nanoHUB U Nanophotonic Modeling/Lecture 4.6: Basis Choices for Beam Propagation Method
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[Slide 1] Hey everyone. This is Lecture 4.6. So in the last lecture we just introduced the beam propagation method. And said that although it looks simple, It could actually have a lot of interesting applications both for fiber propagation as well as nonlinear types of problems. 

[Slide 2] And so now we're actually going to explore in more detail our solution method, and so here we have the so called split step method. Because we had rewritten the solution to the BPM equation as basically e to hu over 2 times e to the hw times e to the hu over 2. So then that means there are basically three steps in the propagation. So, that basically would propagate half a step with the Laplacian, which represents the e to the hu over 2. And then we would propagate a linear phase shift over the full distance. So that's e to the hw, and then finally another half step with the Laplacian. And so then that allows us to map our starting psi into psi propagated all the way along the steps of size h. And then this is actually showing basically how we're going back and forth, so propogating half of the Laplacian here, so basically the beam's spreading. And then the phase shift over here, the full distance, and then back to the Laplacian. 

[Slide 3] So, of course, the biggest challenge for this sort of method is not something as simple as like very uniform fiber, which is just propagating forever, but when you have some sort of inhomogeneity or non-trivial physics in the problem. And so in particular like if the lateral features include nonlinearity, or maybe a non-trivial structure in the x and y directions. So there are a few different ways we could try to handle it. One would be, basically, just a uniform spatial grid. So that would be analogous to what we did with finite difference time domain, just in 2D. And then we could also do fast Fourier transform. And then finally the finite element method. 

[Slide 4] And we haven't talked about that before, but we'll explain in more detail, so first of all, how does the Fast Fourier Transform approach work? So it's well-suited for at least one of the steps, the diffraction step, because we can actually re-write the U that comes at the beginning and end of e to the hu over 2. As something that looks as simple as minus j over 2 beta times k plus g perpendicular squared. Okay, so, that allows us to take something that's otherwise a little messy to write and just write it as a scalar quantity. And so here, basically, the trick is that we transform both into the free domain, and then back to propagate the linear phase shift, and then back into FFT to propagate the diffraction again and then back. 

[Slide 5] So we just keep going back and forth basically. In the spatial grid approach, then of course we just keep everything in real space the whole time. And so here basically you would take this Laplacian operator, and then you would split it up into this uniform grid in the following way. So the easiest way to think about is, if you wanted to take the first spacial derivative, it would be phi of i minus 1 minus phi i over h, like the step size, right? And so if you take a second derivative, then it becomes phi of i minus 1, minus 2 phi i plus phi i plus 1. And you can easily prove that for yourself, if you just try to take the derivative of the derivative. Okay, but then if you're doing in 2D, then of course you've got these extra terms, which basically are assumed to be space by N from your kind of center point, because we have like an end by end grid in this uniform spatial grid by assumption. And so then that's why we have this phi of i minus N and phi i plus N terms here. And so this gives a reasonable approximation. Now, of course, there's a catch to this, which is that this only can be really accurate if you have very fine grid. 

[Slide 6] So, this brings us actually to the finite element method. And, of course, finite element method doesn't only apply to the impropagation method, but this is just a nice way to motivate it. So, this basically would be Instead of using the uniform grid actually to divide this spatial domain in any number of dimensions into a mesh. So it's a generalization of our uniform lattice. And generally it has D plus 1 vertices and D dimensions so presumably, three vertices in 2D, which is being shown here, so basically a bunch of little triangles or the tetrahedron in 3D. Okay, and then it can fill up our whole spatial region, and then we can basically have very small triangles, like right near the edges where inhomogeneity is occurring. But also very large ones, where you don't have inhomogeneity. And it should give us more accurate solutions overall, because we have fine mesh where we need it. And then we don't have it where we don't need it. So it allows us to efficiently do a very intense calculation. In the next lecture, we'll actually explore more details of how to set up this finite element problem.