  The synthesis of non-magnetic 2D dielectric cloaks as proper solutions of an
inverse scattering problem is addressed in this paper. Adopting the relevant
integral formulation governing the scattering phenomena, analytic and numerical
approaches are exploited to provide new insights on how frequency and direction
of arrival of the incoming wave may influence the cloaking mechanism in terms
of permittivity distribution within the cover region. In quasi-static
(subwavelength) regime a solution is analytically derived in terms of
homogeneous artificial dielectric cover with $\varepsilon<\varepsilon_0$ which
is found to be a necessary and sufficient condition for achieving
omnidirectional cloaking. On the other hand, beyond quasi-static regime, the
cloaking problem is addressed as an optimization task looking for only natural
dielectric coatings with $\varepsilon>\varepsilon_0$ able to hide the object
for a given number of directions of the incident field. Simulated results
confirm the validity of both analytic and numerical methodologies and allow to
estimate effective bandwidths both in terms of frequency range and direction of
arrival of the impinging field.
