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\end{bmatrix}\]  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 $\tilde{A}(0)\neq 0$. \[ F(z;z_0)=   \begin{bmatrix}  \frac{\beta_c cos \beta_c(z-z_0)-i\delta sin \beta_c(z-z_0)}{\beta_c})e^{i \beta_c(z-z_0)}{\beta_c}e^{i  \delta (z+z_0)} & \frac{i \kappa_{ab}}{\beta_c}sin\beta_c(z-z_0)e^{i \delta (z+z_0)}\\ \frac{i \kappa_{ba}}{\beta_c}sin\beta_c(z-z_0)e^{-i \delta (z+z_0)} &\frac{\beta_c cos \beta_c(z-z_0)+i\delta sin \beta_c(z-z_0)}{\beta_c}e^{-i \delta (z+z_0)} \\   \end{bmatrix} \]  Subject to our initial conditions then \[\begin{bmatrix} \tilde{A}(z)\\