An analytical model for spatial resolution estimation of near-field beamforming

Estimation of spatial resolution is crucial to the development of beamforming imaging methods. The classical Rayleigh formula based on the plane wave assumption is only valid for the estimate of the far-field resolution. However, in near-field beamforming, effects of wavefront curvature must be considered to achieve correct prediction of spatial resolution. To this end, an analytical wavenumber-domain method based on a spherical-wave model is developed in this work, yielding a novel wavenumber-based formula (WBF) for resolution prediction. The proposed formula explicitly characterizes the dependence of spatial resolution on measurement distance, acoustic frequency, and array aperture. When higher-order terms in the Taylor expansion with respect to the numerical aperture are neglected, the present formula reduces to the Abbe formula widely used in optical imaging. Moreover, both the proposed and Abbe formulas asymptotically converge to the classical Rayleigh limit in the far-field region. Numerical simulations are conducted to investigate the variation of beamforming spatial resolution with measurement distance and frequency, as well as its dependence on the choice of steering vector formulation and array geometry. The results demonstrate that the proposed WBF achieves consistently higher prediction accuracy than the Rayleigh and Abbe formulas, particularly in the near-field regime. Experimental validations under both single- and dual-source conditions further confirm the superiority and validity of the proposed formula.