# Recovery and Characterization of Orbital Angular Momentum Modes with Ghost Diffraction Holography
## 2 Principle and Methods
### Experimental Design and Recovery Scheme  
To demonstrate the applicability of the technique, we develop a vortex beam generation and detection experimental system as shown in Figure 2. The system utilizes the time frozen speckle field detection corresponding to various vortex modes at the detector plane. The experimental technique employs a sequential independent recording of the time frozen speckle fields corresponding to various modes and utilizes the space averaging as a replacement of ensemble averaging in the execution of digital cross-correlation of recorded intensity images. The experimental geometry of the proposed technique is shown in Figure 2a. A spatially filtered and collimated He-Ne laser (632.8 nm) is used as the coherent light source. A non-polarized beam splitter (BS1) splits the beam into two parts, where the transmitted part illuminates a rotating ground glass diffuser (GG1) controlled by an aperture (A) of specific size and generates a spatially incoherent light beam with random speckle field at each instant of time. The instantaneous speckled field from the rotating ground glass illuminates the SLM through a beam splitter (BS2). The reflecting type of phase sensitive SLM (Hamamatsu X131318, 1272 \(\times\) 1024 pixels, with a pixel pitch of 12.5 \(\upmu\)m) is used to encode different vortex modes with specific \(l\) values. A half wave plate (HWP) is used to rotate the polarization state of the input beam to match with the polarization sensitivity of the SLM. The reflected beam from BS1 is used to make the holography system by generating an independent reference random field. A point source created by an off-axis microscope objective (MO:10X/0.25 NA) illuminates an independent diffuser (GG2) and produces the respective reference random field. This reference random field propagates and combines with the vortex encoded spatially incoherent field by using BS3. The intensity recording is implemented using a charge coupled device (CCD) camera (Prosilica-GT1910 from Allied Vision having total pixels of 1920 \(\times\) 1080 with a pixel pitch of 5.5 \(\upmu\)m, and 57.5 fps). The random light illuminating the SLM further propagates along with the reference random field and is recorded by a CCD, which is synchronized with RGGD and SLM. The vortex phase mask corresponding to various topological charges and zero order phase mask were encoded in the SLM, and the respective time frozen speckle patterns were recorded for various position shifts of the rotating diffuser. The corresponding resultant speckle patterns obtained with specific topological charges are shown in Figure 2b-e. An enlarged view of the marked specific region is shown in the inset of Figure 2b-e, which clearly indicates the speckle pattern modulation with interference fringes corresponding to various vortex phase masks. A schematic representation of the recovery scheme of OAM modes in the cross-correlation function is shown in Figure 3a. The digital analysis corresponding to vortex phase mask with topological charge \(el=+3\) are shown in Figure 3b-i. The phase masks encoded in the SLM are shown in Figure 3b,c. This phase-encoded reflected field individually combined with the reference random field and a double shot recording system records the time frozen intensity images as shown in Figure 3d,e. In the digital processing of two point correlation of intensity fluctuations, spatial averaging is performed by selecting the distance \(\Delta\mathbf{r}\) with varying the pixel positions \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\) in the double shot recorded intensities \(I_{m}(\mathbf{r})\) and \(I_{0}(\mathbf{r})\). Therefore, the realization of spatial averaging is carried out by pixel-by-pixel movement of \(\Delta I_{m}(\mathbf{r}_{1})\) with 200 \(\times\) 200 pixels over the entire region of \(\Delta I_{0}(\mathbf{r}_{2})\). This cross-correlation of the intensity fluctuations of the recorded intensities retrieves the correlation hologram as given in Equation (4), which is represented in Figure 3f.