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We explore the spatial features of various orders of Fraunhofer diffractionpatterns in a four-level N-type atomic system. The system interacts with a weakprobe light, a standing wave (SW) coupling field in the x-direction, and acylindrical beam of composite optical vortex type. We derive the first-orderlinear and third-order cross-Kerr nonlinear parts of the probe susceptibilityby expanding the probe susceptibility of the system into the second order ofthe SW beam. This allows us to solve the integral equation of Fraunhoferdiffraction, decoding its varying degrees to specific degrees of Besselfunctions containing the nonlinear susceptibility. Notably, the nonlinearsusceptibility exhibits dependence on the Orbital Angular Momentum (OAM) of thelight beam, leading to spatial variations in the Bessel functions and,consequently, in the different orders of Fraunhofer diffraction. Leveraging themanipulation of OAM, we achieve precise control over the spatial mapping ofdiverse diffraction orders at various locations. Our research sheds new lighton the spatial behavior of Fraunhofer diffraction in complex atomic systems. Itpresents exciting prospects for harnessing the OAM characteristics of light infuture optical technologies.

Spatial Characterization of Fraunhofer Diffraction in a Four-Level Light-Matter Coupling System