# BPM Technical Background

## Introduction

The finite difference beam propagating method (BPM) is one of the most powerful techniques to investigate linear and nonlinear lightwave propagation phenomena in axially varying waveguides such as curvilinear directional couplers, branching and combining waveguides, S-shaped bent waveguides, and tapered waveguides. BPM is also quite important for the analysis of ultrashort light pulse propagation in…

## Slowly Varying Envelope Approximation

Suppose Φ is an electric or magnetic component of the optical electromagnetic field. This component is a periodic (harmonic) function of position, it changes most rapidly along the optical axis, z , and has a period that is on the order of the optical wavelength. The slowly varying approximation involves replacing the quickly varying component,…

## Differential Equations of BPM

In this section we show the derivation for the differential equations found in BPM. Of course, more complete accounts of this material can be found elsewhere (see References  – ). This section exists for the convenience of the user of OptiBPM, to define important terms and illustrate the nature of the different levels of approximation.…

## Semi-Vector and Scalar BPM

The above system of equations for BPM is called the Full-Vector form, as it includes both transverse components of the field. Often it is not necessary to have both field components in the simulation. If it is known that the device does not change the polarization of light, then it is sufficient to model one polarization…

## Crank-Nicholson Method and Scheme Parameter

Formally, the solution to the BPM equations (whether Full-Vector, Semi-Vector, or Scalar) is where Δz =  z1 – z0 . If the field et  is assumed known at a transverse plane z0 , then the above equation will calculate the field at some other plane z1 . A rational function is needed to approximate the exponent of…

Usually the most time consuming process in the execution of the BPM simulation is the solution of the linear system in Equation 24. Sometimes, it is possible to speed this calculation by manipulating the formulation so that only simple linear systems result. There are other formulations of the problem which permit the operator to be split…

## Boundary Conditions for BPM

In actual structures, radiated waves are reflected at the boundaries and return to the core area, where they interact with the propagating fields. This interaction disturbs the propagating fields and greatly degrades the calculation accuracy. It is usual to impose boundary conditions when formulating propagation algorithms in such way to avoid radiation from the device to…

## Perfectly Matched Layer (PML)

Berenger introduced the concept of a perfectly matched layer (PML) for reflectionless absorption of electromagnetic waves, which can be employed as an alternative to the transparent boundary condition (TBC). The PML approach defines the truncation of the computation domain by layers (which absorb impinging plane waves) without any reflection, irrespective of their frequency and angle…

## Wide-Angle Beam Propagation Method

The beam propagation method (BPM), originally derived from the paraxial (Fresnel) approximation, has been widely used to study optoelectronics devices. The paraxial limitation was removed in the so called wide-angle beam propagation methods (WA-BPMs). Here WA-BPM focus on extending the range of applicable “angle” using high order Padé approximation in the transverse direction. Such extensions…

## Finite Difference Beam Propagation Method (FD-BPM) with Perfectly Matched Layers

We consider a planar waveguide where x and z are the transverse and propagation directions, respectively, and there is no variation in the y direction ( ∂ ⁄ ∂y ≡ 0 ) . Furthermore we consider that the planar optical waveguide with width W is surrounded by PML regions with thickness d as shown in…