Example 4 As shown in Figure 3, let $F$ be the focus of the parabola $y^{2}=4 x$, and $A, B$ be two points on the parabola different from the origin $O$, satisfying $\overrightarrow{F A} \cdot \overrightarrow{F B}=0$. Extend $A F, B F$ to intersect the parabola at points $C, D$ respectively. Find the minimum value of the area of quadrilateral $A B C D$.
As shown in Figure 3, with $F$ as the pole and $F x$ as the polar axis, the equation of the parabola is
$$
\rho=\frac{2}{1-\cos \theta} \text {. }
$$

Let point $A\left(\rho_{1}, \theta\right)(0<\theta<\pi)$. Then
$$
\begin{array}{l}
B\left(\rho_{2}, \theta+\frac{\pi}{2}\right) 、 C\left(\rho_{3}, \theta+\pi\right) 、 \\
D\left(\rho_{4}, \theta+\frac{3 \pi}{2}\right) .
\end{array}
$$

Thus, $|A C|=\rho_{1}+\rho_{3}$
$$
=\frac{2}{1-\cos \theta}+\frac{2}{1-\cos (\theta+\pi)}=\frac{4}{\sin ^{2} \theta} \text {. }
$$

Similarly, $|B D|=\frac{4}{\cos ^{2} \theta}$.
Therefore, $S_{\text {quadrilateral } A B C D}=\frac{1}{2}|A C| \cdot|B D|$
$$
=\frac{1}{2} \cdot \frac{4}{\sin ^{2} \theta} \cdot \frac{4}{\cos ^{2} \theta}=\frac{32}{\sin ^{2} 2 \theta} \text {. }
$$

When $\sin 2 \theta=1$, i.e., $\theta=\frac{\pi}{4}$, the minimum value of $S_{\text {quadrilateral } A B C D}$ is 32.