Perfectly matched layer

Last updated
A FDTD scheme for a light scattering problem. The striped borders correspond to perfectly matched layers, which are used to simulate open boundaries by absorbing the outgoing waves. FDTD TFSF (English).png
A FDTD scheme for a light scattering problem. The striped borders correspond to perfectly matched layers, which are used to simulate open boundaries by absorbing the outgoing waves.

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. [1] [2] 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.

Contents

PML was originally formulated by Berenger in 1994 [3] 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, [4] 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, [5] 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. [6] [7] 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. [8] [9]

Technical description

Absorption of a pulsed spherical wave through stretched coordinate PML in 2D FDTD method. The white border indicates the simulation boundary.

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 appears in the wave equation, it is replaced by:

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

where we have taken a planewave propagating in the +x direction (for ) and applied the transformation (analytic continuation) to complex coordinates: , or equivalently . The same coordinate transformation causes waves to attenuate whenever their x dependence is in the form 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. 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.

Limitations of perfectly matched layers

PML is widely used and has become the absorbing boundary technique of choice in much of computational electromagnetism. [1] Although it works well in most cases, there are a few important cases in which it breaks down, suffering from unavoidable reflections or even exponential growth.

One caveat with perfectly matched layers is that they are only reflectionless for the exact, continuous wave equation. Once the wave equation is discretized for simulation on a computer, some small numerical reflections appear (which vanish with increasing resolution). 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. [1] In general, any absorber, whether PML or not, is reflectionless in the limit where it turns on sufficiently gradually (and the absorbing layer becomes thicker), but in a discretized system the benefit of PML is to reduce the finite-thickness "transition" reflection by many orders of magnitude compared to a simple isotropic absorption coefficient. [10]

In certain materials, there are "backward-wave" solutions in which group and phase velocity are opposite to one another. This occurs in "left-handed" negative index metamaterials for electromagnetism and also for acoustic waves in certain solid materials, and in these cases the standard PML formulation is unstable: it leads to exponential growth rather than decay, simply because the sign of k is flipped in the analysis above. [11] Fortunately, there is a simple solution in a left-handed medium (for which all waves are backwards): merely flip the sign of σ. A complication, however, is that physical left-handed materials are dispersive: they are only left-handed within a certain frequency range, and therefore the σ coefficient must be made frequency-dependent. [12] [13] Unfortunately, even without exotic materials, one can design certain waveguiding structures (such as a hollow metal tube with a high-index cylinder in its center) that exhibit both backwards- and forwards-wave solutions at the same frequency, such that any sign choice for σ will lead to exponential growth, and in such cases PML appears to be irrecoverably unstable. [14]

Another important limitation of PML is that it requires that the medium be invariant in the direction orthogonal to the boundary, in order to support the analytic continuation of the solution to complex coordinates (the complex "coordinate stretching"). As a consequence, the PML approach is no longer valid (no longer reflectionless at infinite resolution) in the case of periodic media (e.g. photonic crystals or phononic crystals) [10] or even simply a waveguide that enters the boundary at an oblique angle. [15]

See also

Related Research Articles

In physics, the cross section is a measure of the probability that a specific process will take place in a collision of two particles. For example, the Rutherford cross-section is a measure of probability that an alpha particle will be deflected by a given angle during an interaction with an atomic nucleus. Cross section is typically denoted σ (sigma) and is expressed in units of area, more specifically in barns. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.

<span class="mw-page-title-main">Maxwell's equations</span> Equations describing classical electromagnetism

Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, electric and magnetic circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar, etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon. The modern form of the equations in their most common formulation is credited to Oliver Heaviside.

<span class="mw-page-title-main">Uncertainty principle</span> Foundational principle in quantum physics

The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. In other words, the more accurately one property is measured, the less accurately the other property can be known.

<span class="mw-page-title-main">Active laser medium</span> Source of optical gain in a laser

The active laser medium is the source of optical gain within a laser. The gain results from the stimulated emission of photons through electronic or molecular transitions to a lower energy state from a higher energy state previously populated by a pump source.

<span class="mw-page-title-main">Cutoff frequency</span> Frequency response boundary

In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced rather than passing through.

The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the dimensionless change in magnitude or phase per unit length. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.

<span class="mw-page-title-main">Skin effect</span> Tendency of AC current flow in a conductors outer layer

In electromagnetism, skin effect is the tendency of an alternating electric current (AC) to become distributed within a conductor such that the current density is largest near the surface of the conductor and decreases exponentially with greater depths in the conductor. It is caused by opposing eddy currents induced by the changing magnetic field resulting from the alternating current. The electric current flows mainly at the skin of the conductor, between the outer surface and a level called the skin depth.

<span class="mw-page-title-main">Love wave</span> Horizontally polarized surface waves

In elastodynamics, Love waves, named after Augustus Edward Hough Love, are horizontally polarized surface waves. The Love wave is a result of the interference of many shear waves (S-waves) guided by an elastic layer, which is welded to an elastic half space on one side while bordering a vacuum on the other side. In seismology, Love waves (also known as Q waves (Quer: German for lateral)) are surface seismic waves that cause horizontal shifting of the Earth during an earthquake. Augustus Edward Hough Love predicted the existence of Love waves mathematically in 1911. They form a distinct class, different from other types of seismic waves, such as P-waves and S-waves (both body waves), or Rayleigh waves (another type of surface wave). Love waves travel with a lower velocity than P- or S- waves, but faster than Rayleigh waves. These waves are observed only when there is a low velocity layer overlying a high velocity layer/ sub–layers.

<span class="mw-page-title-main">Gaussian filter</span> Filter in electronics and signal processing

In electronics and signal processing, mainly in digital signal processing, a Gaussian filter is a filter whose impulse response is a Gaussian function. Gaussian filters have the properties of having no overshoot to a step function input while minimizing the rise and fall time. This behavior is closely connected to the fact that the Gaussian filter has the minimum possible group delay. A Gaussian filter will have the best combination of suppression of high frequencies while also minimizing spatial spread, being the critical point of the uncertainty principle. These properties are important in areas such as oscilloscopes and digital telecommunication systems.

In electrical engineering, dielectric loss quantifies a dielectric material's inherent dissipation of electromagnetic energy. It can be parameterized in terms of either the loss angleδ or the corresponding loss tangenttan(δ). Both refer to the phasor in the complex plane whose real and imaginary parts are the resistive (lossy) component of an electromagnetic field and its reactive (lossless) counterpart.

The word electricity refers generally to the movement of electrons, or other charge carriers, through a conductor in the presence of a potential difference or an electric field. The speed of this flow has multiple meanings. In everyday electrical and electronic devices, the signals travel as electromagnetic waves typically at 50%–99% of the speed of light in vacuum. The electrons themselves move much more slowly. See drift velocity and electron mobility.

The linear attenuation coefficient, attenuation coefficient, or narrow-beam attenuation coefficient characterizes how easily a volume of material can be penetrated by a beam of light, sound, particles, or other energy or matter. A coefficient value that is large represents a beam becoming 'attenuated' as it passes through a given medium, while a small value represents that the medium had little effect on loss. The (derived) SI unit of attenuation coefficient is the reciprocal metre (m−1). Extinction coefficient is another term for this quantity, often used in meteorology and climatology. Most commonly, the quantity measures the exponential decay of intensity, that is, the value of downward e-folding distance of the original intensity as the energy of the intensity passes through a unit thickness of material, so that an attenuation coefficient of 1 m−1 means that after passing through 1 metre, the radiation will be reduced by a factor of e, and for material with a coefficient of 2 m−1, it will be reduced twice by e, or e2. Other measures may use a different factor than e, such as the decadic attenuation coefficient below. The broad-beam attenuation coefficient counts forward-scattered radiation as transmitted rather than attenuated, and is more applicable to radiation shielding. The mass attenuation coefficient is the attenuation coefficient normalized by the density of the material.

<span class="mw-page-title-main">Stokes wave</span> Nonlinear and periodic surface wave on an inviscid fluid layer of constant mean depth

In fluid dynamics, a Stokes wave is a nonlinear and periodic surface wave on an inviscid fluid layer of constant mean depth. This type of modelling has its origins in the mid 19th century when Sir George Stokes – using a perturbation series approach, now known as the Stokes expansion – obtained approximate solutions for nonlinear wave motion.

When an electromagnetic wave travels through a medium in which it gets attenuated, it undergoes exponential decay as described by the Beer–Lambert law. However, there are many possible ways to characterize the wave and how quickly it is attenuated. This article describes the mathematical relationships among:

In fluid dynamics, Airy wave theory gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mean depth, and that the fluid flow is inviscid, incompressible and irrotational. This theory was first published, in correct form, by George Biddell Airy in the 19th century.

<span class="mw-page-title-main">Substrate-integrated waveguide</span> Waveguide formed by posts inserted in a dielectric substrate

A substrate-integrated waveguide (SIW) is a synthetic rectangular electromagnetic waveguide formed in a dielectric substrate by densely arraying metallized posts or via holes that connect the upper and lower metal plates of the substrate. The waveguide can be easily fabricated with low-cost mass-production using through-hole techniques, where the post walls consists of via fences. SIW is known to have similar guided wave and mode characteristics to conventional rectangular waveguide with equivalent guide wavelength.

In acoustics, acoustic attenuation is a measure of the energy loss of sound propagation through an acoustic transmission medium. Most media have viscosity and are therefore not ideal media. When sound propagates in such media, there is always thermal consumption of energy caused by viscosity. This effect can be quantified through the Stokes's law of sound attenuation. Sound attenuation may also be a result of heat conductivity in the media as has been shown by G. Kirchhoff in 1868. The Stokes-Kirchhoff attenuation formula takes into account both viscosity and thermal conductivity effects.

In computational fluid dynamics, the k–omega (k–ω) turbulence model is a common two-equation turbulence model, that is used as an approximation for the Reynolds-averaged Navier–Stokes equations (RANS equations). The model attempts to predict turbulence by two partial differential equations for two variables, k and ω, with the first variable being the turbulence kinetic energy (k) while the second (ω) is the specific rate of dissipation (of the turbulence kinetic energy k into internal thermal energy).

Menter's Shear Stress Transport turbulence model, or SST, is a widely used and robust two-equation eddy-viscosity turbulence model used in Computational Fluid Dynamics. The model combines the k-omega turbulence model and K-epsilon turbulence model such that the k-omega is used in the inner region of the boundary layer and switches to the k-epsilon in the free shear flow.

<span class="mw-page-title-main">JCMsuite</span> Simulation software

JCMsuite is a finite element analysis software package for the simulation and analysis of electromagnetic waves, elasticity and heat conduction. It also allows a mutual coupling between its optical, heat conduction and continuum mechanics solvers. The software is mainly applied for the analysis and optimization of nanooptical and microoptical systems. Its applications in research and development projects include dimensional metrology systems, photolithographic systems, photonic crystal fibers, VCSELs, Quantum-Dot emitters, light trapping in solar cells, and plasmonic systems. The design tasks can be embedded into the high-level scripting languages MATLAB and Python, enabling a scripting of design setups in order to define parameter dependent problems or to run parameter scans.

References

  1. 1 2 3 Allen Taflove and Susan C. Hagness (2005). Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Artech House Publishers. ISBN   978-1-58053-832-9.
  2. ↑ Johnson, Steven G. (2021). "Notes on Perfectly Matched Layers (PMLs)". arXiv: 2108.05348 [physics.comp-ph]. Tutorial review based on online MIT course notes.
  3. ↑ J. Berenger (1994). "A perfectly matched layer for the absorption of electromagnetic waves". Journal of Computational Physics. 114 (2): 185–200. Bibcode:1994JCoPh.114..185B. doi:10.1006/jcph.1994.1159.
  4. ↑ Fathi, Arash; Poursartip, Babak; Kallivokas, Loukas (2015). "Time-domain hybrid formulations for wave simulations in three-dimensional PML-truncated heterogeneous media". International Journal for Numerical Methods in Engineering. 101 (3): 165–198. Bibcode:2015IJNME.101..165F. doi:10.1002/nme.4780. S2CID   122812832.
  5. ↑ S.D. Gedney (1996). "An anisotropic perfectly matched layer absorbing media for the truncation of FDTD latices". IEEE Transactions on Antennas and Propagation. 44 (12): 1630–1639. Bibcode:1996ITAP...44.1630G. doi:10.1109/8.546249.
  6. ↑ W. C. Chew and W. H. Weedon (1994). "A 3d perfectly matched medium from modified Maxwell's equations with stretched coordinates". Microwave and Optical Technology Letters. 7 (13): 599–604. Bibcode:1994MiOTL...7..599C. doi:10.1002/mop.4650071304.
  7. ↑ 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.
  8. ↑ V. Kalvin (2012). "Limiting absorption principle and perfectly matched layer method for Dirichlet Laplacian in quasi-cylindrical domains". SIAM J. Math. Anal. 44: 355–382. arXiv: 1110.4912 . doi:10.1137/110834287. S2CID   2625082.
  9. ↑ V. Kalvin (2013). "Analysis of perfectly matched layer operators for acoustic scattering on manifolds with quasicylindrical ends". J. Math. Pures Appl. 100 (2): 204–219. arXiv: 1212.5707 . doi:10.1016/j.matpur.2012.12.001. S2CID   119315209.
  10. 1 2 A. F. Oskooi, L. Zhang, Y. Avniel, and S. G. Johnson, The failure of perfectly matched layers, and towards their redemption by adiabatic absorbers, Optics Express16, 11376–11392 (2008).
  11. ↑ E. Bécache, S. Fauqueux and P. Joly (2003). "Stability of perfectly matched layers, group velocities and anisotropic waves" (PDF). Journal of Computational Physics. 188 (2): 399–433. Bibcode:2003JCoPh.188..399B. doi:10.1016/S0021-9991(03)00184-0. S2CID   18020140.
  12. ↑ Cummer Steven A (2004). "Perfectly Matched Layer Behavior in Negative Refractive Index Materials". IEEE Antennas and Wireless Propagation Letters. 3 (9): 172–175. Bibcode:2004IAWPL...3..172C. doi:10.1109/lawp.2004.833710. S2CID   18838504.
  13. ↑ Dong X. T., Rao X. S., Gan Y. B., Guo B., Yin W.-Y. (2004). "Perfectly matched layer-absorbing boundary condition for left-handed materials". IEEE Microwave and Wireless Components Letters. 14 (6): 301–333. doi:10.1109/lmwc.2004.827104. S2CID   19568400.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  14. ↑ Loh P.-R., Oskooi A. F., Ibanescu M., Skorobogatiy M., Johnson S. G. (2009). "Fundamental relation between phase and group velocity, and application to the failure of perfectly matched layers in backward-wave structures" (PDF). Phys. Rev. E. 79 (6): 065601. Bibcode:2009PhRvE..79f5601L. doi:10.1103/physreve.79.065601. hdl: 1721.1/51780 . PMID   19658556.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  15. ↑ Oskooi A., Johnson S. G. (2011). "Distinguishing correct from incorrect PML proposals and a corrected unsplit PML for anisotropic, dispersive media" (PDF). Journal of Computational Physics. 230 (7): 2369–2377. Bibcode:2011JCoPh.230.2369O. doi:10.1016/j.jcp.2011.01.006.
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.