Hello everyone
I'm trying to program the eigenmode expansion method to calculate EM field propagation in photonic integrated devices such as input/output spotsize converters. I wrote a FDFD mode solver and am now trying to normalize all the optical modes found by the mode solver. There are no problems with guided modes and some unguided modes (I use integration by trapezoidal rule, the limits of integration are defined by calculation window). Could you recommend some articles or books where I could find information on how to numerically normalize all the modes that FDF modesolver finds, or some practical recommendations?
Thank you and have a nice day!
This reminds me of the questions I had when I did exactly the same, on FDTD an FDFD.
The thing is, there's not one definition of the normalisation because it all depends on how you define the scalar product you construct your basis with.
Once you've decided on one, you just need to make is so the norm of each mode is equal to 1.
I think modes are orthogonal if your scalar product is the integral of the product of both modes, but do check in a guided optics book. I could dig in the files in my software to see how I did back then.
Dear Loïc O. Le Cunff ,
thank you for your reply. I follow Lorentz reciprocity theorem defines orthogonality relation as the integral of vector products of modes (that reduce to integral of scalar product of modes if we assume modes are scalar wave equation solutions, as you wrote).
I expected that all modes (guided and unguided) could be normalized to the unit power. However, not all unguided modes in my FDFD were able to normalize. I use trapz function in SciPy python package to calculate the integral.
I would appreciate, if you check your FDTD code to find out how did you normalize modes. Of course, if you have time for that.
Alright, I've found the relevant part of my code. Now this stuff is from 10 years ago not it might not be the freshest: https://github.com/KAHR-Alpha/Aether/blob/3d98a926ea9c19a3fb91dbdec50aee36e95c679a/src/multilayers/planar_wgd.cpp#L682
Here's the conclusion I came to back then: the normalization is not actually relevant. What actually matter is having an orthogonal basis of modes, and how to decompose an actual field on it.
Now, the number of modes could be infinite if you consider leaky ones, and your solver will only give you a limited number of them on a finite window.
So, I defined an approximate scalar product over the limited space, and used it to "decompose" the field onto them.
Since the "scalar product" of two modes would not be 0 anymore due to the spatial truncation, it was necessary to account for the cross products between them.
Let's say for instance you have a field F that is a sum of modes M.
So, F = Sum_i a_i M_i
If you take the "scalar product" of that with an arbitrary mode M_j, you get
F . M_j = Sum_i a_i M_i . M_j
Quickly, you see you can write that as a matrix system like
[M_i . M_j] [a] = [F . M_j]
where the first part is the matrix containing of the products of all the modes onto eachother, [a] is a vector of the coefficients you're actually looking for, and [F . M_j] is a vector of the products between your field and the modes.
If it's unclear, I can paste the part of my PhD that described this. It's in french, the maths should be obvious.
Dear Loïc O. Le Cunff,
many thanks, the idea is clear!
Anyway, could you share the part of your PhD?
Have a nice day and thank you again for the discussion and the code.
Dmitrii
Well, here you go:
https://i.imgur.com/yWDVkEy.png
https://i.imgur.com/Uz9P41W.png
It's nearly exactly what I explained above though.
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