A perfectly matched layer (PML) is an artificial absorbing layer for wave equations, commonly used to truncate computational regions in numerical methods to simulate problems with open boundaries, especially in the FDTD and FE methods. The key property of a PML that distinguishes it from an ordinary absorbing material is that it is designed so that waves incident upon the PML from a non-PML medium do not reflect at the interface—this property allows the PML to strongly absorb outgoing waves from the interior of a computational region without reflecting them back into the interior. PML was originally formulated by Berenger in 1994 for use with Maxwell's equations, and since that time there have been several related reformulations of PML for both Maxwell's equations and for other wave-type equations, such as elastodynamics, the linearized Euler equations, Helmholtz equations, and poroelasticity. Berenger's original formulation is called a split-field PML, because it splits the electromagnetic fields into two unphysical fields in the PML region. A later formulation that has become more popular because of its simplicity and efficiency is called uniaxial PML or UPML, in which the PML is described as an artificial anisotropic absorbing material. Although both Berenger's formulation and UPML were initially derived by manually constructing the conditions under which incident plane waves do not reflect from the PML interface from a homogeneous medium, both formulations were later shown to be equivalent to a much more elegant and general approach: stretched-coordinate PML. In particular, PMLs were shown to correspond to a coordinate transformation in which one (or more) coordinates are mapped to complex numbers; more technically, this is actually an analytic continuation of the wave equation into complex coordinates, replacing propagating (oscillating) waves by exponentially decaying waves. This viewpoint allows PMLs to be derived for inhomogeneous media such as waveguides, as well as for other coordinate systems and wave equations.
Technical description
Specifically, for a PML designed to absorb waves propagating in the x direction, the following transformation is included in the wave equation. Wherever an x derivative ∂ / ∂ x {\displaystyle \partial /\partial x} appears in the wave equation, it is replaced by:
∂ ∂ x → 1 1 + i σ ( x ) ω ∂ ∂ x {\displaystyle {\frac {\partial }{\partial x}}\to {\frac {1}{1+{\frac {i\sigma (x)}{\omega }}}}{\frac {\partial }{\partial x}}}
where ω {\displaystyle \omega } is the angular frequency and σ {\displaystyle \sigma } is some function of x. Wherever σ {\displaystyle \sigma } is positive, propagating waves are attenuated because:
e i ( k x − ω t ) → e i ( k x − ω t ) − k ω ∫ x σ ( x ′ ) d x ′ , {\displaystyle e^{i(kx-\omega t)}\to e^{i(kx-\omega t)-{\frac {k}{\omega }}\int ^{x}\sigma (x')dx'},}
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