field within absorber notebook

[simbilod]

Hello Femwell community,

I was working on a new example using a mode solver to estimate the field profile within a photodetector:

1. First I compute an "input mode" confined in a silicon layer:

2. I then calculate a large set of modes in a detector cross-section (silicon slab + germanium):

3. Finally, using the simplest expression for eigenmode expansion, I can know the optical field propagating at any point in the detector as the superposition of the detector modes weighted by their overlap to the input and a phase factor function of their (complex) propagation constant. See e.g. https://en.wikipedia.org/wiki/Eigenmode_expansion

This is very simplified EME that neglects transitions between the input and the detector and any reflections at the back of the detector, but should be useful in some regimes.

Current issues

If I add an imaginary component to the refractive index of the germanium, a few things happen

1. The sum of the overlaps of the detector modes with the input mode do not sum to 1, as if the basis is not complete, or there is an issue in the calculation or normalization of the coefficients
2. The mode at z=0 does not match the input mode (which could be a symptom of 1).

Both of these things are fine if I set all my refractive indices real, which makes me think that there might be a problem with the overlap calculation when the modes are obtained from a non-Hermitian system (with loss). This is known to be tricky in the literature (in fact we don't expect to have an proper basis if the Maxwell operator is non-Hermitian).

Thoughts appreciated! The example is attached, and is set to run with real indices. To see what happens with imaginary indices, change line 113 and run all the cells again.