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Helmoltz equation describes the evolution of the bulk electromagnetic field, even when doing scalar optics as an approximation.

Beam Propagation Method is a common approximation that assumes certain simplifications, in particular only forward propagation is taken in consideration. The benefit of this is that there is that the state of the beam at each point of an optical circuit can be entirely defined by the $A(x,y)$ amplitude of the wavefront in a slice of $z$ distance. propagation in the $z$ axis and in time are the same and one.

in the EigenMode decomposition approach, one takes a region, which in the simplest form is rectangular, and one demands the Helmholtz equation to be satisfied in the bulk of the volume, and demand some Dirichlet or Neumann conditions on the boundary. When the field is decomposed in a complete basis, be that polynomial or trigonometrical, the equation becomes an equation on the eigenmode coefficients $F_{k,\omega}$ which are constant (time dependence is resolved in the decomposition in the $\omega$ frequencies).

my goal is to use the beam state produced by $BPM$ as an adequate input for an eigenmode problem in a neighbouring region. The idea that i want to exploit for this is that, as BPM is time-independent, then we must demand that at least intensities in the eigenmode domain be constant as well (which sounds about right, because we are not interested in transient solutions). The second idea is that if we split the field in a forward-travelling and a backward-travelling component along the axis where the BPM beams comes from, we should be able to match the forward component of the field with the BPM output data. This is the part that i'm not sure is meaningful, and i'll discuss it.

The above two constraints can be expressed as this:

  • As we are working on a scalar model of light, intensities are described as $F^{*} (x,t) F (x,t)$. We assume that our Eigenmode solution satisfies

$$ \frac{ \partial (F^{*} F) }{ \partial t } = 0 $$

Which means we demand that intensity must be constant.

  • Suppose the boundary between the BPM-computed incoming beam is along the z-axis, then in our EigenMode field decomposition, we separate the field in two components, travelling in the positive and negative directions along the z-axis. That decomposition would look like

$$ F = \sum_{k,\omega} { F_{k,\omega} e^{-i(k_x x + k_y y + k_z z - \omega t)}} = F_{++} + F_{--} + F_{-+} + F_{+-}$$

components that travel in the positive z-axis are those that have

1) $k_z > 0$ and $\omega > 0$, which we describe the field component $F_{++}$

2) $k_z < 0$ and $\omega < 0$. this gives $F_{--}$

components that travel in the negative z-axis are these

1) $k_z < 0$ and $\omega > 0$. this we label $F_{-+}$

2) $k_z > 0$ and $\omega < 0$. this we label $F_{+-}$

So, in order to interface with the BPM output beam data $A(x,y)$ at $z = z_{b}$, i demand that

$$(F_{++} + F_{--})^{*} (F_{++} + F_{--})(x,y,z_b) = A^{*}(x,y) A(x,y)$$

This is the part that i'm not able to convince myself is the right/ideal condition to ask in order to obtain a meaningful result. In particular, this seems to not be doing anything with the BPM phase information. I thought about equating

$$ F_{++} + F_{--} = A(x,y)$$

but the problem is that amplitudes do not have to be time-constant, so i'm not sure how to forward from this point. I guess that the whole question boils down to how to physically interpret the $A(x,y)$ time-independent amplitude obtained in BPM, and how to map that concept as an eigenmode expansion boundary condition

any suggestions welcome as always

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