Using recurrence formula Equation 118 for Padé(2,2) we get:

Optical BPM - Equation 143 - 144

Here  a = 2k0nref .

Using Equation 144 into Equation 143, we get:

Optical BPM - Equation 145

From Equation 145, we get:

Optical BPM - Equation 146 - 147

Thus,

Optical BPM - Equation 148

and,

Optical BPM - Equation 149

where:

Optical BPM - Equation 150 - 153

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

Optical BPM - Equation 154

Multistep Method

In order to solve Equation 154, we applied the multistep method developed by Hadley.

First, we rewrite Equation 154 as

Optical BPM - Equation 155

Then, defining the field φ l + 1 ⁄ 2  as

Optical BPM - Equation 156

We rewrite Equation 154 as

Optical BPM - Equation 157

Since φ l  is known, we can obtain φ l + 1 ⁄ 2 by solving:

Optical BPM - Equation 158

If we consider FEM, we get,

Optical BPM - Equation 159

Thus, we can rewrite Equation 158 as

Optical BPM - Equation 160

here

Optical BPM - Equation 161

Using φ l + 1 ⁄ 2 , we rewrite Equation 156 as

Optical BPM - Equation 162

If we are using FEM, we get,

Optical BPM - Equation 163

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

Optical BPM - Equation 164

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:

Compute φ l + 1 ⁄ 2   considering the linear system:

Optical BPM - Equation 165

If we are using FEM we get:

Optical BPM - Equation 166

Finally, to get the field φ l + 1 at z + Δwe solve:

Optical BPM - Equation 167

If we are using FEM, we get,

Optical BPM - Equation 168