## 3 Numerical Results and Discussions
It was shown in [46] that the partially coherent beams carrying twist phase have the ability to resist the turbulence-induced degradation, reducing the scintillation index (intensity fluctuation), and the larger the twist factor is, the more significant the effect is. That may be the reason the image quality and visibility are almost immune to turbulence in our analysis in the case of large twist factor. To characterize the image visibility and quality quantitatively, we adopt the parameter \(V\) used in [31]  
\[V=\frac{\left[g(u_{2x},u_{2y}=0)\right]_{\text{max}}-\left[g(u_{2x},u_{2y}=0) \right]_{\text{min}}}{\left[g(u_{2x},u_{2y}=0)\right]_{\text{max}}+\left[g(u_{ 2x},u_{2y}=0)\right]_{\text{min}}}, \tag{33}\]  
where the subscripts \"max\" and \"min\" denote the maximum value and minimum value of function \(g\), respectively. The larger the value of \(V\) is, the higher the visibility of the image is. A new parameter, the correlation coefficient (CC) is introduced to characterized image quality, defined as  
\[\text{CC}=\frac{E[\left[g-E(g)\right][H-E(H)]]}{\left\{E\left\{\left[g-E(g) \right]^{2}\right\}E\left\{\left[H-E(H)\right]^{2}\right\}\right\}^{1/2}}, \tag{34}\]  
where the function \(g\) and \(H\) are defined in Equations (33) and (34), respectively. \(E\{\}\) is an expectation operator, given by \(E\{f\}=\int f(u_{2x},u_{2y}=0)du_{2x}\). It follows from Equation (34) that the denominator can be regarded as the product of the standard deviation of the function \(g\) and \(H\), and the numerator is the covariance of the function \(g\) and \(H\). According to the Cauchy-Schwarz inequality, the value of CC is in the range from 0 to 1. When the CC approaches the upper limit, it implies that the ideal ghost image is obtained. As the value of CC decreases, the quality of the image deteriorates gradually. Figure 3a,b illustrate the variation of CC and \(V\) as a function of twist factor, respectively. It can be seen that the value of CC decreases monotonously as the twist factor increases both in free space and in oceanic turbulence. However, the differences of CC between in free space and in oceanic turbulence become smaller with the increase of twist factor, implying that the quality of the image with larger twist factor is less affected by the oceanic turbulence. Especially for the twist factor larger than 0.4 m\({}^{-1}\), the quality of the image seems to be immune to the turbulence, but in this situation, the image of the double slits is completely blurred. Thus, we set a critical value of CC which is 0.401 (see the image plotted with blue dashed dot line in Figure 2b). Only when the value is larger than 0.401, can the image of double slits be distinguished. As shown in Figure 3b, the visibility of the image increases as the twist factor increases both in free space and in oceanic turbulence, while in turbulence the visibility is always smaller than that in free space. The reason, as we stated previously, is that the different phase distortions imposed on the beams in two paths results in the visibility degeneration. In order to show the effects of the oceanic turbulence on the quality and visibility with different twist factors, the parameter named relative difference of quality or visibility is introduced  
\[\Delta CC=\frac{\left|CC_{f}-CC_{t}\right|}{CC_{f}}, \tag{35}\]  
\[\Delta V=\frac{\left|V_{f}-V_{t}\right|}{V_{f}}, \tag{36}\]  
where the subscripts \"\(f\)\" and \"\(t\)\" denote the quality or visibility in free space and in turbulence, respectively. The smaller \(\Delta CC\) or \(\Delta V\) is, the fewer effects that are induced by the turbulence. Figure 3c presents the changes in the \(\Delta CC\) and \(\Delta V\) with the twist factor. It clearly shows that the visibility or quality is less affected by the turbulence as the twist factor increases.