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The question 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 $\tilde{A}(0)\neq 0$. \[ F(z;z_0)=   \begin{bmatrix}  \frac{\beta_c cos \beta_c(z-z_0)-i\delta sin \beta_c(z-z_0)}{\beta_c} $ \frac{i \kappa_{ab}}{\beta_c}sin\beta_c(z-z_0)e^{i \delta (z+z_0)\\  \end{bmatrix}  \tilde{B}(z_0) \\   \end{bmatrix}\]