## 2 Theory of GI with a TGSM Beam in Turbulent Ocean  
Figure 1 denotes the typical GI setup in an underwater environment. A TGSM beam generated from the source plane first goes towards a 50:50 non-polarized beam splitter (BS), separating into two equal portions. The reflected portion (idle path) propagates through oceanic turbulence and arrives at a detector (\(D_{2}\)). The transmittance portion (signal path) also passes through oceanic turbulence, and arrives at an object. A bucket detector (\(D_{1}\)) is placed against the object to receive all the light intensity scattered from the object. Note that the beam size of the TGSM beam before illuminating the object should be larger than that of the object and the \(D_{1}\) should put as close as the object to receive all the light from the object if the size of the object is smaller than that of the sensitive area of the detector. Both the distances from the source plane to \(D_{2}\) and from the source plane to \(D_{1}\) are \(z\), i.e., \(z=z_{1}=z_{2}\). The output signals from \(D_{1}\) and \(D_{2}\) are then sent to a coincidence circuit connected to a personal computer to measure the fourth-order correlation function (i.e., intensity fluctuations). According to [35; 40; 42], the cross-spectral density (CSD) function of a TGSM beam in the source plane is given as follows:  
\[W(\mathbf{r}_{1},\mathbf{r}_{2})=\left\langle E_{s}(\mathbf{r}_{1})E_{s}^{ \ast}(\mathbf{r}_{2})\right\rangle_{s}=\exp[-\frac{\mathbf{r}_{1}^{2}+\mathbf{ r}_{2}^{2}}{4\sigma_{I0}^{2}}-\frac{(\mathbf{r}_{1}-\mathbf{r}_{2})^{2}}{2 \sigma_{g0}^{2}}]\exp[ik\mu(\mathbf{r}_{1}\times\mathbf{r}_{2})_{z}]. \tag{1}\]  
Here \(\mathbf{r}_{i}\equiv(x_{i},y_{i})\), (\(i=1\), \(2\)) are two arbitrary position vectors. \(E_{s}\) is the electric field with random fluctuations. The asterisk and angular brackets with subscript \"\(s\)\" denote the complex conjugate and the ensemble average over source fluctuations. \(\sigma_{I0}\) and \(\sigma_{g0}\) are the beam width and transverse coherence width in the source plane. \(k=2\pi/\lambda\) is the wavenumber of the light beam, and \(\lambda\) is the wavelength. \(\mu\) is a real-valued parameter termed as the twist factor, which is a measure of the strength of the twist phase. \((\mathbf{r}_{1}\times\mathbf{r}_{2})_{z}\) denotes the component of the cross product with respect to the propagation z-axis, i.e., \((\mathbf{r}_{1}\times\mathbf{r}_{2})_{z}=x_{1}y_{2}-x_{2}y_{1}\). The twist factor \(\mu\) must satisfy the inequality \(\left|\mu\right|\leq 1/k\sigma_{g0}^{2}\) due to the non-negative requirement of Equation (1). Paraxial propagation of the electric field from the source plane to the output plane in linear turbulent media can be explored by the following extended Huygens-Fresnel integral  
\[E(\mathbf{u},z)=\int_{-\infty}^{\infty}E_{s}(\mathbf{r})h(\mathbf{r}, \mathbf{u})d^{2}\mathbf{r}, \tag{2}\]  
Figure 1: Schematic for ghost imaging with a twisted Gaussian Schell-model (TGSM) beam in a turbulent ocean. BS, beam splitter; \(D_{1},D_{2}\), detectors. where \(E\) denotes the random electric field in the output plane, and \(h(\mathbf{r},\mathbf{u})\) is the response function between the source plane to the output plane. According to Figure 1, the functions \(h_{1}\) in path one and \(h_{2}\) in path two can be written as  
\[h_{1}(\mathbf{r},\mathbf{u}_{1})=\frac{i}{\lambda z}H(\mathbf{u}_{1})\exp \biggl{[}-\frac{i\pi}{\lambda z}(\mathbf{u}_{1}-\mathbf{r})^{2}+\psi(\mathbf{ r},\mathbf{u}_{1},z)\biggr{]}, \tag{3}\]  
\[h_{2}(\mathbf{r},\mathbf{u}_{2})=\frac{i}{\lambda z}\exp\biggl{[}-\frac{i\pi} {\lambda z}(\mathbf{u}_{2}-\mathbf{r})^{2}+\psi(\mathbf{r},\mathbf{u}_{2},z) \biggr{]}, \tag{4}\]  
respectively. \(H(\mathbf{u}_{1})\) in Equation (3) is the transmission function of the object. \(\psi(\mathbf{r},\mathbf{u},z)\) in Equations (3) and (4) represents the random complex phase perturbation of spherical wave propagation from \((\mathbf{r},0)\) to \((\mathbf{u},z)\) induced by the oceanic turbulence.