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Then looking at the power of the two modes when they are completely phase matched, that is when $\delta=0$   \[\frac{P_a(z)}{P_a(0)}=|\frac{\widetilde{A}(z)}{\widetilde{A}(0)}|^2=cos^2 \beta_c z \]   \[\frac{P_b(z)}{P_a(0)}=|\frac{\widetilde{B}(z)}{\widetilde{A}(0)}|^2=sin^2 \beta_c z\] where $\kappa_{ab}=\kappa_{ba}^{*}$   Define the coupling efficiency for a length l as $\eta=\frac{|\kappa_{ba}|^2}{\beta_c ^2} sin^2 \beta_c z$ as well as the coupling length $l_c=\frac{\pi}{2 \beta_c}$