Non-dispersive graded impedance acoustic lenses

Lenses are typically based on refractive index profiles derived from the geometric approximation of high-frequency waves, yet the critical issue of impedance mismatch is often neglected. Mismatched devices suffer from unwanted reflections and dispersion, which can significantly degrade performance in practical applications. In this work, we propose impedance profiles for lenses to achieve efficient wave transmission while maintaining the desired refractive index and minimizing dispersion effects. A family of impedance profiles is derived from the acoustic wave equation such that the phase velocity is preserved. First, the 1D setting is considered to explain how dispersion occurs inside a lens and at its interfaces. Then, the method is applied to 2D axisymmetric configurations where the impedance mismatch is radially redistributed. These profiles are demonstrated in the acoustic setting of a Luneburg lens, but can be easily extended to more general scenarios such as imaging or cloaking in air and water, where matching the impedance of the background poses significant challenges.