Simple formulas of directional amplification from non-Bloch band theory

Green's functions are fundamental quantities that determine the linear responses of physical systems. The recent developments non-Hermitian systems, therefore, call for function formulas bands. This task is complicated by high sensitivity energy spectrums to boundary conditions, which invalidates straightforward generalization Hermitian formulas. Here, based on non-Bloch band theory, we obtain simple general one-dimensional Furthermore, in large-size limit, these dramatically reduce finding roots a algebraic equation. As an application, our formulation provides desirable defining quantities, gain and directionality, directional amplification. Thus provide efficient guide designing amplifiers.