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\begin{bmatrix}  \tilde{A}(z_0)\\  \tilde{B}(z_0) \\   \end{bmatrix}\] The question in hand is to solve this subject to the initial condition of when power is launched only into the mode a at $z_0=0$. This means $\tilde{B}(0)=0$ and $\tilde{A}(0)\neq 0$