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. 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. 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. In certain materials, there are "backward-wave" solutions