Wimbledon, Numerical methods: Perfectly Matched Layer, and the Ashes

So the summer (or what passes for it in Britain) is here. The only way to tell is that Wimbledon has started and the Ashes are round the corner. Since the lack of sunshine and reasonable summer temperatures are not reliable I find these sport events a more fun way to tell the seasons.

Now the problem is that all I want to do is watch the tennis and then cricket when it starts. How am I supposed to get any work done? Is it really fair to ask a body to be chained to their desk when there is some glorious serve and volley on at SW19?

So here is my compromise: I’ll write a blogpost instead of reading papers or running simulations. My excuse is this is closer to work than the sport and is as enjoyable.

And so, today I want to talk about the Perfectly Matched Layer or PML boundary condition.

The PML boundary was proposed by Berenger in 1990s as a way to terminate meshes in Finite Difference Time Domain (FDTD) method and has since become the most popular boundary condition in numerical methods.  But lets take a step back: what is a boundary condition and why do we need them?

Numerical methods such as FDTD and Finite Element method (FEM) create a mesh of points or nodes that represents the optical structure of interest. In the real world each structure has a finite extent and optical fields in these structures slowly die out once they leave the structure (due to leakage, or as radiation modes etc.). However, with the numerical mesh, when an optical wave hits the edge of the finite mesh (or the numerical boundary) it gets reflected back into the main domain. This reflection is unphysical and contaminates the solution. Hence boundary condition operators are used to suppress such reflections.

The PML is an example of the physical absorber approach in which a layer of an artificial medium is placed adjacent to the boundary to absorb all outgoing waves from the structure of interest. The name PML derives from its properties: the ability to absorb waves incident at any angle and frequency. The concept is that a medium with permittivity matching that of the main computational domain material is placed at the end of the domain. Since the impedance of the two media are identical, there is no reflection at the interface, even for change in angle of incidence.

There are two distinct ways to formulate the PML: the anisotropic medium formulation or uniaxial PML, and the complex coordinate stretching approach. The latter regards the PML as a mapping onto complex coordinates. When the transverse variable is a complex number, the wave travelling in the transverse direction gets attenuated. The PML width and absorption profile are calculated to absorb the field and reduce reflection into the domain to the tolerance required.

Some key things to consider when using PML in simulations is to work out the width of the PML region, how far to keep the PML from the structure of interest, and the profile that describes the PML absorptivity. PML width, it’s distance from the structure and number of mesh points to place inside the PML can usually be optimized by some trial and error, and thereafter experience. For example, I usually keep PML width for passive devices to be 10% of the full window.

The PML absorptivity profile is usually taken as a square dependence, though other profiles may give better performance. In a paper that is what I worked on with my colleagues. We found that PML with sin hyperbolic profiles were more efficient than the usual power law profile.

Have a read and let me know what you think!

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