Remodeling of the Localized Approximation： Beam Shape Coefficient Calculation for the Gaussian Beam by Using Scalar Translation Addition Theorem

The accurate and efficient calculation of the Beam Shape Coefficients （BSCs） is one of the key issues in researching the interaction between a structured or shaped beam and the particles. For this reason， different techniques have been established， such as the quadrature method， the Localized Approximation （LA） or its variants， the Finite Series （FS） method， the Angular Spectrum Decomposition （ASD）， and the others. Among these methods， the Localized approximation has been widely used due to its advantages including mainly the efficiency in numerical calculation， the conciseness of the expressions and the straightforwardness in deducing the BSCs for the beams in both the on-axis and off-axis scenarios. However， this method， just as it is named， is an approximate one. The beam remodeling takes place when the beam field is reconstructed from the BSCs， i.e. the reconstructed beam field deviates from the originally given field. Up to date， the remodeling of the beam field has not been studied systematically.In this work， the two-step indirect method is employed to study the remodeling effects of the localized approximation method for the Gaussian beams. In the first step， the expression of the BSCs is obtained， by using the localized approximation， for the Gaussian beam which is centered in the coordinate system （called beam system） and is described by using a scalar potential function （therefore the BSCs are called scalar BSCs）. Subsequently， the scalar translational addition theorem is used to transform the BSCs into another coordinate system whose axes are parallel to those of the beam system. The second coordinate system in which the spherical particle is centered is called particle system. In the second step of the indirect method， the electric and magnetic （EM） fields of the Gaussian beam are described by using the scalar potential function together with a polarization parameter. Based on this relation， the BSCs of the EM field （called vector BSCs or EM BSCs） are expressed as a linear combination of the scalar BSCs. In this way， the EM BSCs for the off-axis located Gaussian beam are obtained indirectly from the scalar BSCs of the centered beam. The two-step indirect method simplifies the analytical derivation of the BSCs. The expressions of the EM BSCs which are directly deduced in the localized approximation for the off-axis located Gaussian beam are used in the numerical calculations for making a comparison. It is found that the indirect method is at least eight times faster than the direct method in numerical calculation of the BSCs.The remodeling of the localized approximation is studied， based on a comparison between the originally given field of the Gaussian beam and the fields which are reconstructed from the directly and/or indirectly calculated EM BSCs. The numerical results reveal that， in the direct LA method， the axisymmetric structure of the Gaussian beam is broken in the reconstructed beam field， exhibiting spurious-peaks in the electric field. When the off-axis distance becomes sufficiently large， a ring structure is produced in the field. The deviation of the reconstructed beam field from the given one increases gradually along the increase of the off-axis distance. However， in the indirect method， the reconstructed field is always axisymmetric and the discrepancy between the reconstructed and given fields is independent of the off-axis distance. In both the direct and indirect methods， the reconstructed beam fields deviate from the given one， showing a close dependence on the beam waist radius. Namely， the reconstructed beam fields agree much better with the given field when the beam waist radius is large， but the quality of the remodeled fields become poorer when the beam waist radius decreases， especially when the beam waist radius is close to or less than the wavelength of the light beam.It is concluded that， compared with the direct localized approximation method， the two-step indirect method has advantages in analytical derivation of the BSCs， in conciseness of the expressions， in efficiency of numerical calculation， and in quality of reconstructed beam field. Besides， the work presented here suggests a warning against the use of the direct/indirect LA methods for strongly focused Gaussian beams and the use of the direct LA for large off-axis located beams.