Given some potential $V(x)$, we can describe the amplitude of incoming and outgoing waves through the scattering matrix $S$ whereby $$\begin{pmatrix} B \\ F \end{pmatrix}= \begin{pmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{pmatrix} \begin{pmatrix} A \\ G \end{pmatrix}.$$ I understand that the coefficients of the $S$ matrix can give us some information regarding the probabilities of transmission and reflection. I believe however it can also give us some information regarding the phase shift of an incoming and outgoing wave. How may we determine this relationship? Would it be a ratio $F/A$ for example, relating the amplitudes of a transmitted incoming wave through a potential?