Fresnel Approximation (Pade 0th Order)

For paraxial approximant, we get the following equation:

here a =2k₀n_re f  ,

From Equation 119, we get

and therefore

where

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

or

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,

Thus, we can rewrite Equation 127 as

here

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