Arbitrary eigenmode reshaping induced by distributed non-reciprocity in non-Hermitian systems

Periodic non-reciprocal systems have attracted significant attention for their striking non-Hermitian skin effect (NHSE) and the development of a comprehensive non-Bloch theory. In contrast, aperiodic non-reciprocal systems have been rarely explored, owing to the intrinsic complexity induced by spatial non-uniformity and absence of predictable physical behavior. Here, we establish a theoretical framework for one-dimensional nearest-neighbor lattices under open boundary conditions, and reveal that the spatially varying non-reciprocal coupling can be interpreted geometrically as a designable imaginary gauge field, providing a tunable metric on the Hilbert space that serves as a powerful and deterministic knob to control the wave-functions of eigenmodes. By tailoring the non-reciprocity distribution in this specific system, we combine theory, proof-of-concept electric-circuit experiments, and high-frequency optical-lattice simulations with micro-ring resonators to demonstrate that the eigenmodes of a non-Hermitian system can be reshaped into arbitrary spatial profiles without altering their spectral distribution, a mode-field engineering strategy termed non-Hermitian reshaping engineering (NHRE). In the special case of a uniform non-reciprocity distribution, NHRE recovers the characteristic exponential mode localization in the conventional NHSE. While the full diagonalization framework underlying the inverse mapping is limited to one dimension, some of its design principles can be partially extended to arbitrary higher-dimensional systems, offering an efficient route to mode modulation and deepening our understanding of non-Hermitian physics.