For paraxial approximant, we get the following equation:

Optical BPM - Equation 119

here a =2k0nre f ,

Optical BPM - Equation

From Equation 119, we get

Optical BPM - Equation 120 - 121

and therefore

Optical BPM - Equation 122 - 123

where

Optical BPM - Equation 124 - 125

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

Optical BPM - Equation 126

or

Optical BPM - Equation 127

Using Equation 127 we finally can get the unknown field φ l + 1 at z + Δz from the known field φ l

If we consider FEM we get,

Optical BPM - Equation 128

Thus, we can rewrite Equation 127 as

Optical BPM - Equation 129

here

Optical BPM - Equation 130

The element matrices [ M e] and [ K e] are defined by Equation 100 and Equation 99, respectively.