The 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 in imaging point-like sources. Here, we exploit an alternative GRIN lens design, which enables a one-to-one correspondence between input and output phase, and allows to determine the focal length using the well-known thin lens equation, effectively establishing the elastic equivalent of the convex lens in optics. This is demonstrated analytically, obtaining a bijective relation between the location of a point-like source and its image, and the results are confirmed numerically and experimentally in an aluminium plate, where the lens is realized by introducing rows of circular cavities of variable diameters. Moreover, a proof-of-concept experiment demonstrates the possibility to image sources of flexural waves at the centimetre scale with subwavelength resolution. This research can extend applications of elastic GRIN lenses to new fields such as imaging and non-destructive testing, where the location of defects can be identified by focusing the scattered field. Multiple sources can be imaged simultaneously, and the combined effect of multiple lenses can also be used to design more complex systems, opening new possibilities in the technological exploitation of elastic wave manipulation.