nanoHUB U Nanophotonic Modeling/Lecture 3.14: MEEP: Kerr Nonlinearities
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[Slide 1] Hey everyone, welcome to Lecture 3.14. So we're continuing with examples in MEEP. In this case, we've already done a straight, bent, and ring resonators and now we're going to  look at some non-linear behavior in materials. 

[Slide 2] So what is a nonlinear material in general? Of course, there are several types, but we're going to  focus particularly on Kerr nonlinearities. And so, Kerr nonlinearities can be described in a few different ways, but usually you say that the refractive index increases as the square of the electric fields. And there's a certain constant associated with that which we can call the Kerr coefficient. So of course, if you have no nonlinearity k equals 0. And so, that would probably be a good description of a lot of materials like Kerr. And physically this can also give rise to some very interesting phenomenon. So broadly speaking it's called four-wave of mixing. So then that means that you can actually do two key processes. One is like called sum or difference frequency generation where you may start with two photons possibly at the same frequency so that would be like a degenerate for wave mixing. And then, you could generate a pair of photons. And then, usually the constraint is energy conservation, so if you have two photons at frequency omega one, then of course the outputs would also have to add up to two omega one. But then you could have an omega signal and an omega idler, and so then this combination could be kind of spaced very close to your source or very far apart. And then, there's another phenomenon that can actually happen in which case you have three waves come together and then generate one higher frequency photon. So basically omega plus omega plus omega goes to a three omega photon. So both are possible in this kind of system, know, relative rates will depend on a lot of factors, so usually field strength is very important. So as you can imagine if E is very small compared to square root of k, or one over root k, I guess, then you could imagine the refractive index is negligible. So it's not as strong a fact. Input profile is important, and then the output photonic density of modes is very important here. 

[Slide 3] So how do we set up an example in MEEP to actually test this? So here we're going to  basically create a very long cell in the z direction. So length of 100. And we're going to  choose a central frequency of one-third. So which is very deliberate because remember that we said we could have a third harmonic generation. So that would imply the output of three one-third frequency photons would be exactly one. So you should be able to pick that up. And then, we can see that the frequency width of the source should be narrow enough so you won't just get like the frequency of one generated automatically by your source. So we don't want to confuse ourselves with that. We set the amplitude to the source, we set this Kerr coefficient to be 0.01. Of course, this is in somewhat arbitrary units. So we always have to scale it by what the units of the E field that we're interpreting are so keep that in mind that this is a very reduced set of units you can't necessarily directly take this from literature without some careful thought. And then, we create a pml which will of course absorb any excess optical waves that we don't want, as they leave the system. And we set it up as a 1D problem, so it's just very long and thin in the z direction. And then, we have a pml in both sides, set our resolution and then here is how we basically create the non-linear material. So you make a dielectric material, with the base index of 1, but it has a chi 3, as it's also called, of k, which is the Kerr coefficient. 

[Slide 4] And then, here, we basically set up our sources. And the sources are all based on what we had earlier. So this shouldn't be a surprise if you saw the earlier lectures. Set number of frequencies which will basically be an array to measure all the output frequencies coming from our input so we can calculate the full transmission and generation spectrum. And we can see that actually we're setting this to a pretty broad range that goes from half of the central frequencies, so about a frequency one-sixth up to four times central frequency, which would be four-thirds. And so, then, we should be able to see third harmonic generation if that's happening as well as a four wave mixing that results in idler pairs. And then, we define a transmitted flux region, which you can see here, that's basically positioned like towards the end of the system. Then, we set up our system to run. And then, it'll keep going until the fields have fully decayed. Obviously, not much is going to  happen after that. And then, we output the fluxes at the end of what was transmitted. And then, we try to analyze the results. 

[Slide 5] So this actually showing what we get in different circumstances. You can actually see, here in blue, we actually only have a very small Kerr coefficient. In these units, it's ten to minus three. You can see basically what happens is you have very narrow peak at one third, and then you have another narrow peak at 1. So there is third harmonic generation happening even for what appears to be a relatively modest non-linear coefficient. However, as you increase that, you can see that additional width is being added to the single either pair, which is near a frequency one-third, as well as like to the third harmonically generated photons. And of course, this doesn't even necessarily capture everything that's happening outside of this frequency range. Now, once you go to very large Kerr coefficients on the order of say, 0.1 or even 1, you can actually see value start to basically fill in this whole spectrum with all this generation and then subsequent recombinations. Basically multiple cascaded thoroughly mixing processes that are happening. And this is oftentimes called modulation instability, when you start filling up this whole range with instability. So this is a big deal in ultra fast optics, for example, this sort of thing. 

[Slide 6] And so, if we want to quantify the efficiency of third harmonic generation we can actually do so, as well. So we can add the following two terms to our CTL file. Basically defining two flux regions which have different central frequencies. You can see one is centered at fcen and 3 times fcen. And then, we can basically output the harmonic output based on the input and output. Assuming that most of the one-third frequency is from our source, and then almost all of the frequency at one must be from the third harmonic generation instead of the source directly. And then, we just kind of summarize our output here by outputting these two sets of lines it's just a print statement. Basically printing out our results. And so, you can see here actually what this does is it steps through powers of 10 of k so basically stepping through logk as it goes from 0.2 to larger values and then it'll output 10 to that power. And so, we can actually see how the harmonic output varies. And so, here, for example, we're seeing that for a relatively small value of k, we see a relatively small third harmonic generation which is like 10 to the minus 18 as a fraction of the input. 

[Slide 7] And so, if we plot this over a very broad range of chi 3 values, you can see starting from very small chi 3 here of 10 to the minus 6, then going up all the way to 1, the intensity of the input frequency basically is unchanged for almost all the range. But then the third harmonic generation actually goes up fairly rapidly. It actually goes quadratically. So that tells us something important about the physics of the problem. Because the Kerr term itself enters as E squared. It stands to reason that you would have an increase of the power that might go like k squared because you need basically three frequencies interact in order to generate the output photon. So you can see it goes up very rapidly, actually. In the next lecture, we will talk more in details about what can be done with MEEP.