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. 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. 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. 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 As a consequence, the PML approach is no longer valid (no longer reflectionless at infinite resolution) in the case of periodic media (e.g. photonic crystals or phononic crystals) or even simply a waveguide that enters the boundary at an oblique angle. See also Cagniard–de Hoop method References External links Animation on the effects of PML (YouTube) Numerical differential equations Partial differential equations Wave mechanics Computational electromagnetics