nanoHUB U Nanophotonic Modeling/Lecture 3.16: MEEP: Defect Resonant Modes
========================================
 

[Slide 1] Hey everyone, this is Lecture 3.16. In the last lecture we just explained how to calculate band structures in MEEP. Which harken back to Unit 1, but we have some additional flexibility and we can calculate lots of modes very quickly with lossy and eigenmodes. 

[Slide 2] So here's an interesting problem to look at with the band structure. So before we were assuming we had a wave guide which is effectively infinite number of holes repeated in the x direction. Now what happens when we introduce a gap between those holes? And so we can think of that as a defect and so we might expect that. Much like we saw earlier, introducing a defect will create gaps in the band structure and basically create these defect modes that will be localized to the region of the defect. And so if we surround it by finite number of holes, n holes, n could large potentially or it could be a small value then we can look at what's happening inside of that cavity as well as what's happening to the rest of the band structure. 

[Slide 3] So here's the code that we use basically to set this up. This is mostly similar to what we had in the previous case except that we're explicitly repeating the basic unit over and over again, which of course it initially starts off with a waveguide and then it has a hole punched into it. And then we repeat that hole in both the minus x and plus directions using this so-called geometric object duplicates which will put the system at integer multiples. In this case along either the minus x or plus x direction you can see here we have x factor so this displacement between adjacent repetitions of that B6 subunit and then minus n1. basically tells you like how far to go in that direction and then plus n1 tells you to go on the other direction of course. 

[Slide 4] And then here's basically copied from before like setting up the pml layers and then we've narrowed the range of frequencies because we know that. There will be more modes than there were before, and we really want to zero in on the defect mode and that we can assume is within the band gap that we observed earlier. And then we also create this set of sources, impose the odd mirror plane symmetry in the y direction but then we can also impose odd mirror plane symmetry in x, or we can make it even, odd and so on, right, so we have several choices there. But this just basically to make it run faster, and then we can basically run this system outputs some modes using harminv, which you can see here. And then finally run an output the HVLs that are being generated from the system. 

[Slide 5] So this actually, the result, if you turn this whole system into movie, after you've output it. So you can see first, we basically generate this central mode, and you can see it has two things going on at the same time. First it's index guided In the y direction, which is something we expect for all those modes below the white line. But the second thing that's very important is that it's localized to this region where we had a high index gap between the holes. And that was also something that we anticipated. And that's happening within that band gap. And also below the light line. So that's how you can confine it, both in the X and Y directions simultaneously. And so this is analogous also to what might happen in a 3D slab as well. And so you can actually see in this movie, this mode is kind of pulsating back and forth between red and blue, which is what we kind of normally expect for a single harmonic mode. That's highly localized and you can explicitly see the exponential decay in the x direction. You can see that this kind of follows some sort of cosine dependence in the y direction. But as it exits the wave guide there's exponential tail, which is too faint to really see well. 

[Slide 6] And then if we look at explicitly what is the quality factor of this cavity, we can actually plot as a function of the number of sub-units in the x direction, how that quality factor scales. And this is a log plot and basically what we're seeing is that as we add more holes at first, you have a dramatic increase in the quality factor which kind of makes sense because when you only had say, one hole on each side. Then a lot of the mode is tunneling through that single hole outside into the main waveguide. So it's not very well confined. As you add more holes, of course the quality factor goes up, but you can observe something else interesting which is that actually starts to saturate at a certain value in order of like 10 to the 4. And so why is it saturating? So that basically is telling us that there's still losses in the system, but the dominant loss is no longer from outside the waveguide, so when you see here when it was leaking in this direction you could actually see some light was being waveguided here. That's no longer dominating our system so instead it must be something else dominating. 

[Slide 7] And so what is that something else? So, if you look at large N then you can actually observe a second mode. So that may give us some hints about what's going on. If we plot, like, very large N. The frequency of the two modes that are present you can actually see the quality factor of the first one of course which we carried about initially is pretty large but then there is a smaller quality factor mode which you can see is highly leaky. it leaks into the air region. So it is not as well guided. But then second it has like ability to leak through this wave guide, even if there's like a large N, so that tells us that basically you can have coupling of other moods which maybe highly lossy even if your mode itself is not directly that lossy. And also that there maybe other sources of loss such as imperfect index guiding. In the next lecture we'll talk about other things that we can model in MEEP.