Scattering-amplitude equations for one-dimensional band edges
= Scattering-amplitude equations for one-dimensional band edges
If one symmetric cell has reflection and transmission amplitudes $r,t$, then
$$
\operatorname{tr}M
=\frac{(t^2-r^2)e^{ika}+e^{-ika}}t.
$$
At a periodic or antiperiodic band edge, with $\operatorname{tr}M=2\sigma$ and $\sigma=\pm1$, setting $z=e^{-ika}$ gives
$$
z^2-2\sigma tz+t^2-r^2=0,
\qquad
z=\sigma t\pm r.
$$