  We realize a flat lens with graded negative refractive index by a
two-dimensional phononic crystal. The index-grade is achieved by gradual
modification of the filling fraction along the transverse direction to
propagation. We demonstrate that this lens can realize the focusing and
amplification of parallel incident acoustic waves. A formula for the refractive
index profile is derived by which the lens with desired focal length can be
precisely designed and the aberrations of image can be effectively reduced.
This designable lens is expected to bear significance in applications such as
coupling or integration with various types of acoustic devices.
