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| A '''perfectly matched layer''' ('''PML''') is an artificial absorbing layer for [[wave equation]]s, commonly used to truncate computational regions in [[numerical method]]s to simulate problems with open boundaries, especially in the [[Finite-difference time-domain method|FDTD]] and [[finite element method|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.
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| 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 field]]s 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 [[birefringence|anisotropic]] absorbing material. Although both Berenger's formulation and UPML were initially derived by manually constructing the conditions under which incident [[plane wave]]s 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 number]]s; 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 [[waveguide]]s, as well as for other [[coordinate system]]s and wave equations.
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| ==Technical description==
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| 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 <math>\partial/\partial x</math> appears in the wave equation, it is replaced by:
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| :<math>\frac{\partial}{\partial x} \to \frac{1}{1 + \frac{i\sigma(x)}{\omega}} \frac{\partial}{\partial x}</math>
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| where ω is the [[angular frequency]] and σ is some [[function (mathematics)|function]] of ''x''. Wherever σ is positive, propagating waves are attenuated because:
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| :<math>e^{i(kx - \omega t)} \to e^{i(kx - \omega t) - \frac{k}{\omega} \int^x \sigma(x') dx'} ,</math>
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| where we have taken a planewave propagating in the +''x'' direction (for <math>k > 0</math>) and applied the transformation (analytic continuation) to complex coordinates: <math>x \to x + \frac{i}{\omega} \int^x \sigma(x') dx'</math>, or equivalently <math>dx \to dx (1 + i\sigma/\omega)</math>. The same coordinate transformation causes waves to attenuate whenever their ''x'' dependence is in the form <math>e^{ikx}</math> for some [[propagation constant]] ''k'': this includes planewaves propagating at some angle with the ''x'' axis and also [[transverse mode]]s of a waveguide.
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| 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 (electromagnetism)|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).
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| 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 number|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.
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| One caveat with perfectly matched layers is that they are only reflectionless for the ''exact'' wave equation. Once the wave equation is [[discretization|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. [[quadratic function|quadratically]]) over a short distance on the scale of the [[wavelength]] of the wave.
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| ==When perfectly matched layers fail==
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| Perfectly matched layers have shown their efficiency in a lot of situations : acoustic or electromagnetic waves, Schrödinger equation, etc. But PML fail for media where backward waves appear as in [[negative index metamaterials]] or in [[Plasma_(physics)|plasma]], but also for anisotropic acoustic and elastic waves, for aeroacoustic waves etc.
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| Backward waves are waves with opposite [[Group velocity|group]] and [[phase velocity]]. The outgoing waves are the ones with a group velocity which are pointed to the exterior of the domain. But perfectly matched layers attenuate waves according to their phase velocity ''k''/ω and not their group velocity. So perfectly matched layers amplify exponentially backward waves instead of attenuating them (Bécache, Fauqueux and Joly, 2003).
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| == References ==
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| *{{cite journal | author= J. Berenger | title= A perfectly matched layer for the absorption of electromagnetic waves | journal= Journal of Computational Physics | year= 1994 | volume= 114 | pages= 185–200 | doi= 10.1006/jcph.1994.1159 | issue= 2 | bibcode=1994JCoPh.114..185B}}
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| *{{cite journal | author= S.D. Gedney | title= An anisotropic perfectly matched layer absorbing media for the truncation of FDTD latices| journal= Antennas and Propagation, IEEE Transactions on | year= 1996 | volume= 44 | pages= 1630–1639 | doi= 10.1109/8.546249 | issue= 12 | bibcode=1996ITAP...44.1630G}}
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| *{{cite journal | author= W. C. Chew and W. H. Weedon | title= A 3d perfectly matched medium from modified Maxwell's equations with stretched coordinates| journal= Microwave Optical Tech. Letters | year= 1994 | volume= 7 | pages= 599–604 | doi= 10.1002/mop.4650071304 | issue= 13 }}
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| *{{cite journal | author= F. L. Teixeira W. C. Chew | title= General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media| journal= IEEE Microwave and Guided Wave Letters | year= 1998 | volume= 8 | pages= 223–225 | doi= 10.1109/75.678571 | issue= 6 }}
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| *{{cite journal | author= E. Bécache, S. Fauqueux and P. Joly| title= Stability of perfectly matched layers, group velocities and anisotropic waves| journal= Journal of Computational Physics | year= 2003 | volume= 188 | pages= 399–433| doi=10.1016/S0021-9991(03)00184-0 | issue= 2}} [http://hal.archives-ouvertes.fr/docs/00/07/22/83/PDF/RR-4304.pdf]
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| *{{cite book | author=[[Allen Taflove]] and Susan C. Hagness | title=Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. | publisher=Artech House Publishers | year=2005 | isbn=1-58053-832-0 }}
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| * S. G. Johnson, [http://math.mit.edu/~stevenj/18.369/pml.pdf Notes on Perfectly Matched Layers], online MIT course notes (Aug. 2007).
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| *{{cite journal | author= R. M. S. de Oliveira and C. L. S. S. Sobrinho| title= UPML formulation for truncating conductive media in curvilinear coordinates| journal= Numerical Algorithms | year= 2007 | volume= 46 | pages= 295–319| doi= 10.1007/s11075-007-9139-6 }} [http://download.springer.com/static/pdf/502/art%253A10.1007%252Fs11075-007-9139-6.pdf?auth66=1391108927_9cfae44a5698a56067166a18abd13095&ext=.pdf]
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| ==External links==
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| *[http://www.youtube.com/watch?v=XcL9iEK0GDY Animation on the effects of PML (YouTube)]
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| [[Category:Numerical differential equations]]
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| [[Category:Partial differential equations]]
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| [[Category:Wave mechanics]]
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