# Recovery and Characterization of Orbital Angular Momentum Modes with Ghost Diffraction Holography
## 1 Introduction  
Optical beams with helical wavefronts and associated orbital angular momentum (OAM) have become a focus of intense research in recent years owing to its high-capacity information bearing characteristics with noteworthy applications in various domains of imaging and communication [1]. The recognition of OAM of helical phase beams by [PERSON] et al. provides new insights into the theoretical and applied realms of optical vortex beams [2]. The vortex-encoded light beams possess an azimuthal phase of \(\exp(il\phi)\), where \(\phi\) is the angular coordinate and \(l\) is the topological charge that owns a positive or negative sign, and carries an OAM equivalent to \(l\hbar\) per photon (\(\hbar\) is the Planck's constant divided by \(2\pi\)) [3]. The orthogonality and infinite-dimensional basis of OAM modes offer extensive applications in OAM multiplexing-based optical communication in free space and optical fiber, quantum and classical information processing, multi-dimensional coding, and imaging [1, 4, 5]. Following theoretical investigations on the properties of optical vortex beams, several experimental techniques were realized for the generation of helically phased beams by utilizing spiral phase plate (SPP), diffractive optical elements, cylindrical lens with mode converters, q-plates, meta surface, etc. [1]. Recent years have witnessed the extension of the regime of optical vortices from coherent light to low coherent light with the demonstration of helicoidal modes in partially coherent beams [6, 7, 8]. The horizonof partially coherent vortex beams or vortex beams embedded into the randomness has expanded with the demonstration of the existence of phase singularity in the correlation function utilizing the Young-type interference experiment [9], by using broad band light source illumination on a vortex phase mask [10], an irradiance control of an extended quasi monochromatic source [11], with spatial light modulator [12], etc. Moreover, experimental strategies to generate vortices in coherence function by utilizing the partially coherent light arrays were demonstrated by the controlled modulation in the correlation structure of the light fields [13; 14]. Along with the advancements in the generation and transmission of optical vortex beams, the detection and sorting of specific OAM modes are an indispensable part in view of the applied domain. A variety of techniques for detection and sorting were demonstrated in the last two decades by utilizing inverse SPPs, holographic gratings, interferometry, mode sorters, etc. [1]. However, the presence of inhomogeneous or turbulent atmosphere inevitably destroys the helical wavefront of the vortex beam, which limits the execution of conventional detection schemes. Last decade saw the introduction of several research efforts that emphasize the development of sophisticated approaches for the recovery and determination of topological charge of vortex beams propagating through rough scattering surface by exploiting the information bearing properties of the speckle pattern generated from the scattering medium [15; 16; 17; 18; 19]. In addition, several theoretical investigations on the cross-correlation function of partially coherent optical beams with vortices were demonstrated and techniques were developed for helicity and topological charge determination of partially coherent vortex beams [20; 21; 22]. Additionally, the robustness of the low coherent vortex beams in propagation through turbulent and noisy media in comparison to high coherent optical beams was explored [23; 24], and a few techniques were demonstrated recently for the discrimination of incoherent vortex states of light [25] and in the sorting of incoherent optical vortex modes [26]. In this article, we propose a speckle correlation assisted quantitative technique for the recovery of OAM modes with a holographic ghost diffraction system.