# Recovery and Characterization of Orbital Angular Momentum Modes with Ghost Diffraction Holography
## 2 Principle and Methods
### Generation and Recovery of OAM Modes
From Equation (3), it is evident that this intensity cross-correlation retrieves only the modulus of \(W(\mathbf{r}_{1},\mathbf{r}_{2})\); therefore, for the complex correlation function recovery and thereby the simultaneous recovery of amplitude and phase distributions of the OAM mode, we make use of the holography scheme, where a reference random field is combined with the propagated spatially incoherent field as shown in Figure 1. The independent addition of the holographic reference random field [31; 32]  
Figure 1: Conceptual schematic of the generation and propagation of a vortex beam with specific OAM mode. The superposition of an independent random reference field with the ghost diffraction fields makes an off-axis ghost diffraction holography scheme; RGGD: rotating ground glass diffuser; SLM: spatial light modulator. (\(\psi_{R}(\mathbf{r})\)) with the field contributions \(\psi_{m}(\mathbf{r})\) and \(\psi_{0}(\mathbf{r})\) modifies the correlation of intensity fluctuations in Equation (3) to:  
\[\langle\Delta I_{m}(\mathbf{r}_{1})\Delta I_{0}(\mathbf{r}_{2})\rangle=\left|\mathcal{W }(\mathbf{r}_{1},\mathbf{r}_{2})+W^{R}(\mathbf{r}_{1},\mathbf{r}_{2})\right|^{2} \tag{4}\]  
where \(I_{m}(\mathbf{r}_{1})=(\psi_{m}(\mathbf{r}_{1})+\psi_{R}(\mathbf{r}_{1}))^{*}(\psi_{m}(\bm {r}_{1})+\psi_{R}(\mathbf{r}_{1}))\) the resultant intensity contribution from \(\psi_{m}(\mathbf{r})\) and \(\psi_{R}(\mathbf{r})\), \(I_{0}(\mathbf{r}_{2})=(\psi_{0}(\mathbf{r}_{2})+\psi_{R}(\mathbf{r}_{2}))^{*}(\psi_{0}(\bm {r}_{2})+\psi_{R}(\mathbf{r}_{2}))\) the resultant intensity contribution from \(\psi_{0}(\mathbf{r})\) and \(\psi_{R}(\mathbf{r})\) with \(\Lambda I\mathbf{r}\) representing the fluctuation of intensity value with respect to its average value, \(W(\mathbf{r}_{1},\mathbf{r}_{2})=\psi_{m}^{*}(\mathbf{r}_{1})\psi_{0}(\mathbf{r}_{2})\) represents the complex cross-correlation function that bears the OAM mode in its correlation function and \(W^{R}(\mathbf{r}_{1},\mathbf{r}_{2})=\psi_{R}^{*}(\mathbf{r}_{1})\psi_{R}(\mathbf{r}_{2})\) the reference correlation function. Albeit the execution of ensemble averaging in the approach can be performed either with temporal averaging or with space averaging, we rely on space averaging by considering the time frozen speckle fields at the detector plane under the assumption of spatial stationarity [32; 33; 34; 35]. The spatial averaging is performed in the execution of the cross-correlation of intensity images by fixing the distance \(\Delta\mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2}\) equivalent to the pixel size at the detector plane, and by moving a specific window size in the intensity distribution \(I_{m}(\mathbf{r})\) over the entire intensity image \(I_{0}(\mathbf{r})\). Equation (4) is the contribution of two complex correlation functions, and thus described as correlation hologram, as in analogous to the conventional holography, but with the superposition of correlation functions [36; 37]. Therefore, by considering the change of variables \(\mathbf{r}_{1}=\mathbf{r}+\Delta\mathbf{r}\) and \(\mathbf{r}_{2}=\mathbf{r}\) the complex cross-correlation function \(W(\mathbf{r}_{1},\mathbf{r}_{2})\) in Equation (4) is expressed by the relation, (see Supplementary Materials)  
\[ \begin{split}\text{W}(\mathbf{r}+\Delta\mathbf{r},\mathbf{r})& \approx\iint\psi_{0}^{*}(\hat{\mathbf{r}}_{1})V_{0}^{al}(\hat{\mathbf{r}}_{1})\psi_{0 }(\hat{\mathbf{r}}_{2})\delta(\hat{\mathbf{r}}_{1}-\hat{\mathbf{r}}_{2})\text{exp}\left[-i \big{(}\varphi_{g}(\hat{\mathbf{r}}_{1})-\varphi_{g}(\hat{\mathbf{r}}_{2})\big{)} \right]\\ &\times\text{exp}\left[-i\frac{k}{2z}\left(|\hat{\mathbf{r}}_{1}|^{2 }-|\hat{\mathbf{r}}_{2}|^{2}\right)\right]\text{exp}\left[-i\frac{2\pi}{\lambda z} \Delta\mathbf{r}\cdot\hat{\mathbf{r}}_{1}\right]d\hat{\mathbf{r}}_{1}d\hat{\mathbf{r}}_{2} \end{split}  \tag{5}\]  
where \(\delta(\hat{\mathbf{r}}_{1}-\hat{\mathbf{r}}_{2})\propto\int exp\left[-i\frac{2\pi}{ \lambda z}(\hat{\mathbf{r}}_{1}-\hat{\mathbf{r}}_{2})\cdot\mathbf{r}\right]d\mathbf{r}\) the delta function.