15. In a dihedral angle $\alpha-l-\beta$ of $60^{\circ}$, a moving point $A \in \alpha$, a moving point $B \in \beta, A A_{1} \perp \beta$ at point $A_{1}$, and $\left|A A_{1}\right|=a,|A B|=\sqrt{2} a$. Then, the maximum distance from point $B$ to the plane $\alpha$ is $\qquad$.
15. $\frac{(\sqrt{3}+1) a}{2}$