Here, we propose and experimentally demonstrate a new basis for the recovery of the OAM mode using a holographic ghost diffraction scheme. However, the presence of scattering environment and turbulent atmosphere scrambles the helical wavefront and destroys the orthogonality of modes in vortex beam propagation. Orbital angular momentum (OAM) of optical vortex beams has been regarded as an independent physical dimension of light with predominant information-carrying potential. Furthermore, we discuss demonstrated applications of the technique in the imaging various spatially varying complex-valued macroscopic and microscopic samples and the potential application of the technique in the recovery and characterization of orbital angular momentum modes encoded in spatially incoherent speckle field. In addition, the technique exploits the spatial statistics of time-frozen recorded speckle intensity with snapshot detection in ghost framework, which could broaden the applications of the developed microscopy to real-time imaging of two-and three-dimensional biological samples with high resolution. The development of the unconventional correlation-assisted GDH technique by adopting the holography concept in ghost diffraction scheme is described, and the quantitative phase imaging capability is demonstrated in the microscopy. In this chapter, we discuss the recently developed ghost diffraction holography (GDH) system with due emphasis on the capability of quantitative complex-field imaging in the ghost framework. The fascinating domain of ghost imaging has been a subject of interest in the fundamental and applied research for the last two decades with its promising applications in various imaging and characterization scenarios. In addition, recent interesting applications of LC-SLMs have been discussed thoroughly within the framework of polarization holography. This chapter provides comprehensive literature (review) of the LC-SLMs along with their major calibration methods. Therefore, it is recommended to calibrate the modulation characteristics of SLMs prior to their implementation for imaging applications. However, the resolution of the CGHs are sometimes limited by the structural discrepancies (fill factor, spatial anomalies, refresh rate, etc.) of SLM. Apparently SLMs serve a crucial role in the experimental implementation of digital holographic techniques. Replacing the conventional optical elements from the SLM-based computer-generated holograms (CGHs) is a trending approach in modern digital holographic applications due to the optimized phase shift depending on the phase modulation features of SLMs. Liquid crystal spatial light modulators (LC-SLMs) have gained substantial interest of the research fraternity due to their remarkable light modulation characteristics in modern imaging applications. The experimental results are in good agreement with the theoretical basis and numerical results. As an application of the proposed technique, recovery of the complete phase maps of the different vortex beams from the random light is experimentally demonstrated and the twisted modes of the incident light are quantitatively analyzed using orthogonal projections. A complete theoretical basis is developed to retrieve the twisted wavefront of propagating light through random scattering media and is also supported by numerical simulation and experimental demonstration. Out of these 16 elements, only four elements of the matrix are used to retrieve real and imaginary parts of the complex polarization correlation function and are subsequently applied in the reconstruction of twisted wavefronts. Two-point correlations of the Stokes parameters form a 4 × 4 matrix with a total of 16 elements. Recovery of the complex polarization correlation functions and hence twisted modes of the beam are retrieved from the Stokes parameters of a randomly scattered field. Correlation of the orthogonal polarization states of a random field encodes the signature of the twisted mode of the vortex beam. This couples the spatial and polarization modes of the coherent light prior to entering the scattering medium. In the proposed technique, the vortex beam with an arbitrary topological charge coaxially enters into the scattering medium with another nonvortex beam with an orthogonal polarization state. We propose and experimentally demonstrate a noninterferometric method to quantitatively determine the topological charge and complete phase structure of a vortex beam scrambled by a random scattering medium.
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