Analysis On Characteristics Of Electric Field Distributions Of Non-paraxial LG₀₀ Mode And LG₁₀ Mode Gaussian Beams

With the development of laser technology,the Gaussian beams with waist spot size comparable with the wavelength are gained by strong-focusing.Analysis on characteristics of electric field amplitude distributions of the non-paraxial Gaussian beams will provide theoretical basis for the practical application.This paper begins with scalar Helmholtz equation in the oblate spheroidal coordinate system,the expression of electric field of non-paraxial LGio mode Gaussian beam is given by separation variable method and virtual point source method.The expressions of transverse electric field amplitude distributions and far-field electric field amplitude distributions of non-paraxial LG00 mode and LGio mode Gaussian beams are given by coordinate conversion between the oblate spheroidal coordinate system and cylindrical coordinate system.Based on the definition of radial coordinate,which is corresponding to the electric field amplitude intensity distributions of non-paraxial LG00 mode Gaussian beam in the waist plane falls to l/e of amplitude in radial coordinate r=0,the calculation formula of spot size is expressed by half of the focal length d of the oblate ellipsoid coordinate system.Based on the definition of second-order moment of the far-field electric field intensity distributions of non-paraxial LG00 mode Gaussian beam,the calculation formula of the sin value half of far-field divergence angle is given.The characteristics of transverse electric field amplitude distributions and far-field electric field amplitude distributions of low-order(LG00 mode and LG10 mode)non-paraxial and paraxial Gaussian beams are compared.In order to illustrate the relevance of the electric field distributions of the low-order non-paraxial and paraxial Gaussian beams,characteristics of matching efficiency of transverse electric field amplitude distributions and matching efficiency of far-field electric field amplitude distributions between the low-order non-paraxial and paraxial Gaussian beams versus half of the focal length d are analyzed.The calculation formulas of the additive phase shifts of non-paraxial LG00 mode and LG10 mode Gaussian beams on the axis are given.The differences between the additive phase shifts of low-order non-paraxial and paraxial Gaussian beams are analyzed by calculating their absolute error.It is shown that the transverse electric field amplitude distributions and far-field electric field amplitude distributions of the non-paraxial LG00 mode and LG10 mode Gaussian beams tend to be the transverse electric field amplitude distributions and far-field electric field amplitude distributions of the paraxial LG00 mode and LGio mode Gaussian beams when the half of focal length of the oblate ellipsoid coordinate system is much larger than the wavelength.