# Methods of Atmospheric Coherence Length Measurement
## 1 Introduction  
Conventional radio frequency (RF) systems suffer from vulnerability to interference, bandwidth limitations, high power consumption and frequency licensing, while free-space optical (FSO) communications are not subject to these limitations. The laser beam has a highly narrow divergence, which enables immunity against eavesdropping. Even free-space quantum optical communication can provide unconditional information security [1; 2]. Compared to radio spectrum resources, which are highly regulated, optical frequencies are not regulated and can be readily used. Additionally, the terminals in FSO are more compact and consume less energy than those in RF. Accordingly, the FSO system can be deployed in less time. The above advantages make FSO a promising communication technique [3; 4; 5; 6]. However, the FSO links are easily affected by deleterious atmospheric turbulence effects [7; 8; 9; 10; 11]. Atmospheric turbulence is generated due to the continuous mixing of air temperatures and the formation of turbulent motion under the action of wind. Atmospheric turbulence is actually a process of continuous energy transfer. As the wind speed and its Reynolds number increases, locally unstable air masses are generated. This is known as an eddy. The largest eddy, with a size of \(L_{0}\), is called the outer scale of turbulence (typically in dozens to hundreds of meters), while the smallest eddy, with a size of \(l_{0}\), is referred to as the inner scale of turbulence (typically in a few millimeters). Under the influence of inertial forces, the large eddies split into smaller ones. The eddy that is smaller than the inner scale \(l_{0}\) belongs to the viscous dissipation region, where the turbulent eddy is dissipated into heat. Eddies between \(l_{0}\) and \(L_{0}\) form the inertial subrange. The atmospheric turbulence, in turn, causes random perturbations to the atmospheric refractive index in the temporal and spatial domains [12; 13; 14]. After optical signals propagate through the free-space channel, both the wavefront phases and amplitudes will produce random fluctuations due to the presence of atmospheric turbulence. Therefore, the optical signals are distorted, which markedly degradesthe performance of FSO. In this regard, measuring the turbulence intensity and understanding the characteristics and transformation laws of turbulence are of great interest and importance to FSO. The atmospheric coherence length reflects the turbulence intensity and characterizes the magnitude of both the phase and amplitude fluctuations of the optical waves. Thus, the atmospheric coherent length measurement can be used for the evaluation of FSO links [4; 7; 8; 11; 12; 13; 15; 16; 17] as well as for other optical applications operating over atmosphere, such as optical imaging [18; 19], optical remote sensing [20; 21], etc. The atmospheric coherence length is defined as the coherence distance of the phase in the beam cross section after the optical beam is transmitted over the atmospheric channel in the presence of turbulence. This concept was first introduced by [PERSON] in his study of the structure function of the wavefront phase undulation [22]. The atmospheric coherent length is related to the integral over the propagation distance. Given that atmospheric turbulence is statistically homogeneous and isotropic, within the framework of the Kolmogorov model it can be expressed as \(r_{0}=\left(0.423k^{2}\int_{0}^{1}C_{n}^{2}\left(z\right)dz\right)^{-\frac{3} {5}}\), where \(k=2\pi/\lambda\) is the wave number with \(\lambda\) being the optical signal wavelength; \(C_{n}^{2}\) is the atmospheric refractive index structure parameter. This integral is applied along the entire path from the source to the destination \(L\). When considering a horizontal FSO link, \(C_{n}^{2}\) is often assumed to not vary and is a constant with \(r_{0}=\left(0.423C_{n}^{2}Lk^{2}\right)^{-\frac{3}{5}}\). The coherence length \(r_{0}\) captures the total fluctuations induced by atmospheric turbulence.