Week 5: Modeling Framework

Welcome back to my blog! Last blog, I used S4 to analyze the transmission and reflection at an air-water interface. I confirmed that S4 matched with what we’d expect from p-polarized light incident at Brewster’s angle. With a general idea of what S4 does, I needed to understand why.

For the past week or so, I’ve been analyzing the S4 framework. S4 is quite flexible, allowing the modeling of diffraction gratings and crystalline structures. I’ll leave that experiment to another blog. For now, let’s just dive right into the documentation.

The Simulation Object

Most of the following information is found on S4 documentation, please refer there for more details [1].

The bulk of functions are run through a Simulation Object. Each simulation object contains information on every layer, its lattice geometry, material, and thickness. S4 offers a very flexible range of possible structures, including periodic patterns. After specifying the geometry, I send an excited plane-wave with a specified polarization through S4, and S4 produces a graph of transmission and reflective flux.

So that’s how to run S4 on the surface level. What really goes on internally? Remember, S4 uses FMM. Generally, the first step is to solve the eigenmode problem followed by a set of linearized Maxwell’s equations for a single layer. This involves solving a series of boundary conditions. Next, S4 continues to solve the other layers with the S-matrix method.

For the purposes of nonlinear optics, I’ll need to eventually figure out the relation between the effective linear susceptibility with a materials permittivity/permeability. For now, this blog will be relatively short as I continue to dig into S4s sourcecode.

[1] Victor Liu and Shanhui Fan, “S4: A free electromagnetic solver for layered periodic structures,” Computer Physics Communications 183, 2233-2244 (2012) http://dx.doi.org/10.1016/j.cpc.2012.04.026. 2. Michael Ghebrebrhan, Peter Bermel, Yehuda Avniel, John D. Joannopoulos, Steven G. Johnson, “Global optimization of silicon photovoltaic cell front coatings”, Optics Express 17, 7505 (2009).