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

Optical BPM - Equation 169 - 170

Using Equation 145 into Equation 169, we get:

Optical BPM - Equation 171

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

Optical BPM - Equation 172

Here a = 2k0nre .

From Equation 172, we get:

Optical BPM - Equation 173 - 174

Thus,

Optical BPM - Equation 175

and

Optical BPM - Equation 176

where

Optical BPM - Equation 177-179

Optical BPM - Equation 180-182

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 183

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

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

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

Optical BPM - Equation 184

In case of using FEM solve,

Optical BPM - Equation 185

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

Optical BPM - Equation 186

In case of using FEM compute,

Optical BPM - Equation 187

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

Optical BPM - Equation 188

In case of using FEM compute,

Optical BPM - Equation 189