Cut-Dependent Topology Optimization for Enhancing Shear-Mode Purity in Lithium Niobate Wafers

We present a topology-optimization methodology for designing single-sided, tri-state electrode patterns (+V, 0, electrode-free) that maximize shear-mode purity in lithium niobate (LiNbO₃) wafers. The framework combines a complex-Hermitian adjoint sensitivity formulation based on Wirtinger calculus with a coarse-fine design-mesh decomposition and Heaviside projection, and treats the bottom-face electrical boundary condition as an explicit design variable. Applying the method to five crystal cuts (X, Y, Z, 41°Y, 128°Y) over 3.30-4.10 MHz under grounded and single-sided configurations, we find that the optimal boundary condition is jointly determined by crystal cut and frequency through the rotated piezoelectric tensor and that topology optimization improves purity by up to 28.8 percentage points when the baseline is poorly matched but can be counterproductive when it is already optimal (Z-cut). We distil these behaviors into a three-regime taxonomy that predicts, a priori, when optimization is worthwhile. The result is a reusable design methodology, together with per-cut design rules, for shear-mode LiNbO₃ transducers.