Objective  Affected by atmospheric turbulence, the laser produces scintillation, beam wander, and fluctuation of the angle-of-arrival. Beam wander causes the beam to deviate from the target detector and reduces the efficiency of energy transmission, affecting applications such as laser tracking, laser positioning, and optical communications, especially over long distances. Under weak fluctuation conditions, the traditional theory of laser atmospheric propagation can be relatively accurately described. There is still a lack of reliable experimental evidence as to whether the theory is accurate in several kilometers of propagation under deep turbulence near the ground. Most of the current theories and simulations are based on existing assumptions, and there is often a certain deviation from the actual situation, especially under deep turbulence, where the theory is not yet perfect. The distances of the beam wander experiments are all relatively short and do not support the theory associated with long distance propagation. Therefore, it is of great significance to summarize the relevant theories of beam wander under deep turbulence and carry out long-distance laser propagation experiments to reveal the differences between the theoretical model and the reality. The relevant data provide an experimental basis for evaluating the model accuracy or error, and optimizing or correcting the model.

  Methods  Firstly, the theory of the variance of the drift angle for different refractive index power spectral models in the moderately to strongly turbulent region is summarized and derived. Then, the experiment system of fold path laser propagation (Fig.1) was built to collect the echo spot data using a large-aperture telescope and the atmospheric coherence length by an atmospheric coherence length meter. Four days of experiments were carried out on the 1 km propagation path and three days on a 7 km propagation path. The daily variation of the standard deviation of drift angle is obtained by calculating the echo spot data and studied in comparison with the theory. Simultaneously, the variation of turbulence anisotropy during the experiment period is analyzed.

  Results and Discussions   On the 1 km and 7 km propagation paths, the daily variations of the standard deviation of drift angle and the atmospheric coherence length show some periodicity, with synchronized changes but opposite trends (Fig.2). The standard deviation of drift angle decreases with increasing atmospheric coherence length, showing good agreement with the theoretical variation curve (Fig.3). The Rytov variance is used to classify the turbulence into three turbulence states of weak, medium and strong. Thereinto, the 1 km propagation path passes through three turbulence states, and the degree of proximity to the different theoretical models varies in different states (Tab.2). The relative deviation of turbulence in different states is further analyzed. On the 1 km propagation path, the measured values are closest to the theory based on the Von Karman refractive index model, with an average relative deviation of about 18.20%. While on the 7 km propagation path, the theory based on the modified Rytov shows a good agreement with the measured values, with an average relative deviation of about 21.09%. On the 1 km and 7 km propagation path, the anisotropy factor R of the laser beam wander gradually converges to 1 as atmospheric coherence length decreases (Fig.6), which means the beam wander tends to be isotropic. The anisotropy factor can reflect the contribution of thermal convection and transverse wind to the turbulent energy injection, and its specific influence mechanism needs to be further explored. The appropriate model and method of simulation needs to be chosen according to the specific situation, which will help to improve the accuracy and reliability.

  Conclusions  In this paper, the experimental system of laser fold path propagation is decomposed into laser propagation and spot image propagation, and formulas for the variance of the drift angle under moderate to deep turbulence are summarized and derived. The measured values in the 1 km propagation path are closer to the theoretical values based on the Von Karman refractive index model, and the average value of the relative deviation over four days is 18.2%. The measured values on the 7 km propagation path are closer to the theoretical values based on the modified Rytov theoretical refractive index model, and the average value of the relative deviation over three days is 21.09%. The anisotropy factor R of the laser beam wander tends to 1 as r_0 decreases, which means the beam wander tends to be isotropic. Through this experiment, we have a clearer understanding of the laser echo beam wander characteristics under different propagation conditions, the range of the standard deviation of drift angle intervals and their deviation from the theoretical expectations, which is of great significance for the model selection and engineering application evaluation. The relevant data have a certain reference value for revealing the target-in-loop laser propagation mechanism under deep turbulence conditions, and provide indispensable experimental data for optoelectronic systems involving laser aiming, positioning, coupling, and other applications.