## 1 Introduction  
In recent years, underwater wireless optical communication have attracted a lot of attention due to the demands of underwater sensor networks, underwater unmanned devices, offshore exploration vessels and submarines [1; 2; 3; 4; 5]. However, turbulence in the seawater channel is an inevitable obstacle in the underwater wireless communication process, so it is necessary to reduce the signal attenuation and transmission capacity loss of the communication system caused by turbulence (diffraction) broadening of the beam [6; 7; 8]. To reduce the beam broadening caused by the beam edge diffraction and oceanic turbulence, quasi-diffracting free finite energy vortex beam become a new carrier of underwater wireless optical communication system. Because quasi-diffracting free finite energy vortex beam can be realized in experiments [9; 10; 11; 12; 13; 14]. Because for that reason, there have been many reports on the performance of quasi-diffracting free finite energy vortex beam resist the beam expansion and channel capacity loss caused by oceanic turbulence [15; 16; 17; 18; 19; 20; 21; 22; 23]. Such as, the crosstalk probability of vortex modes of Lommel-Gaussian beam in anisotropic oceanic turbulence [15] and the beam wander of Lommel Gaussian-Schell beam in unstable stratification weak ocean-turbulence [16] were investigated. The capacity of a free space optical communication link with vortex modes carrier that is owned by Bessel-Gaussian beam in weak turbulent ocean [17] was explored, and the transmitting probability of vortex modes carried by Bessel-Gaussian localized wave in anisotropic turbulent ocean was derived by developing the spectrum average mutual coherence function of the Bessel-Gaussian localized wave and the spectrum average coherence length of spherical wave [18]. The results in [17] show that Bessel-Gaussian beam has significant advantages over Laguerre-Gaussian beam in mitigating the effects of turbulence and enhancing the performance of the free-space optical communication link, as well as [18] show that Bessel-Gaussian localized wave is a more suitable beam for the vortex modes communication than conventional vortex beams. [19] indicates that the frozen wave has an optimal wavelength window in \"seawater transmission window\". The transmitting probability of vortex modes of finite energy Airy vortex beam in anisotropic weak oceanic turbulence was analyzed [20]. The model shows that finite energy Airy beam having vortex topological charge larger than 5 is more resistant to turbulent disturbance than Laguerre-Gaussian beam [20]. Beam wander model of a partially coherent and finite energy Airy beam in weak to strong oceanic turbulence was derived [21]. The self-healing property of Bessel-Gaussian beam in an underwater environment was verified experimentally [22]. Besides, the performance of underwater optical communications based on optical vortices in the presence of water flow, stagnant water, air bubbles, and temperature difference was evaluated [23]. The non-diffracting beams have an infinite energy, thus they are difficult to realize in experiment. It is worth noting that the quasi-diffracting free beams of Lommel-Gaussian, Lommel Gaussian-Schell, Bessel-Gaussian, localized Bessel-Gaussian, and finite energy Airy solve the experimental generation dilemma of non-diffracting beams. However, due to the introduction of Gaussian truncation function [24], the ability of the Gaussian truncation non-diffracted beam to resist diffraction is greatly reduced. The Mathieu beam, which was introduced by [PERSON] et al. [25], is a family of non-diffracting beams. Mathieu-Gaussian (MG) beam can be generated experimentally and is a very good approximation of non-diffracting Mathieu beam [26]. Therefore, this kind of beam should be the preferred one for underwater optical communication. That is, it is particularly necessary to study the anti-turbulence propagation performance of MG beam. However, to the best of our knowledge, the research for the propagation of MG beam in ocean turbulence was few reports.