Transmission, * Reflection (10min)

We define the transmission coefficient $T$ and the reflection coefficient $R$ as

$$T:=\frac{k_{N+1}}{k_1}\left\vert\frac{a_{N+1}}{a_1}\right\vert^2,\quad R:=\left\vert\frac{b_1}{a_1}\right\vert^2,$$ (18)

where the scattering condition $b_{N+1}=0$ is assumed. Formulate this scattering condition in words. Show

$$T=\frac{k_{N+1}}{k_1}\frac{1}{\vert M_{11}\vert^2},\quad R =\left\vert\frac{M_{21}}{M_{11}}\right\vert^2,$$ (19)

where $M_{ij}$ are the matrix elements of the transfer matrix.