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# BPM Technical Background

## Finite Difference Beam Propagation Method (FD-BPM) with Transparent Boundary Conditions

We can include the left-hand TBC expressing the field of the hypothetical node φ–1 (see Figure ) outside the analysis domain, (shown in “Transparent Boundary Condition” on page 23), as Here the parameter γL  is determined by the known fields at z . Following similar way described we can include the right-hand TBC expressing the field φ…

## Finite Element Beam Propagation Method (FE-BPM) with Perfectly Matched Layers

We proceed now with the solution of Equation 50 on the basis of the Finite Element Method [29] and [30]. As a first step we introduce the residual which must be zero in accordance with the state problem. However, it is impractical to enforce R( x ) = 0 at every point in the domain from…

## Wide-Angle Beam Propagation Method based on Pade Approximant Operators

Here, we follow the Padé approximant approach to get the wide angle beam propagation [31]. It is worth it to point out that expansion via Padé is more accurate than Taylor expansion for the same order of terms. When Padé is employed, larger angles, higher index contrast and more complex mode interference can be analyzed in…

## Fresnel Approximation (Pade 0th Order)

For paraxial approximant, we get the following equation: here a =2k0nre 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…

Using the recurrence formula Equation 118, we get: Here a = 2k0nref . Using Equation 119 into Equation 131, we get: Thus, and, where Thus, we can get the unknown field φ l + 1  at z + Δz from the known field φ l  solving the triagonal linear system: If we consider FEM we get, Thus, we can…

Using recurrence formula Equation 118 for Padé(2,2) we get: Here  a = 2k0nref . Using Equation 144 into Equation 143, we get: From Equation 145, we get: Thus, and, where: Thus, the unknown field φ l + 1  at z + Δz is related to the known field φ l at z as follows: Multistep Method In order to…