Cross-modal Willis metasurfaces for perfect control of elastic wave refraction in solids

Ultrasonic wave-front shaping is crucial for biomedical and industrial applications but remains severely constrained by longitudinal-transverse mixed-mode scattering at solid interfaces, resulting in substantial energy leakage and consequently poor transmission. Perfect refraction under these conditions remains unattainable even with conventional generalized Snell's law approaches, due to the difficulty of fully addressing complex volumetric-shear coupling in solids. Here, we reveal that ideal power efficiency cannot be achieved without a novel material concept, the cross-modal Willis approach, and propose a rationally designed Willis-type metasurface that enables perfect wave steering in arbitrary directions while eliminating undesired parasitic modes. Our finding uncovers a critical yet previously overlooked mechanism: cross-modal anisotropy in Willis media, which fundamentally governs wave mode coupling behavior. We develop a systematic design methodology for Willis unit cells through intentional unbalancing of fully anisotropic effective masses, providing precise control over wave propagation characteristics. Experimental validation using ultrasonic measurements confirms successful implementation across mode-preserving and mode-converting refraction scenarios, demonstrating the versatility and robustness of our approach. This work not only extends the theoretical framework of Willis materials but also establishes paradigms for ideal controllability in ultrasound engineering applications.