  We consider the high frequency Helmholtz equation with a variable refraction
index $n^2(x)$ ($x \in \R^d$), supplemented with a given high frequency source
term supported near the origin $x=0$. A small absorption parameter
$\alpha_{\varepsilon}>0$ is added, which somehow prescribes a radiation
condition at infinity for the considered Helmholtz equation. The semi-classical
parameter is $\varepsilon>0$. We let $\eps$ and $\a_\eps$ go to zero {\em
simultaneaously}. We study the question whether the indirectly prescribed
radiation condition at infinity is satisfied {\em uniformly} along the
asymptotic process $\eps \to 0$, or, in other words, whether the conveniently
rescaled solution to the considered equation goes to the {\em outgoing}
solution to the natural limiting Helmholtz equation. This question has been
previously studied by the first autor. It is proved that the radiation
condition is indeed satisfied uniformly in $\eps$, provided the refraction
index satisfies a specific {\em non-refocusing condition}, a condition that is
first pointed out in this reference. The non-refocusing condition requires, in
essence, that the rays of geometric optics naturally associated with the
high-frequency Helmholtz operator, and that are sent from the origin $x=0$ at
time $t=0$, should not refocus at some later time $t>0$ near the origin again.
In the present text we show the {\em optimality} of the above mentionned
non-refocusing condition, in the following sense. We exhibit a refraction index
which {\em does} refocus the rays of geometric optics sent from the origin near
the origin again, and, on the other hand, we completely compute the asymptotic
behaviour of the solution to the associated Helmholtz equation: we show that
the limiting solution {\em does not} satisfy the natural radiation condition at
infinity. More precisely, we show that the limiting solution is a {\em
perturbation} of the outgoing solution to the natural limiting Helmholtz
equation, and that the perturbing term explicitly involves the contribution of
the rays radiated from the origin which go back to the origin. This term is
also conveniently modulated by a phase factor, which turns out to be the action
along the above rays of the hamiltonian associated with the semiclassical
Helmholtz equation.
