13. In the plane rectangular coordinate system $x O y$, the hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ has its right focus at $F$. A line $l$ passing through point $F$ intersects the hyperbola $C$ at points $A$ and $B$. If $O F \cdot A B=F A \cdot F B$, find the eccentricity $e$ of the hyperbola $C$.
Consider the line $l:\left\{\begin{array}{l}x=c+t \cos \alpha, \\ y=t \sin \alpha,\end{array}\right.$ and substitute it into the hyperbola equation $b^{2} x^{2}-a^{2} y^{2}=a^{2} b^{2}$ to get
$$
\begin{array}{l}
\left(b^{2} \cos ^{2} \alpha-a^{2} \sin ^{2} \alpha\right) t^{2}+2 b^{2} c t \cos \alpha+b^{2} c^{2}-a^{2} b^{2}=0 . \\
\text { Then } O F \cdot A B=F A \cdot F B \Rightarrow c \cdot \frac{\sqrt{4 b^{4} c^{2} \cos ^{2} \alpha-4 b^{4}\left(b^{2} \cos ^{2} \alpha-a^{2} \sin ^{2} \alpha\right)}}{\left|b^{2} \cos ^{2} \alpha-a^{2} \sin ^{2} \alpha\right|}=\frac{b^{4}}{\left|b^{2} \cos ^{2} \alpha-a^{2} \sin ^{2} \alpha\right|} \\
\Rightarrow c \cdot 2 \sqrt{c^{2} \cos ^{2} \alpha-\left(b^{2} \cos ^{2} \alpha-a^{2} \sin ^{2} \alpha\right)}=b^{2} \Rightarrow 2 a c=c^{2}-a^{2} \Rightarrow e^{2}-2 e-1=0 \Rightarrow e=\sqrt{2}+1 .
\end{array}
$$