Angular spectrum decomposition method for evaluating the beam shape coefficients of the scalar Gaussian beams with...

This paper adopts a scalar approach to evaluate the beam shape coefficients (BSCs) and delve into the intrinsic relationship between angular spectrum decomposition (ASD) and the finite-series (FS) method (or radial quadrature method (RQ)) as well as localized approximation (LA) method. By introducing the finite-series expressions of the normalized associated Legendre functions, we investigate the interconnections of these methods in describing BSCs and successfully approximate the angular spectrum representation of BSCs under the paraxial condition into the forms obtained through the LA and/or the FS methods. We demonstrate the derivation of BSCs for Gaussian beams, prove the consistency between the FS and RQ methods, verify the connection between the ASD and LA methods, and to the best of our knowledge, for the first time, confirm the equivalence of the ASD and FS methods in describing the BSCs of scalar Gaussian beams. The achievement provides new methodologies and deep insights for describing and analyzing the BSCs in practical applications.