Tuning of non-paraxial effects of the Laguerre-Gaussian beam interacting with the two-component Bose-Einstein condensates

We present the theory of microscopic interaction of the spin-orbit coupled focused Laguerre-Gaussian (LG) beam with the two-component Bose-Einstein condensate (BEC), composed of two hyperfine states of $^{87}$Rb in a harmonic trap. We have shown that Raman Rabi frequency distributions over the inter-component coupling identify phase separation coupling strength. A significant enhancement of side-band transitions due to non-paraxial nature of vortex beam is observed for particular values of inter-component coupling around 1.25 and 0.64 in unit of 5.5nm for $10^5$ and $10^6$ number of atoms, respectively. The uncertainty in the estimation of these coupling strengths is improved with the focusing angles of the beam. We discuss an experimental scheme to verify this non-paraxial effect on ultra-cold atoms.