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For cases of interest, perturbation will either be independent of z or be a periodic funciton of z. This further reduces our coupling coefficients to \[ \pm \frac{\partial \widetilde{A}}{\partial z}=i\kappa_{ab} \widetilde{B}e^{i 2 \delta z} \] \[ \pm \frac{\partial \widetilde{B}}{\partial z}=i\kappa_{ba} \widetilde{A}e^{-i 2 \delta z}\]  The parameter of $2\delta$ is the phase mismatch between the two coupled modes.   \section{Codirectional Coupling}  Codirectional coupling is when the coupling of two propagating modes are in the same direction, over some length $l$, where $\beta_a$ and $\beta_b$ are both greater than zero. Coupling equations to be used are \[\frac{\partial \widetilde{A}}{\partial z}=i\kappa_{ab} \widetilde{B}e^{i 2 \delta z} \] \[ \frac{\partial \widetilde{B}}{\partial z}=i\kappa_{ba} \widetilde{A}e^{-i 2 \delta z}\] The general solution of this system is solved as an initial value problem in matrix form is given as \begin{bmatrix}   \widetilde{A}(z)  \widetilde{B}(z)  \end{bmatrix}