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, 