3. Let the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a, b>0)$ have its left and right foci at $F_{1}$ and $F_{2}$, respectively. A line passing through point $F_{2}$ with an inclination angle of $\frac{\pi}{4}$ intersects the hyperbola at point $A$. If $\triangle F_{1} F_{2} A$ is an isosceles right triangle, then the eccentricity of the hyperbola is ( ).
(A) $\frac{\sqrt{3}+1}{2}$
(B) $\sqrt{3}+1$
(C) $\frac{\sqrt{2}+1}{2}$
(D) $\sqrt{2}+1$
3. D.

To make $\triangle F_{1} F_{2} A$ an isosceles right triangle, then
$$
\left|A F_{1}\right|-\left|A F_{2}\right|=2 a,\left|A F_{2}\right|=\left|F_{1} F_{2}\right|=2 c \text {. }
$$

By the Pythagorean theorem, we get
$$
(2(a+c))^{2}=2(2 c)^{2} \Rightarrow \frac{c}{a}=\sqrt{2}+1 .
$$