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Beam propagation method
The beam propagation method (BPM) is an approximation technique for simulating the propagation of light in slowly varying optical waveguides. 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.
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.
==Principles==
BPM is generally formulated as a solution to Helmholtz equation in a time-harmonic case,
〔Okamoto K. 2000 Fundamentals of Optical Waveguides (San Diego, CA: Academic)〕
〔EE290F: BPM course slides, Devang Parekh, University of Berkeley, CA〕
:
(\nabla^2 + k_0^2n^2)\psi = 0

with the field written as,
:$E\left(x,y,z,t\right)=\psi\left(x,y\right)\exp\left(-j\omega t\right)$.
Now the spatial dependence of this field is written according to any one TE or TM polarizations
:$\psi\left(x,y\right) = A\left(x,y\right)\exp\left(+jk_o\nu y\right)$
,
with the envelope
:$A\left(x,y\right)$
following a slowly varying approximation,
:
\frac = 0

Now the solution when replaced into the Helmholtz equation follows,
:
\left(ウィキペディア（Wikipedia）

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