Gain loss in metasurfaces caused by reflection-coefficient quantization: an error vector magnitude approach

Metasurfaces with a finite number of discrete reflection states inevitably suffer a gain loss compared with their ideal continuous-phase counterparts. This paper proposes an error vector magnitude (EVM)-based metric, denoted Γ_EVM, for the complex reflection coefficient of a metasurface unit, where Γ_EVM is defined as the root-mean-square difference between the ideal and realizable reflection coefficients over [0,  2π). The metric enables quantitative evaluation of the unit's quantization precision and prediction of the resulting gain loss in the entire array. Assuming a uniform phase probability density function (PDF) over [0,  2π), a closed-form expression for Γ_EVM is derived together with an empirical relation that relates Γ_EVM to the gain loss, so that the array radiation efficiency can be estimated directly from Γ_EVM. For practical metasurfaces with nonuniform phase statistics, a statistical method based on empirical phase PDFs is introduced, resulting in closed-form expressions for the expectation and variance of the gain-loss statistics in terms of Γ_EVM and the total number of metasurface units. The proposed method is validated by Monte Carlo simulations and on five 20 × 20 prototypes operating at 28 GHz and covering quantization levels from 1-bit to multiple-bit resolutions. Measurements confirm agreement with theoretical estimations, with a maximum deviation of 0.34 dB. These results demonstrate that the proposed metric provides a generalizable and easy-to-use tool for estimating metasurface gain loss, facilitating efficient design in 5G and satellite communication systems.