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PML attenuation vanishes at grazing and excludes evanescent modes

Example

For a propagating angle $0 \leq θ < π / 2$ , $q = k \cos θ$ and the one-way PML amplitude after accumulated stretch $S$ is $e^{- k S \cos θ}$ . It equals $e^{- k S}$ at normal incidence and tends to $1$ as $θ ↑ π / 2$ . For a purely evanescent normal branch $q = i α$ with $α > 0$ , the factor $e^{i q X} = e^{- α x - i α S (x)}$ has modulus $e^{- α x}$ , so this imaginary stretch adds no modulus decay.

Facts & Assumptions

Given: The Cartesian stretch [F1] and propagating-mode theorem [F2].

[F1]

Cartesian complex-stretch PML for a homogeneous Helmholtz mode: $X = x + i S (x)$ and a separated normal mode has factor $e^{i q X}$ .

[F2]

Perfect matching and attenuation for non-grazing propagating modes: for real $q > 0$ , the one-way amplitude factor is $e^{- q S}$ .

Verification

Given: $k, S \geq 0$ and the modes in [F1]–[F2].

1.1F1F2algebra

For a propagating mode, substitute $q = k \cos θ$ into [F2]. At $θ = 0$ the factor is $e^{- k S}$ ; as $θ ↑ π / 2$ , $\cos θ \to 0$ , so it tends to $1$ .

2.1F1step 1.1algebra∎

For $q = i α$ , direct substitution into [F1] gives $e^{i q X} = e^{i (i α) (x + i S)} = e^{- α x - i α S}$ , whose modulus is $e^{- α x}$ . The added stretch contributes only a phase, confirming that the propagating attenuation theorem does not cover evanescent modes.

Depends on

Used by

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Sources

Steven G. Johnson, Notes on Perfectly Matched Layers (PMLs), §§6, 7.2 (standard reference, not scraped)