Here ˜ V$_{j}$ = ( α β ) T ˜ f ( k$_{x}$, k$_{y}$ ) ( j = 1 , 2) are the scaler Whittaker potentials. The amplitude of the angular spectrum is referred to as where w$_{0}$ is beam-waist size. The matrix ( α β ) T denotes the Jones vector which satisfies the normalization condition αβ ∗ + α ∗ β = 1. The coefficients α and β satisfy the relation σ = i ( αβ ∗ − α ∗ β ). The polarization operator σ = ± 1 corresponds to left and right circularly polarized light, respectively. It is well known that a circularly polarized beam can carry spin angular momentum σ ℏ per photon due to its polarization state [15]. From the viewpoint of Fourier optics [16], the Whittaker potentials V$_{j}$ are given by the relation It can be verified that both V$_{1}$ and V$_{2}$ satisfy the scalar Helmholtz equation: On substituting Eq. (5) into Eq. (2), the electric field can be expressed in terms of the Whittaker potentials In fact, after the angular spectrum on the plane z = 0 is known, Eq. (9) together with Eqs. (4) and (8) provides the expression of the electric field vectors as Here the superscript [W] represents the vectorial model given by the Whittaker potentials. We find that the electric components can be written as the Whittaker potentials and their first- and second-order derivatives. Up to now, we have established a general propagation model to describe the vector field structure. 5 E ( r ) = ∇ × ∇ × ( u V$_{1}$ ) − ik ∇ × ( u V$_{2}$ ) . (9) ( ∇ 2 + k $^{2}$) V$_{j}$ ( r ) = 0 . (8) V$_{j}$ ( r ) = 1 k 2 ∫ dk$_{x}$dk$_{y}$ ˜ V$_{j}$ ( k$_{x}$, k$_{y}$ ) × exp[ i ( k$_{x}$x + k$_{y}$y + k$_{z}$z )] . (7) E [W] x = sin θ ( k 2 + ∂ 2 ∂x 2 ) V$_{1}$ + cos θ ∂ $^{2}$V$_{1}$ ∂x∂z − i cos θk ∂V$_{2}$ ∂y , (10) E [W] y = sin θ ∂ $^{2}$V$_{1}$ ∂x∂y + cos θ ∂ $^{2}$V$_{1}$ ∂y∂z − i sin θk ∂V$_{2}$ ∂z + i cos θk ∂V$_{2}$ ∂x , (11) E [W] z = sin θ ∂ $^{2}$V$_{1}$ ∂x∂z + cos θ ( k 2 + ∂ 2 ∂z 2 ) V$_{1}$ + i sin θk ∂V$_{2}$ ∂y . (12) ˜ f ( k$_{x}$, k$_{y}$ ) = w$_{0}$ √$_{2}$$_{π}$ exp [$_{−}$ k 2 x + k 2 y 4 w 2 0 ] , (6)