  We prove that the finite field Fourier extension operator for the paraboloid
is bounded from $L^2\to L^r$ for $r\geq \frac{2d+4}{d}$ in even dimensions
$d\ge 8$, which is the optimal $L^2$ estimate. For $d=6$ we obtain the optimal
range $r> \frac{2d+4}{d}=8/3$, apart from the endpoint. For $d=4$ we improve
the prior range of $r>16/5=3.2$ to $r\geq 28/9=3.111\ldots$, compared to the
conjectured range of $r\geq3$. The key new ingredient is improved additive
energy estimates for subsets of the paraboloid.
