Using recurrence formula Equation 118 for Padé (4,4) we get:

Optical BPM - Equation 190 - 191

Using Equation 191 into Equation 190 we get:

Optical BPM - Equation 192

From Equation 192, we get:

Optical BPM - Equation 193 - 194

Thus,

Optical BPM - Equation 195

and

Optical BPM - Equation 196

here

Optical BPM - Equation 197 - 200

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

Optical BPM - Equation 201

Multistep Method

In order to solve Equation 201, we use the multistep method, that is, the unknown field

φ l + 1  is obtained with the following the steps:

1          Compute φ l + 1 ⁄ 4  solving the linear system:

Optical BPM - Equation 202

In case of using FEM compute,

Optical BPM - Equation 203

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

Optical BPM - Equation 204

In case of using FEM compute,

Optical BPM - Equation 205

3          Compute φ l + 3 ⁄ 4   solving the linear system:

Optical BPM - Equation 206

In case of using FEM compute,

Optical BPM - Equation 207

4          Finally, knowing φ l+ 3 ⁄ 4, compute the known field φ  l + 1 at z + Δzl solving linear system:

Optical BPM - Equation 208

In case of using FEM compute,

Optical BPM - Equation 209

The advantage of the multistep is that the matrix equation to be solved in each step is the same size as the Fresnel equation and for 2D problems is tridiagonal when we consider Finite Difference method (FD) or Finite Element Method (FEM) considering linear element.