  We consider approximate cloaking from a regularization viewpoint introduced
in [13] for EIT and further investigated in [12] [17] for the Helmholtz
equation. The cloaking schemes in [12] and [17] are shown to be (optimally)
within $|\ln\rho|^{-1}$ in 2D and $\rho$ in 3D of perfect cloaking, where
$\rho$ denotes the regularization parameter. In this paper, we show that by
employing a sound-hard layer right outside the cloaked region, one could
(optimally) achieve $\rho^N$ in $\mathbb{R}^N,\ N\geq 2$, which significantly
enhances the near-cloak. We then develop a cloaking scheme by making use of a
lossy layer with well-chosen parameters. The lossy-layer cloaking scheme is
shown to possess the same cloaking performance as the one with a sound-hard
layer. Moreover, it is shown that the lossy layer could be taken as a finite
realization of the sound-hard layer. Numerical experiments are also presented
to assess the cloaking performances of all the cloaking schemes for
comparisons.
