nanoHUB U Nanophotonic Modeling/Lecture 4.2: Time-Domain Laser Simulation
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[Slide 1] Hey everyone, this Lecture 4.2. So we're continuing on the topic of lasers. And now we're going to actually use our background knowledge of the basic processes that are happening in interaction of light and matter to perform time-domain laser simulations. 

[Slide 2] Actually predict what would happen in the laser. So just to briefly summarize our theory of how laser amplification works, remember that it depends on selecting out a stimulated emission as our dominant process. And we need to have a situation where we have what's called population inversion. Which basically means that the number of excited level atoms would exceed the number of ground state atoms. So then that means that our stimulated emission rate would have to exceed the absorption rate because those two terms have essentially the same form factor except they have a dependence that's different on number of atoms. So remember that we had N-bar over t spontaneous times plus N1 for absorption and then minus N2 for simulated emission. So then if N2 is greater than N1, then that means we have a net depopulation of the atom population and thus emission into the laser mode. Okay, and then that means the medium is acting as a amplifier. However, if it was the opposite, so basically N2 was less than N1, then it would act as attenuator. And absorb photons, which is okay, but it's not a laser. 

[Slide 3] So, now in order to establish this population inversion I was talking about, we need to have a cover scheme. And so I mentioned that you can do a three-level laser like a ruby, but it's actually usually more efficient to do four level pumping system. And so if we think of states 1 and 2 as our laser transition, then we have to have two states around it, one below and then one above that key lasing transition. And so the way that we'll basically set it up is that we pump from this ground state, R, at state 0 all the way up to state 3. And then state 3 actually decays very rapidly into state 2. And then state 2 primarily goes into state 1 through the process of stimulated emission. And then there may be a small amount of background spontaneous emission or whatever. But then once you get into this state 1, then you actually rapidly decay back to state 0. So the net effect is that you're pumping lots of state 2 system up here, but then you're losing this state 1 very quickly so that means you're very favorable for creating a population inversion. And is much better than just trying to directly work with a two state system or even a three state system. So four states is usually considered the best. 

[Slide 4] So here are the equations that we're going to use to stimulate all this. And so this is essentially adopted from the set of equations I was talking about earlier. And so we basically have a pump transition which goes from state 0 to state 3. And then this transition of course depends on the relative populations of states 0 and state 2 here. And this is tracked by this polarization term, dP2/dt, okay? And then we actually have a spontaneous emission and non-radiative transitions from three to two, two to one, and then one to zero. And then we also have a relatively strong stimulated emission as we go from two to one, so that's the laser transition. So, that shows that at least in principle, this kind of setup should be able to give us the behavior that we're looking for. 

[Slide 6] And now if we look at the population inversion as the function of the electric field amplitude, you can see first of all, just like with a uniform medium that has no cavities surrounding it. That as you increase the electric field amplitude, then the upper level population goes up and then you can approach inversion even for a two state system. But what's interesting is that if you put a photonic crystal cavity around that then the level of upper state population actually goes up more than an order of magnitude. So then there's an advantage to doing that sort of thing compared to the no cavity state. 

[Slide 6] So, then if we look at the lasing transition for this system, then you can see that if we plot, basically, the input power on the x-axis, and then the output power on the y-axis. You can see that initially the rate at which you convert input power to output power is very weak, namely because there's very little stimulated emission happening. But then you can see that there's actually at certain input power, which here in this case, it appears to be on the order of 0.003. But of course it would depend on your units and a lot of details to your system where it actually occurred exactly. You cross what's the lasing threshold. And so then when you cross the lasing threshold, that means that you just have achieved enough population inversion that a stimulated emission will become a dominant process, in general. And so of course there is a upper limit to this as well, which is that as you absorb more and more basically upper level state systems, you start to saturate the upper level and you can't get any more population inversion. So that's going to ultimately limit the efficiency and then that limit is shown as this dotted line here as a function of increasing power. And of course you can see that you're getting very close to this upper possible limit as you go to higher input powers. And so in the next lecture, we'll talk in more detail about what kind of behaviors you see in these photonic crystal laser structures.