## 2 Principle of Operation
Specifically, the phase shift \(\varphi_{XPM}\) induced by XPM is characterized as Equation (8), where \(\gamma\) and \(L_{eff}\) represent the nonlinear coefficient and the effective length of the bidirectional HNLF, respectively. \(P_{XPM}\) is the pump power required to make \(2\gamma L_{eff}P_{XPM}=\pi\). \[\varphi_{XPM}=2\gamma L_{eff}P_{XPM} \tag{8}\]  
Equation (8) defines the phase shift \(\varphi_{XPM}\) induced by XPM, which is proportional to the fiber's nonlinear coefficient \(\gamma\), effective interaction length \(L_{eff}\), and pump power \(P_{XPM}\). By controlling the pump power and fiber parameters, we can ensure that the probe signal is properly modulated, thereby achieving signal reloading. Here, we adjust \(P_{AMI-R}\) and \(P_{AMI-L}\) by setting the proper gain of the EDFA and controlling the output power of the target, such that \(P_{AMI-R}\) and \(P_{AMI-L}\) are either 0 or \(P_{XPM}\) for different consecutive bit sequences. After interacting inside the bidirectional HNLF, \(P_{AMI-R}\) = 0 is reloaded as \(\varphi_{R-in}\) = 0, where \(\varphi_{R-in}\) denotes the phase of the reloaded BPSK signal, while \(P_{AMI-R}\
eq\) 0 is reloaded as \(\varphi_{R:in}=\pi\). Meanwhile, \(P_{AMI-L}\) = 0 is reloaded as \(\varphi_{R-t}\) = 0, where \(\varphi_{R-t}\) denotes the phase of the reloaded BPSK target, while \(P_{AMI-L}\
eq\) 0 is reloaded as \(\varphi_{R:t}=\pi\). Here, the phase shift brought by the self-phase modulation (SPM) effect is ignored due to the low power of the probe. In this way, the phase-amplitude-phase conversion is completed. Two reloaded BPSK signals are coherently interfered at the output of the 50:50 coupler. Assuming the electrical field of the forward output which is close to the right isolator is denoted as \(E_{fw}\), and the electrical field of the backward output which is close to the left isolator is denoted as \(E_{bw}\), the above process can be expressed as Equation (9), where \(E_{AMI-L}\) and \(E_{AMI-R}\) are the electrical fields of the output of the left DI and right DI, respectively. \[ \begin{bmatrix}E_{fw}\\ E_{bw}\end{bmatrix}  = \begin{bmatrix}\sqrt{2}/2&j\sqrt{2}/2\\ j\sqrt{2}/2&\sqrt{2}/2\\ \end{bmatrix}\begin{bmatrix}E_{C}\cdot e^{j\varphi_{R-t}}+E_{AMI-L}\\ E_{CC}\cdot e^{j\varphi_{R-in}}+E_{AMI-R}\end{bmatrix}  \tag{9}\] \[= \begin{bmatrix}\sqrt{2}/2&j\sqrt{2}/2\\ j\sqrt{2}/2&\sqrt{2}/2\\ \end{bmatrix}\begin{bmatrix}\sqrt{2}/2E_{CW}\cdot e^{j\varphi_{R-t}}+E_{AMI-L} \\ j\sqrt{2}/2E_{CW}\cdot e^{j\varphi_{R-in}}+E_{AMI-R}\end{bmatrix} \] \[= \begin{bmatrix}1/2E_{CW}\cdot e^{j\varphi_{R-t}}+\sqrt{2}/2E_{ AMI-L}-1/2E_{CW}\cdot e^{j\varphi_{R-in}}+j\sqrt{2}/2E_{AMI-R}\\ j1/2E_{CW}\cdot e^{j\varphi_{R-t}}+j\sqrt{2}/2E_{AMI-L}+j1/2E_{CW}\cdot e^{j \varphi_{R-in}}+\sqrt{2}/2E_{AMI-R}\end{bmatrix} \]  
Equation (9) calculates the forward output electric field \(E_{fw}\) near the right isolator after passing through the NOLM. It incorporates interference between clockwise/counterclockwise carriers and AMI signals, as well as XPM-induced phase shifts. Next, the forward output \(E_{fw}\) is filtered through the band pass filter (BPF) with a center frequency of \(\lambda_{\text{CW}}\). The electrical field of the output of the BPF, \(E_{BPF}\), is shown in Equation (10). \[E_{BPF} =1/2E_{CW}\cdot e^{i\varphi_{R-t}}-1/2E_{CW}\cdot e^{i\varphi_{R-in}} \tag{10}\] \[=1/2E_{CW}\cdot\left(e^{i\varphi_{R-t}}-e^{i\varphi_{R-in}}\right)\] \[=\left\{ \begin{aligned} 0,\varphi_{R-t}& =\varphi_{R-in}\pm 2k\pi\\ \pm E_{CW},\ \varphi_{R-t}&=\varphi_{R-in}\pm(2k+1)\pi \end{aligned} \right.\]  
Equation (10) represents the filtered output \(E_{BPF}\) after the BPF, which isolates the wavelength of the local CW \(\lambda_{CW}\). Equations (7)-(10) model the signal transmission, interference in NOLM, and filtering processes, which are critical for achieving phase-locking-free bit-recognition. It can be noted that the reloaded BPSK signal has \(\pi\) phase deflection because it goes through the 50:50 coupler twice.