Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 39A / c / Solution

Matching at $z=L$ first expresses the middle solution in terms of $A_T$. Propagating its value and derivative across the layer of thickness $2L$ uses the transfer matrixMatching this result to $A_I+A_R$ and $-im{A}_I+im{A}_R$ at $z=-L$, then eliminating $A_R$, gives