  In this paper, we design a truly exact and optimal perfect absorbing layer
(PAL) for domain truncation of the two-dimensional Helmholtz equation in an
unbounded domain with bounded scatterers. This technique is based on a complex
compression coordinate transformation in polar coordinates, and a judicious
substitution of the unknown field in the artificial layer. Compared with the
widely-used perfectly matched layer (PML) methods, the distinctive features of
PAL lie in that (i) it is truly exact in the sense that the PAL-solution is
identical to the original solution in the bounded domain reduced by the
truncation layer; (ii) with the substitution, the PAL-equation is free of
singular coefficients and the substituted unknown field is essentially
non-oscillatory in the layer; and (iii) the construction is valid for general
star-shaped domain truncation. By formulating the variational formulation in
Cartesian coordinates, the implementation of this technique using standard
spectral-element or finite-element methods can be made easy as a usual coding
practice. We provide ample numerical examples to demonstrate that this method
is highly accurate, parameter-free and robust for very high wave-number and
thin layer. It outperforms the classical PML and the recently advocated PML
using unbounded absorbing functions. Moreover, it can fix some flaws of the PML
approach.
