## 2 Theoretical Model for Active Polarization Control System  
### Mathematical Model of Polarization Controller  
In this section, we establish the mathematical model of polarization controller (PC) to calculate the PER of the output laser of an active polarization control system. Figure 1 shows the schematic diagram of active polarization control. The seed source is a phase-modulated single-frequency (SF) linearly polarized fiber laser. The phase modulator (PM) is employed for the suppression of the stimulated Brillouin scattering (SBS) effect, which is one of the key factors that limit the power scaling of fiber lasers. The SF seed laser is phase-modulated through the radio frequency amplified signal, such as sinusoidal signal, white noise signal (WNS), and pseudo-random bit sequence (PRBS) signal, as a result of which the linewidth of the seed laser is broadened and therefore, the threshold of SBS is improved [24; 25]. In addition, in the master oscillator power-amplifier (MOPA)-structured fiber laser based on phase-modulated SF seed laser, other nonlinear effects such as self-phase modulation (SPM) and four-wave mixing (FWM) can also be effectively suppressed [24]. After passing through a PC and being amplified by the amplifiers, the laser is finally collimated into free space through the high-power collimator (CO). The output laser is split into two orthogonally polarized beams by a polarization beam splitter (PBS). The S-polarization beam is reflected by the PBS, and the P-polarization beam passes through it and then is sampled by a high-reflectivity mirror M1. A photo detector (PD) is employed to detect the sampled beam of M1 and generates a feedback voltage signal, which will be sent to the algorithm controller, and is processed by RMS-Prop algorithm. By performing the algorithm, the controller generates four-channel voltage signals to PC for controlling the SOP of the seed laser to compensate the effect of polarization noise during amplification process, and finally realize the linearly polarized output laser with high PER. In the active polarization control system, one type of all-fiber PC, which is composed of four piezoelectric ceramics, is preferred for its fast response and high precision. The fast axis azimuths of four piezoelectric ceramics are fixed. Controlled by the feedback voltage signals, the piezoelectric ceramics squeeze the fiber, and the induced birefringence produces phase retardation in two polarization directions. The retardations of four piezoelectric ceramics vary from 0 to 2\(\pi\), which is proportional to the applied voltages. Each piezoelectric ceramic can be described by the Jones matrix [15], as follows:  
\[A(\alpha,\delta)= \begin{bmatrix}\cos\alpha&-\sin\alpha\\ \sin\alpha&\cos\alpha\end{bmatrix}\begin{bmatrix}\exp(i\delta/2)&0\\ 0&\exp(-i\delta/2)\end{bmatrix}\begin{bmatrix}\cos\alpha&\sin\alpha\\ -\sin\alpha&\cos\alpha\end{bmatrix} , \tag{1}\]  
where \(\alpha\) is the azimuth angle of the fast axis and \(\delta\) is the retardation caused by feedback voltage. Usually, \(\alpha\) is 0\({}^{\circ}\) for the first and third piezoelectric ceramics, and 45\({}^{\circ}\) for the two others. In fact, adopting three piezoelectric ceramics is enough to realize the control of arbitrary SOP [26], and the fourth one is used to prevent the mutation of SOP during the  
Figure 1: Schematic diagram of active polarization control system of fiber laser: PM, phase modulator; PC, polarization controller; CO: collimator; PBS: polarization beam splitter; PD: photo detector; RF-AMP, radio frequency amplifier.