## 3 Results  
### PWFs for Plane Waves  
Consider Plane Waves (L\({}_{\rm S}\) Well above the Turbulent Atmosphere) for Which \(\gamma=1\). W\({}_{\rm T}\)(u) for Different d, F\({}_{\rm N}\), and R is Plotted in Figure 2.  
For a point source such as a star, for which \(R=0\), the \(W_{T}(u)\) for different \(F_{N}\) and \(d\) values is shown in Figure 2. It can be seen that \(F_{N}\) influences the PWF of the tilt phase variance (\(W_{T}(u)\) for \(d=0\)) as expected. Due to the effect of diffraction [3, 4], the \(W_{T}(u)\) for \(d\) = 0 decreases slowly with \(u\), which deviates from the near-field value that is constant along the path, and the deviation becomes notable when \(F_{N}<1\), suggesting that the smaller the \(F_{N}\) is, the bigger the deviation is. This explains why the DIMM operates at the near-field  
Figure 1: The geometry and variables for the covariance functions.  
Figure 2: Normalized path-weighting function \(W_{T}(u)\) for plane waves. **(a)** Point sources with various \(F_{N}\) (\(F_{N}=D^{2}/\lambda L\)) and \(d\) (the distance between the aperture centers) values. **(b)** Distributed sources with various \(R\) (\(R=D_{\rm S}L/DL_{S}\)) and \(d\) values when \(F_{N}=1\).  
approximation (big \(F_{N}\)). Comparing \(W_{T}(u)\) for \(F_{N}=1\), \(d=0D\) (dash line) and \(F_{N}=1\), \(d=2D\) (dash dot line) shows that the effect of diffraction on the tilt covariance is far smaller than that on the tilt variance, which indicates that the effect of diffraction appears to be partly counteracted by the covariance term \(J_{0}(\gamma_{K}\,d/D)\) (see Equation (8)). For a distributed source such as a planet, the \(W_{T}(u)\) for different \(R\) and \(d\) values with a fixed Fresnel number \(F_{N}=1\) is shown in Figure 2b. It can be seen that the \(W_{T}(u)\) for \(d=0\) decreases significantly with \(u\). For larger \(R\) values, the downward trend of \(W_{T}(u)\) is more obvious (dot line versus solid line). By comparing the \(W_{T}(u)\) for \(d=0\) and \(d=2D\) with the same \(R\) value, again we can see that the effect of diffraction on the tilt covariance is smaller than that on the tilt variance.  
For plane waves, the \(W_{P}(u)\) values for different \(d\), \(F_{N}\), and \(R\) values are plotted in Figure 3.  
For a point source such as a star, for which \(R=0\), the \(W_{P}(u)\) for different \(F_{N}\) and \(d\) values is shown in Figure 3a. It can be seen immediately that the asymptotic behavior of the \(W_{P}(u)\) for \(d=0\) with respect to \(F_{N}\) is consistent with well-known asymptotic behavior [5], i.e., the PWFs of stellar scintillations (\(W_{P}(u)\) for \(d=0\)) are approximate to \(u^{5/6}\) for a small \(D\) (i.e., small \(F_{N}\) when \(\lambda L\) is fixed) and \(u^{2}\) for a large \(D\). When the selected \(F_{N}\) is satisfactory (approximately equals one, dash line), the \(W_{P}(u)\) for \(d=0\) is approximate to \(u^{5/3}\), which is needed to calculate the isoplanatic angle [5; 6]. In Figure 3a, there is a remarkable place where the curves of \(W_{P}(u)\) for \(d=0.5D\), \(F_{N}=1\) and \(d=0D\), \(F_{N}=5\) seem almost identical, both approximating \(u^{2}\), which is needed to calculate the tilt isoplanatic angle (\(\theta_{TA}\)) [1; 3]. For a distributed source such as the sun or a planet, the \(W_{P}(u)\) for different \(R\) and \(d\) values with a fixed Fresnel number \(F_{N}=1\) is shown in Figure 3b. It can be seen that the peak position of \(W_{P}(u)\) with the same \(d\) value moves toward the start of the path with the increase in the \(R\) value, and that the peak position of \(W_{P}(u)\) with the same \(R\) value shifts slightly toward the end of the path with the increase in the \(d\) value. When the receiver size \(D\) is very small and the distributed source is the sun or the moon, we will obtain the PWFs that are used in SHABAR [11] (SHAdow BAnd and Ranging) or LuSci [12] (Lunar Scintillometer), both of which are used to measure surface layer turbulence.