Beam Propagation Method - Principles

Principles

BPM is generally formulated as a solution to Helmholtz equation in a time-harmonic case,

(\nabla^2 + k_0^2n^2)\psi = 0

with the field written as,

.

Now the spatial dependence of this field is written according to any one TE or TM polarizations

\psi(x,y) = A(x,y)exp(+jk_o\nu y) ,

with the envelope

A(x,y) following a slowly varying approximation,
\frac{\partial^2( A(x,y) )}{\partial y^2} = 0

Now the solution when replaced into the Helmholtz equation follows,

\leftA(x,y) = \pm 2 jk_0 \nu \frac{\partial A_k(x,z)}{\partial z}

With the aim to calculate the field at all points of space for all times, we only need to compute the function for all space, and then we are able to reconstruct . Since the solution is for the time-harmonic Helmholtz equation, we only need to calculate it over one time period. We can visualize the fields along the propagation direction, or the cross section waveguide modes.

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