nanoHUB U Nanophotonic Modeling/Lecture 1.3: 1D Bandstructures
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[Slide 1] Hey everyone, welcome to Lecture 1.3. So in the last lecture, we were just talking about how we can create a box solution to the bandstructure problem. And so now we're actually going to figure out what kind of basis should we use to solve it. 

[Slide 2] We just talked about last time, I just wanted to illustrate it in more detail, that the real space basis is not positive definite. So in other words, you'll get negative eigenvalues for that system because you are subtracting in the eigenvalue side, the value of h-bar squared k squared over 2m. And so then that actually shows up if you run the eigenfunction in MATLAB as follows, then you can actually see explicitly that there are a bunch of negative values. So it's not as ideal of a solution method. Of course, we could try it. 

[Slide 3] But it will kind of slow us down. So instead, we want to consider another basis which is the Fourier basis. And of course, this was alluded to in the quotation by Felix Bloch himself, who of course came up with the Bloch solution that we should use Fourier analysis and more specifically, we should use a Fourier series. And the Fourier series, the advantage is because we know what the periodicity is, that also means that we know what all the coefficients that are associated with the periodicity should be. Or in other words, we can create a set of G values that match with the lattice that we're interested in. And we'll explain a little bit later in more detail how those work. But taking this as a given that we kind of know the set of G values. Then we can obtain the nice recursion relationship from the Schrodinger's equation. Which relates basically the G prime component of the potential in Fourier basis times this coefficient of c of G minus G prime to this kind of value E of k minus h-bar squared over 2m. k plus G squared times the original coefficient, c of G for the periodic function. And so that of course harkens back to this. 

[Slide 4] And so then we can actually solve it numerically. Again in MATLAB, setting up the problem, we get something like this. Now of course, this is a lot of details. So you don't have to read in detail. But you can see basically, what we're doing is we're performing a pass Fourier transform on the potential that we've represented in the system spatially into the Fourier domain. So that gives us all of the v of g prime components like that would exist in that recursion relationship. And then we can actually set up the Hamiltonion in the Fourier basis, which kind of looks like this. And then we can actually solve the Hamiltonian which has, of course, the potential component v plus the kinetic energy t which is on the diagonal. And then we can actually solve for the wave function as well as the wave vectors. And so then our psi kind of looks like this. And then our eigenvalues look like this. And so you can see a very nice property has emerged from the Fourier solution, that all the values that we're obtaining are actually greater than zero, which is something that we had said was important. So in other words, this kind of formulation is positive definite. 

[Slide 5] And so when we look at the kind of solutions that we get from this sort of approach, then of course we know from before that the energy as a function of k would go quadratically with k. So that was just sort of the free particle, but then once we've introduced the periodicity, then what happens actually is we create this gap. And so this gap actually depends on the magnitude of the Fourier component. Of course if the Fourier component with periodicity associated with the structure is 0, then of course it doesn't show up. But as the potential component or variation becomes larger, with that wavelength, then it becomes a bigger factor and creates a bigger gap between, as you can see it's called A and B. The first and second allowed bands. So that gap is something that you can also derive from nearly degenerate or degenerate perturbation theory in quantum mechanics. And this is discussed in a lot more detail in Charles Kittel's book on solid state physics. 

[Slide 6] So in the context of photonic band structures, you get very similar phenomenon, of course which is that of course, the biggest difference is just that the dispersion relationship is linear because you probably know that in photons, omega equal ck so in other words, the frequency is proportional to wave factor. And then in free material or basically homogenous dielectric, it basically looks like a diamond type structure. But then if we introduce like a small but nonzero periodicity into the structure, we actually open up the very small bandgap. If, for example, you had two materials which had dielectric constants of 12 and 13 alternate. Okay, but then if you alternate between dielectric constants of say, 13 and 1, then you actually start to see a much bigger band gap. And that's the yellow region. And then this big band gap just means that you have a larger potential Fourier component in the periodic direction. Then the practical implication of this, which is also kind of well known from perturbation theory is nearly degenerate perturbation theory specifically, is that you'll have some field localization in the so-called blue regions which are gallium arsenide type dielectric. And then you'll have other field localization at the bottom of the second band which is up here or up here, which is in the low dielectric constant. Because of course, it takes less energy to be in the high index material. So then that kind of illustrates the general trend and the separation between modes and why there's this physical gap. So basically the localization of the modes into different regions gives rise to that energy difference. And that's why it's a so-called forbidden gap. Because there's no way to kind of hedge the difference between those two. In the next lecture, we'll talk in more detail about how we can set ourselves up for more challenging problems in 1D photonics.