When a wave propagates along a waveguide for a large distance (larger compared with the wavelength), rigorous numerical simulation is difficult.
These one-way models involve only a first order derivative in the variable z (for the waveguide axis) and they can be solved as "initial" value problem.
[1] The original BPM and PE were derived from the slowly varying envelope approximation and they are the so-called paraxial one-way models.
Namely, one can find rational approximations to the so-called one-way propagator (the exponential of the square root operator) directly.
It is typically used only in solving for intensity and modes within shaped (bent, tapered, terminated) waveguide structures, as opposed to scattering problems.
Basic implementations are also inaccurate for the modelling of structures in which light propagates in large range of angles and for devices with high refractive-index contrast, commonly found for instance in silicon photonics.
Advanced implementations, however, mitigate some of these limitations allowing BPM to be used to accurately model many of these cases, including many silicon photonics structures.
The BPM method can be used to model bi-directional propagation, but the reflections need to be implemented iteratively which can lead to convergence issues.