# Experimental Study on Phase Scintillation of Optical Transmission in Atmospheric Turbulence
## 3 Experimental Measurements
### Gaussian Curvature  
The Gaussian curvature map shows the local curvature variations of the wavefront phase at each point, which can indirectly reflect the degree of wavefront distortion. In differential geometry, a positive curvature indicates that the surface at that point is spherical, a negative curvature indicates that the surface is saddle-shaped, and a zero curvature indicates that the surface is cylindrical or planar at that point [16]. Regions with greater distortion typically correspond to areas with higher absolute values of Gaussian curvature. The magnitude of the Gaussian curvature can be used to assess the degree of wavefront distortion. The calculation of Gaussian curvature generally relies on the second partial  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Sample**} & **Maximum \(RMS\) Value of the** & **Minimum \(RMS\) Value of the** & **Mean \(RMS\) Value of the** \\  & **Wavefront (\(\upmu\)m)** & **Wavefront (\(\upmu\)m)** & **Wavefront (\(\upmu\)m)** \\ \hline T1 & 17.98 & 11.72 & 16.50 \\ T2 & 39.46 & 9.42 & 14.98 \\ T3 & 16.99 & 15.73 & 16.12 \\ T4 & 16.55 & 12.37 & 15.16 \\ T5 & 19.21 & 15.13 & 17.11 \\  
derivatives of the surface. The measured phase data is converted into a matrix form for ease of calculation. Let the wavefront phase matrix be denoted by \(\Phi(x,y)\). Then, the Gaussian curvature can be calculated using the following Equation (19). \[K=\frac{\Phi_{xx}\Phi_{yy}-\Phi_{xy}^{2}}{\left(1+\Phi_{x}^{2}-\Phi_{y}^{2} \right)^{2}} \tag{18}\]  
Here, \(\Phi_{xx},\Phi_{yy},\Phi_{xy},\Phi_{x}\), and \(\Phi_{y}\) represent the second and first partial derivatives of the wavefront phase matrix \(\Phi(x,y)\), respectively, where \(x\) and \(y\) are the indices of the valid elements in the matrix rows and columns. The Gaussian curvature distribution map is drawn by calculating the measured data using Equation (18), and the wavefront distortion under different weather conditions is analyzed. Figures 10-12 show the Gaussian curvature maps for three links under different weather conditions, where (a-f) represent the Gaussian curvature maps for different weather conditions. The X and Y axes in the Figures 10-12 represent the horizontal and vertical positions on the wavefront sensor's receiving plane, with the units in pixels. The color in the right-side color scale indicates the magnitude of the Gaussian curvature at the corresponding position, with curvature units in \(\upmu\)m\({}^{-1}\). The brighter the color (yellow), the larger the curvature value. The darker the color (black), the smaller the curvature value, with negative values. Bright spots in the figure indicate areas  
curvature distribution under cloudy conditions shows little change compared to Figure 10a (sunny), remaining relatively uniform. This indicates that wavefront distortion is also minimal under cloudy conditions, similar to sunny conditions, with a minimal impact from atmospheric turbulence. In Figure 10c (light rain), the distribution is similar to that in Figure 10a,b (sunny and cloudy), but with increased curvature in the central region, as indicated by brighter central spots. This suggests that wavefront distortion increases under light rain compared to sunny and cloudy conditions. In Figure 10d (heavy rain), the Gaussian curvature map shows significantly increased fluctuations in the central region, indicating stronger wavefront distortion. The edge regions also show marked fluctuations, with distinct negative curvature areas. Heavy rain causes strong turbulence and scattering effects, significantly impacting wavefront quality. In Figure 10e (dust storm), the Gaussian curvature map exhibits significant local fluctuations, particularly in the central part of the image. Wavefront distortion is more pronounced under dusty conditions, likely due to the atmospheric dust particles causing more noticeable wavefront aberrations.