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
$$\begin{array}{l} \left(b^{2} \cos ^{2} \alpha-a^{2} \sin ^{2} \alpha\right) t^{2}+2 b^{2} c t \cos \alpha+b^{2} c^{2}-a^{2} b^{2}=0 . \\ \text { Then } O F \cdot A B=F A \cdot F B \Rightarrow c \cdot \frac{\sqrt{4 b^{4} c^{2} \cos ^{2} \alpha-4 b^{4}\left(b^{2} \cos ^{2} \alpha-a^{2} \sin ^{2} \alpha\right)}}{\left|b^{2} \cos ^{2} \alpha-a^{2} \sin ^{2} \alpha\right|}=\frac{b^{4}}{\left|b^{2} \cos ^{2} \alpha-a^{2} \sin ^{2} \alpha\right|} \\ \Rightarrow c \cdot 2 \sqrt{c^{2} \cos ^{2} \alpha-\left(b^{2} \cos ^{2} \alpha-a^{2} \sin ^{2} \alpha\right)}=b^{2} \Rightarrow 2 a c=c^{2}-a^{2} \Rightarrow e^{2}-2 e-1=0 \Rightarrow e=\sqrt{2}+1 . \end{array}$$