# Recovery and Characterization of Orbital Angular Momentum Modes with Ghost Diffraction Holography
## 2 Principle and Methods  
### Generation and Recovery of OAM Modes  
Let us consider a light source created by the coherent light illumination on a rotating ground glass diffuser (RGGD). The speckled beam proceeding from the diffuser illuminates a spatial light modulator (SLM), which encodes the phase mask for vortex beam generation. The beam emanating from the SLM carries the signature of the OAM mode, which is spatially distorted into the random pattern without any direct signature of the mode. A conceptual schematic of the generation of a vortex beam with the proposed technique is illustrated in Figure 1. The speckle field carries the vortex information (\(V_{0}^{el}(\hat{\mathbf{r}})=exp(i\mathrm{i}\mathrm{i}\ell\phi)\)) with zero radial index and \(l\) azimuthal index, where \(\varepsilon\) and \(l\) represent the sign and magnitude of the topological charge (TC) of the specific OAM mode. The helicity of the specific mode is determined by \(\varepsilon\) with \(\varepsilon=1\) for an anticlockwise helical structure direction and \(\varepsilon=-1\) for a clockwise direction. As shown in Figure 1, the speckle field encoded with the specific modes propagates in the positive Z-direction in free space. The field distribution in a transverse plane at distance \(z\) from the source plane is given by [27],  
\[\psi_{l}(\mathbf{r})=\frac{\exp(i\mathbf{k}z)}{i\lambda z}\int\psi_{0}(\hat{\mathbf{r}}) \exp\left(i\sigma_{g}(\hat{\mathbf{r}})\right)V_{0}^{el}(\hat{\mathbf{r}})\exp\left[ \frac{i\mathbf{k}}{2z}\left(|\mathbf{r}|^{2}-2\mathbf{r}\cdot\hat{\mathbf{r}}+|\mathbf{r}|^{2} \right)\right]d\hat{\mathbf{r}} \tag{1}\]  
where \(\psi_{0}(\hat{\mathbf{r}})\) is the incident light field at the exit plane of source, \(\varphi_{g}(\hat{\mathbf{r}})\) the random phase introduced by the ground glass diffuser, \(\lambda\) the wavelength of the light source, \(k=\frac{2\pi}{\lambda}\) the wave number of light, \(z\) the propagation distance, and \(\hat{\mathbf{r}}\) and \(\mathbf{r}\) are the position coordinates at the source plane and detector plane, respectively. The free space propagating field distribution reaching the detector plane consists of speckle field characterized by Gaussian field statistics [28], where the information of the OAM modes is indistinguishable and scrambled in the random pattern. The cross-correlation of two identical copies of the spatially incoherent field distribution at the detector plane, with one copy \(\psi_{m}(\mathbf{r})\) with \(m\) having any positive or negative integer values and the other \(\psi_{0}(\mathbf{r})\) with a zero-order vortex mode can retain the signature of the OAM modes in its cross-correlation function [29]. Therefore, the cross-correlation of the field contributions \(\psi_{m}(\mathbf{r})\) and \(\psi_{0}(\mathbf{r})\) is expressed via the field correlation function,  
\[W(\mathbf{r}_{1},\mathbf{r}_{2})=\langle\psi_{m}^{*}(\mathbf{r}_{1})\psi_{0}(\mathbf{r}_{2})\rangle \tag{2}\]  
where \(\langle\ \rangle\) is the ensemble averaging and \(W(\mathbf{r}_{1},\mathbf{r}_{2})\) the second order correlation function of the field at the detector plane. Now, let us turn our attention to the measurement of the field correlation function \(W(\mathbf{r}_{1},\mathbf{r}_{2})\). By assuming the random field distribution at the detector plane obeys Gaussian statistics, the correlations of the spatial intensity fluctuations or fourth-order field correlation function can be expressed in terms of its second-order correlation function [28; 30] as,  
\[\langle\Delta I_{m}(\mathbf{r}_{1})\Delta I_{0}(\mathbf{r}_{2})\rangle=|W(\mathbf{r}_{1}, \mathbf{r}_{2})|^{2} \tag{3}\]  
where \(I_{m}(\mathbf{r}_{1})=\psi_{m}^{*}(\mathbf{r}_{1})\psi_{m}(\mathbf{r}_{1})\) and \(I_{0}(\mathbf{r}_{2})=\psi_{0}^{*}(\mathbf{r}_{2})\psi_{0}(\mathbf{r}_{2})\) are the respective intensities at the detector plane, \(\Delta I_{m}(\mathbf{r})=I_{m}(\mathbf{r})-I_{m}(\mathbf{r})\).