The vertices of a hyperbola \( H \) are \( (\pm 6, 0) \) and its eccentricity is \( \frac{\sqrt{5}}{2} \). Let \( N \) be the normal to \( H \) at a point in the first quadrant and parallel to the line \( \sqrt{2}x + y = 2\sqrt{2} \). If \( d \) is the length of the line segment of \( N \) between \( H \) and the \( y \)-axis then \( d^2 \) is equal to ______ .
