Using recurrence formula Equation 118 for Padé(2,2) we get:
Here a = 2k0nref .
Using Equation 144 into Equation 143, we get:
From Equation 145, 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:
Multistep Method
In order to solve Equation 154, we applied the multistep method developed by Hadley.
First, we rewrite Equation 154 as
Then, defining the field φ l + 1 ⁄ 2 as
We rewrite Equation 154 as
Since φ l is known, we can obtain φ l + 1 ⁄ 2 by solving:
If we consider FEM, we get,
Thus, we can rewrite Equation 158 as
here
Using φ l + 1 ⁄ 2 , we rewrite Equation 156 as
If we are using FEM, we get,
Solving Equation 162 or Equation 162, we can obtain the unknown field φ l + 1 . It is apparent from the form of Equation 154 that an n th-order Padé propagator may be decomposed into an n -step algorithm for which the i th partial step takes the form
when i = 1, 2, …, n .
The run time for an n th-order propagator is obviously n times the paraxial run time. Therefore, for Padé (2,2) we follow the steps:
1 Compute φ l + 1 ⁄ 2 considering the linear system:
If we are using FEM we get:
2 Finally, to get the field φ l + 1 at z + Δz we solve:
If we are using FEM, we get,