2. A line $l$ with an inclination angle of $45^{\circ}$ is drawn through the left focus $F$ of an ellipse, intersecting the ellipse at points $A$ and $B$. If $|B F|=2|A F|$, then the eccentricity of the ellipse is ( ).
(A) $\frac{1}{3}$
(B) $\frac{\sqrt{2}}{3}$
(C) $\frac{1}{2}$
(D) $\frac{\sqrt{2}}{2}$
2. B.

As shown in Figure 1, let $A F=d$. Then $B F=2 d$.
Figure 1
By the second definition of an ellipse, we know $A A^{\prime}=\frac{d}{e}, B B^{\prime}=\frac{2 d}{e}$. Therefore, $B C=\frac{d}{e}$.
Also, $B C=A C=\frac{3 \sqrt{2}}{2} d$, so $e=\frac{\sqrt{2}}{3}$.