  Let $\{X(s,t):s,t\geqslant 0\}$ be a centered homogeneous Gaussian field with
a.s. continuous sample paths and correlation function
$r(s,t)=Cov(X(s,t),X(0,0))$ such that
\[r(s,t)=1-|s|^{\alpha_1}-|t|^{\alpha_2}+o(|s|^{\alpha_1}+|t|^{\alpha_2}),
\quad s,t \to 0,\] with $\alpha_1,\alpha_2\in(0,2],$ and $r(s,t)<1$ for
$(s,t)\neq(0,0)$. In this contribution we derive an exact asymptotic expansion
(as $u\to \infty$) of $$\mathbb{P}\left(\sup_{(s n_1(u),t
n_2(u))\in\left[0,x\right]\times\left[0,y\right]}X(s,t)\leqslant u\right),$$
where $n_1(u)n_2(u)=u^{2/\alpha_1+2/\alpha_2}\Psi(u)$, which holds uniformly
for $(x,y) \in [ A , B ]^2$ with $ A , B $ two positive constants and $\Psi$
the survival function of an $N(0,1)$ random variable. We apply our findings to
the analysis of asymptotics of extremes of homogeneous Gaussian fields over
more complex parameter sets and a ball of random radius. Additionally we
determine the extremal index of the discretised random field determined by
$X(s,t)$.
