Perfectly matched layer

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 FEM 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 equations. 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 (Gedney, 1996), 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 (Chew and Weedon, 1994; Teixeira and Chew, 1998). 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 $\partial /\partial x$ appears in the wave equation, it is replaced by:

${\frac {\partial }{\partial x}}\to {\frac {1}{1+{\frac {i\sigma (x)}{\omega }}}}{\frac {\partial }{\partial x}}$

where ω is the angular frequency and σ is some function of x. Wherever σ is positive, propagating waves are attenuated because:

$e^{i(kx-\omega t)}\to e^{i(kx-\omega t)-{\frac {k}{\omega }}\int ^{x}\sigma (x')dx'},$

where we have taken a planewave propagating in the +x direction (for $k>0$) and applied the transformation (analytic continuation) to complex coordinates: $x\to x+{\frac {i}{\omega }}\int ^{x}\sigma (x')dx'$, or equivalently $dx\to dx(1+i\sigma /\omega )$. The same coordinate transformation causes waves to attenuate whenever their x dependence is in the form $e^{ikx}$ for some propagation constant k: this includes planewaves propagating at some angle with the x axis and also transverse modes of a waveguide.

The above coordinate transformation can be left as-is in the transformed wave equations, or can be combined with the material description (e.g. the permittivity and permeability in Maxwell's equations) to form a UPML description. Note also that the coefficient σ/ω depends upon frequency—this is so the attenuation rate is proportional to k/ω, which is independent of frequency in a homogeneous material (not including material dispersion, e.g. for vacuum) because of the dispersion relation between ω and k. However, this frequency-dependence means that a time domain implementation of PML, e.g. in the FDTD method, is more complicated than for a frequency-independent absorber, and involves the auxiliary differential equation (ADE) approach (equivalently, i/ω appears as an integral or convolution in time domain).

Perfectly matched layers, in their original form, only attenuate propagating waves; purely evanescent waves (exponentially decaying fields) oscillate in the PML but do not decay more quickly. However, the attenuation of evanescent waves can also be accelerated by including a real coordinate stretching in the PML: this corresponds to making σ in the above expression a complex number, where the imaginary part yields a real coordinate stretching that causes evanescent waves to decay more quickly.

One caveat with perfectly matched layers is that they are only reflectionless for the exact wave equation. Once the wave equation is discretized for simulation on a computer, some small numerical reflections appear. For this reason, the PML absorption coefficient σ is typically turned on gradually from zero (e.g. quadratically) over a short distance on the scale of the wavelength of the wave.

References

• J. Berenger (1994). "A perfectly matched layer for the absorption of electromagnetic waves". Journal of Computational Physics. 114 (2): 185–200. doi:10.1006/jcph.1994.1159.
• S.D. Gedney (1996). "An anisotropic perfectly matched layer absorbing media for the truncation of FDTD latices". Antennas and Propagation, IEEE Transactions on. 44 (12): 1630–1639. doi:10.1109/8.546249.
• W. C. Chew and W. H. Weedon (1994). "A 3d perfectly matched medium from modified Maxwell's equations with stretched coordinates". Microwave Optical Tech. Letters. 7 (13): 599–604. doi:10.1002/mop.4650071304.
• F. L. Teixeira W. C. Chew (1998). "General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media". IEEE Microwave and Guided Wave Letters. 8 (6): 223–225. doi:10.1109/75.678571.
• Allen Taflove and Susan C. Hagness (2005). Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Artech House Publishers. ISBN 1-58053-832-0.
• S. G. Johnson, Notes on Perfectly Matched Layers, online MIT course notes (Aug. 2007).