## 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 IS\({}_{1}\) 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 IS\({}_{1}\) and the fiber coupler (FC\({}_{1}\)). 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 FPBS\({}_{1}\). The y-PC is converted into x-PC by FR\({}_{1}\) and HWP\({}_{1}\), and then these two x-PCs are coupled into a single x-PC beam by the FC\({}_{1}\). Arbitrarily polarized light emitted from the DFB laser is initially split into x-PC and y-PC, using the FPBS\({}_{1}\). The y-PC is converted into x-PC through the use of FR\({}_{1}\) and HWP\({}_{1}\). Subsequently, these two x-PC beams are combined into a unified x-PC beam using  
FC\({}_{1}\). The unified x-PC beam is divided into three separate beams using a 1 \(\times\) 3 FBS\({}_{1}\). The amplitudes of these beams are individually denoted as \(E_{\rm inj1}\), \(E_{\rm inj2}\) and \(E_{\rm inj3}\). Each beam's intensity is controlled by NDF\({}_{1}\), NDF\({}_{2}\) and NDF\({}_{3}\), respectively. Here, the amplitudes \(E_{\rm inj1}\) and \(E_{\rm inj2}\) are modulated as optical logic inputs \(I_{1}\) and \(I_{2}\), respectively, while the amplitude \(E_{\rm inj3}\) is modulated as the optical clock signal \(I_{3}\). The three beams of x-PC are combined into a single beam using FC\({}_{2}\). This unified x-PC beam is then split into two separate x-PC beams using a 1 \(\times\) 2 FBS\({}_{2}\). One of these x-PC beams is directed into the M-VCSEL after passing through NDF\({}_{4}\). The second x-PC beam is converted into y-PC using FR\({}_{2}\) and HWP\({}_{2}\), before being injected into the M-VCSEL following passage through NDF\({}_{5}\). The polarized light emitted by the M-VCSEL is separated into x-PC and y-PC by the FPBS\({}_{2}\) 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 FR\({}_{3}\) and HWP\({}_{3}\). Under the influence of the applied electric field \(E_{\rm A}\), 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 OA\({}_{1}\). Conversely, the e-light output is converted into y-PC using FR\({}_{4}\) and HWP\({}_{4}\), before being introduced into the slave VCSEL (S-VCSEL) subsequent to amplification by OA\({}_{2}\). The time delay between the light emission from the PPLN to its arrival at the S-VCSEL is denoted as \(\tau\). Subsequently, the light emitted by the S-VCSEL undergoes polarization separation into x-PC and y-PC using FPBS\({}_{3}\), following its transmission through the IS\({}_{3}\). Here, the logic outputs \(O_{1}\) and \(O_{2}\) are derived by demodulating the x\({}_{\rm P}\)-PC and y\({}_{\rm P}\)-PC outputs from the PPLN and the x\({}_{\rm S}\)-PC and y\({}_{\rm S}\)-PC outputs from the S-VCSEL, respectively. The subscripts P and S denote the origins from the PPLN and S-VCSEL, respectively.