content_image/356824.jpg content_image/356820.jpg content_image/356819.jpg content_image/356818.jpg content_image/356821.jpg content_image/356822.jpg content_image/356823.jpg
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 

<img src='content_image/356824.jpg'>

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 

<img src='content_image/356820.jpg'>

It can be verified that both V$_{1}$ and V$_{2}$ satisfy the scalar Helmholtz equation: 

<img src='content_image/356819.jpg'>

On substituting Eq. (5) into Eq. (2), the electric field can be expressed in terms of the Whittaker potentials 

<img src='content_image/356818.jpg'>

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 

<img src='content_image/356821.jpg'>

<img src='content_image/356822.jpg'>

<img src='content_image/356823.jpg'>

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. 

