Hi everyone,
I have one part general info that maybe others will find helpful, and one part question. Here goes:
I've been trying to compute the eigenvalues of dielectric GaAs disk structures (embedded in air or some other dielectric cladding) in the RF module.
In particular, I wanted to find the Q-factor (due to radiation alone) of whispering gallery modes in these structures. The type of mode isn't important, just that I wanted Q-factor for a dielectric structure that can radiate, so the Q factor is not infinity even though all materials are lossless.
For some reasons specific to my problem, I can't do this in the 2D axi-symmetry approach as you'll find in some of the comsol related papers on whispering gallery modes in aix-symmetry.
I'm forced to do 3D. Anyway, I had assumed that simply using perfectly matched layers (PMLs) of one wavelength thickness would do the trick and allow any radiation associated with the eigenmode to be absorbed w/o reflection and just die in the PMLs.
I've had lots of success with comsol default PMLs in say, steady state (frequency domain, stationary) analysis. So I was really puzzled why the choice of the size of my cladding layer before PMLs affected DRAMATICALLY the Q factor reported by comsol. The resonant frequency (and consequently, wavelength) were relatively unaffected by how big a domain I drew around my (nm to micron sized) disk.
Turns out, comsol drops a scaling factor in the "coordinate stretching in the complex plane" that is equal to the wavelength. This means (and I worked through the math here, given comsol's definition of the PML stretching) that effectively my physically 1 wavelength (~1200 nm) long PML "looks" like it's 1 METER long after PML stretching. This is no good--especially when you only have 5 elements in that 1 wavelength long PML. It means that it is TERRIBLY undermeshed. This normally just leads to "Singular Matrix" for most cases, but in the off chance you DO get a solution, the Q factor is nonsense, but yet the wavelength is in the ballpark.
The fix:
Set the scaling factor, F, as it appears in the PML subnode equal to 1 wavelength (of the mode you're interested in, so you kinda have to take a guess, just like you do at where to search for eigenmodes, you can tune later once comsol reports a wavelength from a first run...remember, the wavelengths [frequencies] I think are trustworthy regardless of PML choice).
This worked like a charm for me, and I believe is a general approach for getting Comsol to report Quality factors truthfully. I compared my results for a disk that's 2 microns long made out of GaAs (easy to find in literature--it's all over a google search for "whispering gallery modes + comsol"), and got decent agreement with the Q factor they report from axi-symmetry analysis.
Anyway, I hope the above may help others, since I've struggled for a month now before I read the fine print in comsol and found they drop the "lambda" term in their coordinate stretching in eigenvalue calculations so as to avoid nonlinear dependence (b/c wavelength and frequency are connected, the stretching would be eigenvalue dependent otherwise)...
I do have one question, if anyone can help. Comsol reports 1 Q-factor from taking the imaginary part of the eigenvalue and dividing by twice the real part of that same eigenvalue calculation. If it were REALLY REALLY correct,then I should be able to go into post processing mode and do the following:
1. Know Q =(defined)=omega_0*(Time_Avg_Energy_at_resonance)/(Time_avg_power_disssipated_at_resonance)
2. Extract omega_0, the resonant ANGULAR frequency, from the imaginary part of the eigenvalue
3. [3D Volume] Integrate energy density over entire model space (all domains) to get the numerator
4. [3D Volume] Integrate resistive losses (or power dissipation density in V4.2, they're the same) to get the denominator.
I do this and I get the computed quality factor from (1) to be EXACTLY 2 times that which comsol reports.
Anybody have any ideas on where this is coming from. I'm happy knowing ONE of the answers is correct and that only a factor of 2x separates me from truth, but I'd very much appreciate it if someone could tell me what one is the right one. I'm inclined to believe the computed answer from number (1) above, because it agrees better with numbers I find in the literature for my test case...But even comsol's direct answer isn't too far off those values either (well, a factor of 2....but maybe comsol IS right).
Thanks!
--Matt