  We consider regularized approximate cloaking for the Helmholtz equation.
Various cloaking schemes have been recently proposed and extensively
investigated. The existing cloaking schemes in literature are (optimally)
within $|\ln\rho|^{-1}$ in 2D and $\rho$ in 3D of the perfect cloaking, where
$\rho$ denotes the regularization parameter. In this work, we develop a
cloaking scheme with a well-designed lossy layer right outside the cloaked
region that can produce significantly enhanced near-cloaking performance. In
fact, it is proved that the proposed cloaking scheme could (optimally) achieve
$\rho^N$ in $\mathbb{R}^N$, $N\geq 2$, within the perfect cloaking. It is also
shown that the limit of the proposed lossy layer corresponds to a sound-hard
layer. We work with general geometry and arbitrary cloaked contents of the
proposed cloaking device.
