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# Finite Difference Optical Waveguide Mode Solver

## Finite Difference Optical Waveguide Mode Solver

(OP)
When solving the eigenvalue problem [A]*E =(beta^2/k0^2)*E, where beta is the prop constant, E the electric field and k0 the free space wavenumber:
Is there a limit on how small a discretisation step size you can take?

In my matrix [A] the main diagonal for a 2-D mesh is constructed by the expression:

n(x,y)^2 - 2/(xstep^2*k0^2) - 2/(ystep^2*k0^2)

Obviously when xstep or ystep and reduced to a small enough size this expresion produces a negative result, which gives purely imaginary eigenvalues which are incorrect. Does anyone know have I discretized the mesh incorrectly? Any help would be greatly appreciated. Thanks K.J
Replies continue below

### RE: Finite Difference Optical Waveguide Mode Solver

All I know is that one time I did a sim based on cc chen's narrow slits and plane waves, the number of points was a critical parameter that was not clear in the paper., Plus it was an waste of time as in varying the geometric parameters of the problem, was, pure and simple random "colored" noise. sorry, I can not help, other than your sampling issue is not a trivial one. good luck, hope someone with some brains will step up to the plate.

### RE: Finite Difference Optical Waveguide Mode Solver

(OP)
Thanks GOTWW,

As it turned out my discretisation was fine, it was how I was getting the eigenvalues. I was looking for the 3 eigenvalues that were largest in magnitude. This was fine when the main diagonal was positive but when xstep was reduced further it went negative, the eigenvalues I was after were no longer lagest in magnitude, the negative ones were. It was one of those cases where you leave something for a while and come back to it the problem reveals itself.

All the best!

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