# Recovery and Characterization of Orbital Angular Momentum Modes with Ghost Diffraction Holography
## 3 Results and Discussion
### Experimental Results  
The sequential encoding of specific phase mask in the SLM generate the vortex beams with various OAM modes. The cross-correlation of CCD recorded intensity fluctuations retrieves the correlation hologram at the detector plane, and the acquired experimental results corresponding to the simulation results (Figure 4) are shown in Figure 5. The SLM is encoded with different vortex phase masks having \(el=+1\) & \(-1\), \(el=+4\) & \(-4\) and \(l=0\) for generating the respective vortex beams. Figure 5a-d represent the correlation hologram retrieved from the cross-correlation of intensity fluctuations of CCD recorded intensities. The recovered amplitude and phase distributions of the vortex modes for respective topological charges are shown in Figure 5e-h and Figure 5i-l. The amplitude distributions shown in Figure 5e-h show a doughnut shape. The simultaneous recovery of both phase and amplitude provides direct advantage of quantitative determination of magnitude and sign of the topological charge of the generated OAM modes in terms of the recovered phase. The recovered phase distributions shown in Figure 4i-l in simulation and  
Figure 4: Theoretical simulation results; Simulation results: (**a–d**) retrieved correlation holograms; (**e–h**) recovered amplitude distributions of the OAM modes; (**i–l**) recovered phase distributions of OAM modes; (**m–p**) OAM distribution with \(X\)-axis representing the topological charge (\(el\)) and \(Y\)-axis the OAM power spectrum (\(P(I)\)); Scale bar: 110 μm. Figure 5i-1 in experiment exhibit a phase variation around the singularity in the order of \(2l\pi\) with the number of phase jumps equal to the value of \(l\). Moreover, the quantitative examination of the phase value provides the direct information of the sign of the topological charge from the direction of rotation of the recovered phase distribution of OAM modes. An anticlockwise direction of rotation of phase variation corresponds to the positive and a clockwise direction of rotation corresponds to the negative topological charge. Furthermore, to highlight the applicability of the approach in more applied scenarios, a quantitative analysis of the recovered OAM modes is carried out using orthogonal projection method [19; 39]. The OAM power spectrum of recovered modes is determined, and a decomposition is carried out for each OAM component in terms of azimuthal modes. The simulation and experimental results corresponding to the orthogonal projection analysis shown in Figures 4m-p and 5m-p, respectively, exhibits a good agreement. Also, we carry out experimental recovery and characterization of OAM modes with different TCs, and the quantitative characterization results corresponding to \(|el|=1\)_to_\(11\) are represented in the plot shown in Figure 6. The inner and outer radii of OAM modes were retrieved from the respective amplitude distribution of correlation function [40]. The doughnut structure of the amplitude distribution of OAM mode has a direct dependence with the TC, which is demonstrated in the plot. The plot illustrates the experimentally generated OAM modes of \(|el|=1\)_to_\(11\) and their corresponding simulation results. Figure 6a represents the variation of inner and outer radii of the recovered modes with various TC. Figure 6b shows the variation of areas of dark core and the annular bright rings of the doughnut structure  
Figure 5: Experimental results: (**a–d**) retrieved correlation holograms; (**e–h**) recovered amplitude distributions of the OAM modes; (**i–l**) recovered phase distributions of the OAM modes; (**m–p**) OAM distribution with \(X\)-axis the topological charge (\(el\)) and \(Y\)-axis the OAM power spectrum (\(P(l)\)); Scale bar: \(110\) μm. with the TC. The quantitative comparison of experimental and theoretical results shows a good agreement for all the recovered modes with various \(l\) values.