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Scattering-amplitude equations for one-dimensional band edges

If one symmetric cell has reflection and transmission amplitudes

r, t

, then

tr M = \frac{( t ^{2} - r ^{2} ) e ^{ika} + e ^{- ika}}{t} .

At a periodic or antiperiodic band edge, with

tr M = 2 σ

and

σ = \pm 1

, setting

z = e^{- ika}

gives

z^{2} - 2 σ t z + t^{2} - r^{2} = 0, z = σ t \pm r .

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