Thin-lens equation in elasticity: Imaging with gradient-index phononic crystals

Many works in elasticity have exploited the concept of gradient-index (GRIN) lenses, borrowed from optics, for wave focusing and control. These effects are particularly attractive for cloaking, absorption, or energy-harvesting applications. Despite their potential, current lens designs suffer from limitations, mainly related to the difficulty of imaging pointlike sources. Here, we exploit an alternative GRIN lens design that enables a one-to-one correspondence between input and output phase and allows the focal length to be determined using the well-known thin-lens equation, effectively establishing an elastic equivalent of the convex lens in optics. This is demonstrated analytically, obtaining a bijective relation between the location of a pointlike source and its image. The results are also confirmed numerically and experimentally in an aluminum plate, in which the lens is realized by introducing rows of circular cavities of variable diameter. Moreover, a proof-of-concept experiment demonstrates the possibility of imaging sources of flexural waves at the centimeter scale with subwavelength resolution. This research can extend applications of elastic GRIN lenses to new fields such as imaging and nondestructive testing, in which the locations of defects can be identified by focusing the scattered field. Multiple sources can be imaged simultaneously, and the combined effects of multiple lenses can also be used to design more complex systems, opening new possibilities for the technological exploitation of elastic wave manipulation.