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The '''beam propagation method''' ('''BPM''') is an approximation technique for simulating the propagation of [[light]] in [[slowly varying envelope approximation|slowly varying]] [[optical waveguide]]s. It is essentially the same as the so-called [[parabolic equation]] (PE) method in underwater [[acoustics]]. Both BPM and the PE were first introduced in the 1970s. When a wave propagates along a waveguide for a large distance (larger compared with the wavelength), rigorous numerical simulation is difficult. The BPM relies on approximate differential equations which are also called the one-way models. 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. The "initial" value problem does not involve time, rather it is for the spatial variable z.<ref>{{citation|title=Integrated Photonics|author=Clifford R. Pollock, Michal. Lipson|year= 2003|publisher=Springer |url=https://books.google.com/books?id=DNJEoypcI6oC
The original BPM and PE were derived from the [[slowly varying envelope approximation]] and they are the so-called paraxial one-way models. Since then, a number of improved one-way models are introduced. They come from a one-way model involving a square root operator. They are obtained by applying rational approximations to the square root operator. After a one-way model is obtained, one still has to solve it by discretizing the variable z. However, it is possible to merge the two steps (rational approximation to the square root operator and discretization of z) into one step. Namely, one can find rational approximations to the so-called one-way propagator (the exponential of the square root operator) directly. The rational approximations are not trivial. Standard diagonal Padé approximants have trouble with the so-called evanescent modes. These evanescent modes should decay rapidly in z, but the diagonal Padé approximants will incorrectly propagate them as propagating modes along the waveguide. Modified rational approximants that can suppress the evanescent modes are now available. The accuracy of the BPM can be further improved, if you use the energy-conserving one-way model or the single-scatter one-way model.
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==Numerical methods==
Both ''spatial ___domain'' methods, and ''frequency (spectral) ___domain'' methods are available for the numerical solution of the discretized master equation. Upon discretization into a grid, (using various [[central difference|centralized difference]], [[
==Applications==
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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.
==See also==
*[[Computational electromagnetics]]
*[[Finite-difference time-___domain method]]
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*[[Light]]
*[[Photon]]
==References==
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