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Chris Spencer edited untitled.tex
about 9 years ago
Commit id: 865a1e2cd0cc88a7b25a9800a93a31a71447bab6
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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 $z=0$
\[\widetilde{A}(z)=\widetilde{A}(0)\left( cos \beta_c z -\frac{i \delta}{\beta_c} sin \beta_c z \right) e^{i \delta z} \]
\[\widetilde{B}(z)=\widetilde{B}(0)\left(\frac{i \kappa_{ba}}{\beta_c} sin \beta_c z \right) e^{-i \delta z} \] where $\beta_c=\left( \kappa_{ab}\kappa_{ba}+\delta^2\right)^\frac{1}{2}$
Then looking at the power of the two modes
$\frac{P_a(z)}{P_a(0)}=|\frac{\widetilde{A}(z)}{\widetilde{A}(0)}|^2$ when they are completely phase matched, that is when $\delta=0$
\[\frac{P_a(z)}{P_a(0)}=|\frac{\widetilde{A}(z)}{\widetilde{A}(0)}|^2=cos^2 \beta_c z \]