## 1 Introduction
The isotropic Kolmogorov turbulence model is used in the above publications on ground-satellite and satellite-ground backhaul communication. However, HAP is located at a height of 17 to 32 km in the stratosphere. Suppose the isotropic Kolmogorov turbulence model in the troposphere is used. In this case, it cannot better describe the influence of eddy currents on light wave transmission in turbulent random media. There is an error compared with the actual situation. Primarily, when the beam propagates in the vertical or non-horizontal direction, the non-Kolmogorov effect of atmospheric turbulence is more apparent [1; 17]. It has not been reported that the HAP in the stratosphere adopts an anisotropic non-Kolmogorov turbulence model in the backhaul optical communication between ground and satellite. Additionally, the modulation technologies used in satellite laser communication include differential phase shift keying (DPSK), minimum frequency shift keying (MSK), Gaussian minimum frequency shift keying (GMSK), etc. For example, [PERSON] et al. used binary shift keying subcarrier intensity modulation and differential phase shift keying subcarrier intensity modulation (DPSK-SIM) to evaluate the bit error rate (BER) performance of free-space optical satellite downlink [18]. We studied the application of minimum frequency shift keying (MSK) modulation with coherent detection and Gaussian minimum frequency shift keying (GMSK) in satellite laser communication systems [19; 20]. So far, quadrature amplitude modulation (QAM) has attracted attention due to its advantages of achieving a high communication data rate without increasing system bandwidth. Compared with traditional quadrature amplitude modulation (QAM), rectangular quadrature amplitude modulation (RQAM) improves power efficiency and data transmission rate [21]. [PERSON] studied the BER performance of the FSO communication system using a rectangular quadrature amplitude modulation (RQAM) scheme in ground-to-satellite uplinkunder weak turbulence [1]. Then, based on rectangular quadrature amplitude modulation (RQAM), two new modulation techniques emerged, namely cross quadrature amplitude modulation (XQAM) and hexagonal quadrature amplitude modulation (HQAM). The peak-to-average power ratio (PAPR) of cross quadrature amplitude modulation (XQAM) with higher flexibility and hexagonal quadrature amplitude modulation (HQAM) with the densest two-dimensional (2D) encapsulated hexagonal lattice structure is lower than that of rectangular quadrature amplitude modulation (RQAM), which can improve spectral efficiency and energy efficiency [22; 23]. The rest of this paper is organized as follows: In Section 2, the system and channel model are presented. In Section 3, the bit error rate of the system is analyzed. In Section 4, the effects of hexagonal quadrature amplitude modulation (HQAM) and cross quadrature amplitude modulation (XQAM) on the bit error rate (BER) performance of the system are simulated and analyzed and compared with the ground-satellite (G-S) laser communication system using modulation technology and without a HAP in the satellite-ground laser system. The paper is concluded in Section 5.