Relevance of longitudinal fields of paraxial optical vortices

Kayn A Forbes,Garth A Jones,Dale Green

doi:10.1088/2040-8986/abff96

Kayn A Forbes, Garth A Jones + Show 1 more

Open Access

https://doi.org/10.1088/2040-8986/abff96

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Abstract

Longitudinal electromagnetic fields generally become comparable with the usually dominant transverse components in strongly focused, non-paraxial beams. For paraxial optical vortex modes it is highlighted here how their angular momentum properties produce longitudinal fields that in general must be accounted for. First-order longitudinal components of quantized Laguerre–Gaussian modes are derived and numerically studied with respect to the paraxial parameter, highlighting light-matter and spin-orbit interactions that stem from the longitudinal fields of paraxial beams in free space. New restrictions are cast on the validity of neglecting longitudinal fields for paraxial optical vortices interacting with atoms, molecules and other nanostructures.

Highlights

The idealized plane-wave solutions to the Maxwell and Helmholtz equations are more often than not utilised to provide a theoretical understanding of light–matter interactions [1]
We highlight numerically that there are two distinct factors that influence the magnitude of longitudinal fields of optical vortices; firstly the well-known fact that a larger degree of focusing increases the longitudinal fields through a paraxial parameter−1weighting factor; but secondly that the angular momenta of optical vortices significantly influences the importance of longitudinal fields, highlighting both the quantitative and qualitative necessity of their inclusion even for paraxial optical vortices
Just as it is well-established that longitudinal fields cannot be neglected when unstructured light is strongly focused because terms dependent on factors like 2r kw20 become important, figure 1 shows that certain contributions to longitudinal fields for paraxial optical vortices cannot likewise be neglected

Summary

Introduction

The idealized plane-wave solutions to the Maxwell and Helmholtz equations are more often than not utilised to provide a theoretical understanding of light–matter interactions [1]. In their seminal study, Lax et al [3] highlighted how the paraxial solutions to the scalar wave equation consist of a purely transverse zeroth-order field and smaller, first-order, longitudinal components whose magnitude for Gaussiantype beams depends on the paraxial factor (kw0)−1 where k = 2π/ λ is the wave number and w0 the beam waist (at the focal point). Most studies have been concerned with the angular momentum properties of non-paraxial and longitudinal fields of twisted light, such as spin-orbit interactions of light (SOI) and the transfer to particles to cause mechanical motion [17,18,19]. We highlight numerically that there are two distinct factors that influence the magnitude of longitudinal fields of optical vortices; firstly the well-known fact that a larger degree of focusing increases the longitudinal fields through a paraxial parameter (kw0)−1weighting factor; but secondly that the angular momenta of optical vortices significantly influences the importance of longitudinal fields, highlighting both the quantitative and qualitative necessity of their inclusion even for paraxial optical vortices

Transverse and longitudinal fields of Laguerre–Gaussian modes

The paraxial approximation for optical vortices

Application: single-photon absorption

Discussion and conclusion