# Methods of Atmospheric Coherence Length Measurement
## 2 Retrieval of \(r_{0}\) Based on \(C_{n}^{2}\)
### Scintillation Method  
Due to the presence of atmospheric turbulence in the FSO link, the optical signal will experience arrival angle fluctuation at the receiving site. The temporal statistics related to the arrival angle fluctuation are known as the scintillation effect. It has been concluded that the scintillation is closely related to the atmospheric refractive index structure parameter \(C_{n}^{2}\). Within the framework of the Kolmogorov model, \(C_{n}^{2}\) can be expressed by [23; 24]  
\[C_{n}^{2}=E\sigma_{\lambda}^{2}D_{t}^{7/3}L^{-3}, \tag{4}\]  
Figure 1: Schematic sketch of the temperature structure measurement method. where \(E=4.48D_{r}/D_{t}\) with \(D_{r}\) and \(D_{t}\) representing the diameter of the receiving aperture and the diameter of transmitting aperture, respectively; \(L\) is the propagation distance from source to destination; \(\sigma_{\chi}^{2}\) indicates the log-amplitude variance of the receiving optical field. Generally, the scintillation index denoted by \(\beta_{I}^{2}\), which characterizes the light intensity scintillation intensity, is readily adopted instead of the log-amplitude variance. Under weak turbulence conditions (\(\hat{\rho}_{I}^{2}<1\)), the scintillation index can be expressed by the log-amplitude variance \(\sigma_{\chi}^{2}\). It follows that [12]  
\[\beta_{I}^{2}=\exp(4\sigma_{\chi}^{2})-1. \tag{5}\]  
In practice, the scintillation index is measured by  
\[\beta_{I}^{2}=\frac{\left\langle I^{2}\right\rangle-\left\langle I\right\rangle ^{2}}{\left\langle I\right\rangle^{2}}, \tag{6}\]  
where \(I\) indicates the intensity of optical signal. By measuring \(\beta_{I}^{2}\), \(C_{n}^{2}\) can be calculated based on Equations (4)-(6). As such, the laser scintillation method measures the path-integrated atmospheric refractive index structure parameter \(C_{n}^{2}\), and then performs the retrieval of the atmospheric coherence length \(r_{0}\). The scintillometer is a widely-used instrument to measure the scintillation index of optical signals traveling along a specific atmosphere path [12; 23; 24; 25; 26; 27; 28]. It is also commonly used as the ground-based equipment to evaluate new measurement approaches associated with \(C_{n}^{2}\)[25]. Up to now, multiple scintillometers with diverse configurations have been proposed, including the large area scintillometer with two separate LED discs [25], the saturation-resistant scintillometer with large incoherent transmitting and receiving optics [23], the inner-scale-measurement scintillometer with two sets of transceivers [27], etc. Some of them are already commercially available [29], but all scintillometers are designed by following the principle mentioned above. A schematic sketch of a traditional scintillometer is shown in Figure 2. The instrument adopts a layout that consists of a transmitter, receiver, and data processing module. The specific workflow of the traditional scintillometer is as follows. The optical beam is emitted by the transmitter. After the beam propagates over the turbulent atmosphere in the middle, it reaches the receiver on the right side and is converted to an electrical signal. Eventually the data-processing module carries out an analysis of the demodulated electrical signals, and provides an evaluation of the scintillation index. In concrete terms, the produced electrical signal \(U\) is proportional to the emitted optical signal intensity \(I\), thus Equation (5) becomes  
\[\beta_{I}^{2}=\frac{\left\langle I^{2}\right\rangle-\left\langle I\right\rangle ^{2}}{\left\langle I\right\rangle^{2}}=\frac{\left\langle U^{2}\right\rangle -\left\langle U\right\rangle^{2}}{\left\langle U\right\rangle^{2}}. \tag{7}\]  
Additionally, a deep machine learning algorithm has been presented to evaluate \(\beta_{I}^{2}\)[30]. With the scintillation index \(\beta_{I}^{2}\), one can calculate \(C_{n}^{2}\) by merely substituting Equation (6) into Equation (4). Now based on \(C_{n}^{2}\), the retrieval of the atmospheric coherence length \(r_{0}\) is performed.