## 3 Numerical Results and Discussions  
In this section, as numerical examples, we will study the effects of the twist factor and the oceanic turbulence (i.e., the rate of dissipation of mean-square temperature and temperature-salinity balance parameter) on the image visibility and quality in a lensless GI system based on the formulae derived in Section 2. For the convenience of analysis, the object in the calculation is the double slits whose transmission function is therefore given by  
\[H(\mathbf{u}_{1})=\left\{ \begin{array}{c}1,\frac{d}{2}-\frac{a}{2}\leq|u_{1 x}|\leq\frac{d}{2}+\frac{a}{2}\\ 0,\text{ others}\end{array} \right. \tag{29}\]  
where \(a\) is the width of each slit and \(d\) is the distance between two slits. Thus, the normalized ghost image function associated with background noise is defined as  
\[\mathrm{g}(\mathbf{u}_{2})=\frac{\int_{-\infty}^{\infty}\Gamma(\mathbf{u}_{1}, \mathbf{u}_{2})d^{2}\mathbf{u}_{1}}{\int_{-\infty}^{\infty}G^{(2)}(\mathbf{u} _{1},\mathbf{u}_{2})d^{2}\mathbf{u}_{1}}. \tag{30}\]  
The integration over \(\mathbf{u}_{1}\) in Equation (30) means that the detector \(D_{1}\) is the bucket detector without resolution. On substituting Equations (27)-(29) into Equation (30), we can numerically calculate the image of double slits contained in \(g\) function. Figure 2a presents the image of double slits in the absence of turbulence (\(M=0\)) for several different values of the twist factor. The parameters used in the calculation are chosen to be \(a=0.6\) mm, \(d=1.2\) mm, \(\sigma_{I0}=5.0\) mm, \(\sigma_{g0}=0.4\) mm, \(\lambda=632.8\) nm, and \(z=10\) m, and kept fixed unless other values are specified. For comparison, we plot in Figure 2b the corresponding image in the presence of oceanic turbulence. The turbulence parameters are \(\chi_{T}=10^{-8}\,\mathrm{K}^{2}/\mathrm{s}\), \(\omega=-2.5\), \(\varepsilon=10^{-7}\) m\({}^{2}/\mathrm{s}^{3}\) One can see that the twist factor has a significant effect on the image visibility and quality, regardless of whether it is in free space or oceanic turbulence. As the value of twist factor increases, the image quality decreases whereas the visibility increases. When the oceanic turbulence is introduced, it has negative effects on the formed image, i.e., the quality and visibility of the image were further degraded compared to those without turbulence. In general, the visibility and quality of the image are closely related to the transverse coherence width in the planes \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\). The smaller the coherence width is, the lower the visibility is and the higher the quality is [7]. In oceanic turbulence, the evolution of the coherence width of the TGSM beam with propagation distance is closely related to the second-order statistics of the complex phase perturbation induced by turbulence. According to [45], the relations between the coherence width \(\sigma_{gz}\) at distance z and the turbulence parameters can be expressed by the following analytical formula  
\[\sigma_{gz}=\sigma_{g0}\!\left(\frac{1}{\Delta(z)}+\frac{2\sigma_{g0}^{2}\pi^{ 2}k^{2}Tz}{3}\!\left(1+\frac{2}{\Delta(z)}\right)-\frac{2\sigma_{g0}^{2}\pi^{4 }k^{2}Tz^{2}4}{18\Delta(z)\sigma_{I0}^{2}}\right)^{-1/2}, \tag{31}\]  
with  
\[\Delta(z)=1+\left[\mu^{2}+\frac{1}{4k^{2}\sigma_{I0}^{4}}+\frac{1}{k^{2}\sigma_ {I0}^{2}\sigma_{g0}^{2}}+\frac{2\pi^{2}Tz}{3\sigma_{I0}^{2}}\right]\!z^{2}, \tag{32}\]where \(T=\int_{0}^{\infty}x^{3}\Phi_{n}(x,\eta,\varepsilon,\chi_{T},\omega)dx\); \(\Phi_{n}\) is the power spectrum of refractive index fluctuations in oceanic turbulence. Though Equation (31) describes the behavior of the coherence width in atmospheric turbulence in [45], it also applicable for the case in oceanic turbulence due to that the atmospheric and oceanic turbulence in the analysis all belong to linearly random media. It follows from Equation (31) that when the turbulence is very weak \(T<<1\) or without turbulence \(T=0\), it reduces to \(\sigma_{gz}=\sigma_{g0}\sqrt{\Delta(z)}\).