## 1 Introduction  
With the massive growth of cloud computing, 5G/6G, web-based applications, and other new types of services in recent years, the Internet traffic has been steadily expanding, which promotes optical interconnection networks (OINs) developing towards 400 G/800 G [1, 2]. In this trend, an optical transmission system with high spectral efficiency is required to accommodate a higher transmission rate. There are numerous strategies to significantly increase transmission rate, among which advanced coding and modulation, multi-dimensional multiplexing and forward error correction (FEC) coding play a vital role [3, 4]. Researchers have conducted substantial research on the topic of increasing system capacity. In terms of multi-dimensional multiplexing, ref. [5] proposed a method to realize a space-division multiplexing network in data center to overcome the optical network capacity crunch by using multi-core optical fiber. It performs effectively in data centers with short distances. However, it is unfavorable for OINs since inter-core crosstalk is becoming increasingly severe in long-haul transmission. Another significant aspect of improving capacity is using advanced coding and modulation. The author of [6] proposed a novel multi-band approach based on intensity modulation and direct detection (IM/DD), which combines the use of carrier-less amplitude-phase (CAP). This strategy not only decreases the transceiver's complexity, but it also enhances spectrum efficiency. Furthermore, forward error correction (FEC) coding has been broadly applied in OINs, which can tolerate the accumulated optical effects of optical fiber as well as the impairment induced by the defects of optoelectronic devices [7]. When employing the aforementioned advanced modulation format or FEC, the Shannon limit, on the other hand, cannot be typically achieved and the gap between the system capacity and the Shannon limit is roughly 1.53 dB [8; 9; 10]. To narrow that gap within the constraints of optical signal-to-noise ratio (OSNR), constellation shaping is a promising technique [11]. Constellation shaping is divided into probabilistic shaping (PS) and geometric shaping (GS). Both of them have the advantages of improving capacity and low complexity. PS approximates the Shannon limit by using Maxwell Boltzmann distribution. Among the adopted PS algorithms, probabilistic amplitude shaping (PAS) [12] is the most regularly adopted PS algorithm currently and was first proposed by [PERSON]. PAS is implemented by a fixed-to-fixed length encoder and decoder, called distributed matcher (DM)/de-matcher, which can convert independent and uniformly distributed input bits into a desired distribution symbol [13; 14]. PAS has been shown to operate in optical communication systems both theoretically and experimentally, however it is only appropriate for square constellations, which drastically limits its application. Unlike PS, GS leverages mutual information maximization as the objective function to determine optimal constellation coordinates. Its implementation is straightforward and unnecessarily requires the establishment of a sophisticated hardware framework. In the studies of GS, the authors of [15] proposed the geometrically shaped 32 QAM based on the generalized pair-wise optimization (PO). The experimental results indicated that GS-32 QAM outperforms PS-32 QAM in the low SNR scenario. Researchers in [16] designed a circular quadrature amplitude modulation (CQAM), which generalizes PS to finite prime fields, and enabling non-binary codes. The CQAM scheme could achieve joint shaping gain by combining PS and GS. Nevertheless, previous GS algorithms have a large complexity and are unable to overcome fiber nonlinear issues. Addressing the above challenges, iterative polar modulation was first introduced in [17] and demonstrated in [18; 19; 20].