input and control logic signal, respectively. Utilizing the outstanding mean comparison mechanism, we achieve two identical logic outputs through demodulating the polarized light from the PPLN and S-VCSEL. This enables the realization of high-speed optical chaotic data selection logic operations, achieving operational speeds of up to approximately 114 Gb/s. Through an examination of noise effects on the logic outputs, it was observed that neither logic output displayed any error symbols even under a noise strength as high as 180 dBw. Furthermore, the anti-noise performance of logic output O₁ was found to be superior to that of optical chaotic logic output O₂. Error symbols in O₂ can be detected and corrected by comparison with O₁ when the noise strength does not exceed 185 dBw.