Graded negative index lens with designable focal length by phononic crystal

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.