## 3 Numerical Results and Discussions
Obviously, the parameter \(\Delta(z)\) stands for the increase of coherence width due to the diffraction effect. The first, second, and third term in square brackets in Equation (32) denote the diffractions induced by the twist factor, initial beam width, and initial coherence width during propagation, respectively. The last term is the turbulence-induced extra diffraction effect. However, when the turbulence is strong or the propagation distance is large, the behavior of the coherence width becomes much complicated. The last two terms, which denote the turbulence-induced de-coherence effect, have appreciate values. This effect results in the decrease of the coherence on propagation. Therefore, compared to the case in free space, the coherence width in turbulence increases slowly with propagation distance at first, reaches the maximum value, then starts to decrease with further increases of the propagation distance. One can determine the critical propagation distance where the coherence width reaches the maximum value through the formula: \(d(\sigma_{gz})/d(z)=0\). This critical value is related by the initial beam parameters and the turbulence parameters. In general, the stronger the turbulence is, the smaller the critical distance is. When the turbulence is strong enough or the propagation distance is sufficiently large, i.e., \(1/\Delta(z)<<1\), Equation (31) has an asymptotic expression \(\left(2/\pi^{2}k^{2}Tz\right)^{1/2}\), implying that the coherence width only depends on the turbulence and decreases with the increase of the propagation distance with \(z^{-1/2}\). Figure 2c shows the dependence of coherence width on the twist factor both with and without turbulence at \(z=10\) m, and Figure 2d shows the evolution of the coherence width with propagation distance in turbulence at several twist factors. The parameters in the calculation are the same in Figure 2a,b. It can be seen that the behavior of the evolution is consistent with our aforementioned analysis. As shown in Figure 2c, the coherence width increases with the increase of twist factor both with and without turbulence, while in turbulence, it increases much slower than that without turbulence. That is the reason why the quality of image deteriorates and visibility improves as the twist factor increases (see Figure 2a or Figure 2b). However, the visibility and quality in turbulence are worse than  
Figure 2: Image of the double slits (**a**) in free space and (**b**) in oceanic turbulence; (**c**) the variation of the coherence width with the twist factor both with and without turbulence. (**d**) Evolution of the coherence width with the propagation distance in turbulence at several values of twist factors. that without turbulence under the condition of the same beam parameters when the twist factor is small (\(\mu<0.2\) m\({}^{-1}\)). As the twist factor increases further, i.e., \(\mu>0.3\), the quality and visibility of the image in turbulence are almost the same as those without turbulence (compared to the blue dotted curve in Figure 2a,b). However, the quality of the image seems too low to be applied in practice for \(\mu\)\(>0.3\) m\({}^{-1}\). This phenomenon may be explained by the fact that, in our analysis, the statistics of the turbulence in two paths of GI setup are assumed to be statistically independent, which means that the idle and signal paths suffer the different phase distortions induced by turbulence during propagation at the same time. As a result, the statistical correlations of the two beams at \(z=z_{1}\) (just before the object) in the signal path and at \(z=z_{2}\) in the idle path are partially destroyed (without turbulence, the statistics of the two beams are exactly the same). Therefore, the quality and visibility all become worse in turbulence compared to those without turbulence.