  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.
