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\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)} & \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 \[\begin{bmatrix} \tilde{A}(z)\\  \tilde{B}(z)\\  \end{bmatrix} = F(z;0)   \begin{bmatrix}  \tilde{A}(0)\\  0 \\   \end{bmatrix}\]