# Perfectly Matched Layer

The perfectly matched layer (**PML**) approach to implementing absorbing boundary conditions in FDTD simulation was originally proposed in J. Computational Physics, Vol. 114, pp. 185-200, 1994. The approach involves surrounding the computational cell with a medium that in theory absorbs without any reflection electromagnetic waves at *all* frequencies and angles of incidence. The original PML formulation involved "splitting" Maxwell's equations into two sets of equations in the absorbing layers, appropriately defined. These split-field equations produce wave attenuation but are unphysical. It was later shown in IEEE Transactions on Antennas and Propagation, Vol. 43, pp. 1460-3, 1995 that a similar reflectionless absorbing medium can be constructed as a lossy anisotropic dielectric and magnetic material with "matched" impedance and electrical and magnetic conductivities. This is known as the uniaxial PML (UPML).

The finite-difference implementation of PML requires the conductivities to be turned on *gradually* over a distance of a few grid points to avoid numerical reflections from the discontinuity. It is also important when using PMLs to make the computational cell sufficiently large so as not to overlap the PML with evanescent fields from resonant-cavity or waveguide modes (otherwise, the PML could induce artificial losses in such modes). As a rule of thumb, a PML thickness comparable to half the largest wavelength in the simulation usually works well. The PML thickness should be repeatedly doubled until the simulation results are sufficiently converged.

For a more detailed discussion of PMLs in Meep, see Chapter 5 ("Rigorous PML Validation and a Corrected Unsplit PML for Anisotropic Dispersive Media") of the book Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology. In particular, there are two useful references:

- Notes on Perfectly Matched Layers by S. G. Johnson: a general introduction to PML concepts
- J. Computational Physics, Vol. 230, pp. 2369-77, 2011: a description of the precise PML formulation that is used in Meep, which is slightly different from many PML formulations described elsewhere in order to properly handle arbitrary anisotropy. This paper also describes a general strategy to validate PML.

As shown in the figure below, there are two important cases involving *inhomogeneous* media where the fundamental coordinate stretching idea behind PML breaks down giving rise to reflection artifacts. The workaround is to replace the PML with an adiabatic absorber. Additionally, if you operate near a low-group velocity band edge in a periodic media, you may need to make the PML/absorber very thick (overlapping many periods) to be effective. For more details on why PML breaks down in these cases, see Optics Express, Vol. 16, pp. 11376-92, 2008. Also, Physical Review E, Vol. 79, 065601, 2011 describes a separate case where PML fails involving backward-wave modes which may occur at metal-dielectric interfaces (i.e., surface-plasmon polaritons).