## 3 Numerical Results and Discussions
Let us now turn to investigate the influences of the oceanic turbulence such as the dissipation rate of mean-square temperature \(\chi_{T}\), the rate of dissipation of turbulent kinetic energy per unit mass of fluid \(\varepsilon\), and the ratio of temperature and salinity \(\omega\) on the characteristics of the image of double slits. Figure 4 illustrates the image of double slits, the coherence width, the quality and the visibility as a function of \(\chi_{T}\). The other turbulence parameters used in the calculation are \(\omega=-2.5\), \(\varepsilon=10^{-7}\)m\({}^{2}\)\(/\)s\({}^{3}\). As the value of \(\chi_{T}\) increases, the image of double slits blurs gradually. From the change of the coherence width with propagation distance (see in Figure 4b), the de-coherence effect becomes more and more apparent with the increase of \(\chi_{T}\), indicating that the strength of turbulence increases. As a consequence, the quality of image drops and so does the visibility. The beam with larger twist factor forms the image with higher visibility but with lower quality. Figures 5 and 6 show the image of double slits, coherence width, image quality, and visibility against the parameters \(\varepsilon\) and \(\omega\), respectively. The turbulence parameters used in Figures 5 and 6 are \(\omega=-2.5\), \(\chi_{T}=10^{-8}\) K\({}^{2}\)/s and \(\varepsilon=10^{-7}\)m\({}^{2}\)\(/\)s\({}^{3}\), \(\chi_{T}=10^{-8}\) K\({}^{2}\)/s, respectively. Compared to \(\chi_{T}\), the influences of the \(\varepsilon\) and the \(\omega\) on the quality or visibility of the image are milder. The formed image is not totally blurred in the varying range of \(\varepsilon\) or \(\omega\). Only when the value of \(\varepsilon\) is smaller than \(10^{-7}\) m\({}^{2}\)/s\({}^{3}\) or the \(\omega\) is larger than \(-2\), the quality of the image starts to drop. This varying is consistent with the changes of the coherence width shown in Figures 5b and 6b. The beam with a large twist factor always keeps the large coherence width at the detected plane, which leads to the quality of the image being lower than that of the small twist factor, while in the case of visibility this is reversed. Therefore, one can use the twist phase to improve the visibility of the image, under the situation that the quality of the image is acceptable. At last, we investigate the effects of the propagation distance on the quality and visibility of the image of the double slits. Figure 7 shows the variation of the image of the double slits and the quality and visibility with propagation distance both in free space and in turbulence. In the calculation, the  
Figure 3: (**a**) Variation of CC (quality of the image) with the twist factor; (**b**) Variation of \(V\) with the twist factor \(\mu\). (**c**) Changes of the relative difference of quality and visibility of the image of double slits with the twist factor. turbulent parameters are chosen to be \(\omega=-2.5\), \(\varepsilon=10^{-7}\,\mathrm{m}^{2}/\mathrm{s}^{3}\), \(\chi_{T}=10^{-8}\,\mathrm{K}^{2}/\mathrm{s}\). For the case of free space, the quality/visibility of image decreases/increases as the propagation distance increases, as expected. The reason is that the coherence width of the beam increases as the propagation distance increases due to the diffraction effect. Under the same propagation distance, the coherence width with a large twist factor is larger than that with a small twist factor (see in Figure 7d). Thus, the beam with a large twist factor has high visibility and low quality. In the presence of oceanic turbulence, the change rules of the image quality and visibility are similar to those in free space. However, the image quality for \(\mu=0\) in turbulence drops much faster than that in free space. As the twist factor increases, the variations of the image quality with propagation distance in turbulence are gradually close to their variations in free space, which means that the image with large twist factor is less affected by the turbulence.