A less-for-more metamaterial paradigm via Laplace-Helmholtz correspondence

The Laplace and Helmholtz equations dominate two fundamentally distinct physical processes: steady diffusion with a single constitutive parameter versus dynamic wave propagation with two. Manipulating complex wave dynamics through simple steady fields is typically deemed infeasible. Here, we bridge this gap by establishing the Laplace-Helmholtz correspondence, enabling a 'less-for-more' paradigm for metamaterial design. Rooted in a wave invariant, this framework allows the rigorous derivation of the two parameters and oscillatory field for the Helmholtz equation (the 'more') solely from the single parameter and steady field of the Laplace equation (the 'less'). We demonstrate this capability by unlocking functionalities previously considered intractable, such as hyperbolic invisibility and three-dimensional freeform conformal cloaking with broadband performance. We further experimentally validate our strategy via a waveguide cloak. Our results bridge static and dynamic physics, establishing a versatile gateway for manipulating waves beyond conventional constraints.