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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 = 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 + Δwe solve:

If we are using FEM, we get,