## 1 Introduction  
Considering the issues of cost and complexity, data centers (DCs) have traditionally adopted the intensity modulation direct detection (IMDD) system. However, this scheme only uses the optical signal amplitude information for encoding, which greatly limits improving transmission efficiency [1]. With the explosive growth of network traffic, DCs face increasing pressure as IMDD systems struggle to improve the transmission bit rate to 200 Gbit/s. Therefore, establishing a simple efficient, and low-power interconnected optical network is extremely significant in this era of explosive data growth. The traditional coherent optical transmission system can make full use of all dimensions of the optical signal to transmit information and has been used in long-dual transmission, but it is not suitable for DCs due to its high cost and complexity.  
From the perspective of reducing power consumption and cost, the self-homodyne coherent optical communication system (SHC) is very attractive for the next-generation DC. In the SHC scheme, the modulated signal and local oscillator (LO) come from the same laser at the origin, and they are transmitted to the coherent receiver through the fiber. This method can eliminate the influence of laser frequency offset and minimize the impact caused by the linewidth of the laser. So a large linewidth laser with low power consumption can be used at the transceiver [2]. However, in the SHC optical communication system, random polarization state rotation will occur in LO transmission through optical fiber, and the rotated LO may experience power attenuation in a certain polarization state (such as the X polarization state), which cannot guarantee successful demodulation by the coherent receiver [3]. Therefore, the integrated coherent receiver (ICR) in the SHC system should be tailored for DC. To confront the polarization fading of LO, a polarization diversity coherent receiver (PDCR) using an automatic polarization controller (APC) [4; 5; 6; 7] or complementary PDCR [8] is deployed. The complementary PDCR structure composed of three hybrids is more complicated compared with APC. Meanwhile, compared to the traditional polarization controller which through cascaded waveplates or equivalent waveplates caused by electro-optical effects controls the input light to any state of polarization (SOP) [9; 10], the role of polarization control in the SHC system is to prevent the LO power from fading. Therefore, a simple, low-power automatic feedback polarization control loop can accomplish the goal of outputting equal powers of two orthogonal polarization states of light. There are two types of APC structures: waveplate-type polarization controllers (PC) and Mach-Zehnder interferometer (MZI) control loops. In this paper, these two types of APC structures are studied for the LO polarization control in the SHC system. Additionally, a robust and efficient optimization algorithm is crucial for practical applications of a polarization control system. Commonly used algorithms include the gradient descent (GD) algorithm, simulated annealing algorithm, and swarm iterative algorithm [11; 12; 13; 14]. Due to the simplified polarization control goal in the SHC system, the control algorithm can also be predigested. On the base of theoretical analysis and derivation of the control principle, the polarization control of the GD algorithm and greedy linear descent (GLD) algorithm are realized. According to this simplified polarization control objective, a variable-step GLD algorithm is proposed to automatically adapt an arbitrary input polarization state into the target location. Meanwhile, it is demonstrated that it has better tracking ability than GD and GLD algorithms.