Beam shape coefficients calculation for an elliptical Gaussian beam with 1-dimensional quadrature and localized approximation methods

Highlights: • 1-dimensional integrations are developed instead of 2-dimensional integrations to improve the efficiency of computing EGB's (Elliptical Gaussian beam) BSCs. • The normalized associated Legendre functions (NALF) are employed to prevent from numerical calculation difficulties. • The methods are checked by reconstructing the beam fields and limitations of the concerned methods are discussed in detail. • Limitations of the quadrature and the LA methods in the numerical calculation are analyzed in detail. The use of a shaped beam for applications relying on light scattering depends much on the ability to evaluate the beam shape coefficients (BSC) effectively. Numerical techniques for evaluating the BSCs of a shaped beam, such as the quadrature, the localized approximation (LA), the integral localized approximation (ILA) methods, have been developed within the framework of generalized Lorenz-Mie theory (GLMT). The quadrature methods usually employ the 2-/3-dimensional integrations. In this work, the expressions of the BSCs for an elliptical Gaussian beam (EGB) are simplified into the 1-dimensional integral so as to speed up the numerical computation. Numerical results of BSCs are used to reconstruct the beam field and the fidelity of the reconstructed field to the given beam field is estimated. It is demonstrated that the proposed method is much faster than the 2-dimensional integrations and it can acquire more accurate results than the LA method. Limitations of the quadrature method and also the LA method in the numerical calculation are analyzed in detail.