Wide Angle (WA), Pade(3,3)

Using recurrence formula Equation 118 for Padé (3,3), we get:

Using Equation 145 into Equation 169, we get:

Finally, using Equation 171 into Equation 169, we get:

Here a = 2k₀n_{re f} .

From Equation 172, we get:

Thus,

and

where

Thus, the unknown field φ ^{l + 1}  at z + Δz is related to the known field φ ^l at z as follows

In order to solve Equation 183, we use the Multistep method, that is, the unknown field

φ ^{l + 1}  can be obtained from the known field φ ^l  by successively solving Equation 164 for

i = 1, 2, 3 . Therefore for Padé(3,3) we follow the steps:

1          Using φ ^l , compute φ ^{l + 1 ⁄ 3}  considering the linear system:

In case of using FEM solve,

2          Using φ ^{l+ 1 ⁄ 3}, compute φ ^{l + 2 ⁄ 3} considering the linear system:

In case of using FEM compute,

3          Finally, using φ ^{l+ 2 ⁄ 3}, compute the following linear system to obtain the unknown field φ ^{l + 1}  at z + Δz :

In case of using FEM compute,