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\end{bmatrix} = F(z;z_0) \begin{bmatrix}  \widetilde{A}(z_0)\\  \widetilde{B}(z_0)\\  \end{bmatrix}$$ Where $F(z;z_0)$ is the forward coupling matrix and it relates field amplitudes at an intial value of $z_0$ to those at z. For the most simple case when power is launched into only mode a at $z=0$ , giving $\widetilde{B}(0)=0$. With $z=0$   \[\widetilde{A}(z)=\widetilde{A}(0)\left( cos \beta_c z -\frac{i \delta}{\beta_c} sin \beta_c z \right) e^{i \delta z} \]