Creating Complex Optical Longitudinal Polarization Structures

Abstract

In this Letter, we show that it is possible to structure the longitudinal polarization component of light. We illustrate our approach by demonstrating linked and knotted longitudinal vortex lines acquired upon nonparaxially propagating a tightly focused subwavelength beam. The remaining degrees of freedom in the transverse polarization components can be exploited to generate customized topological vector beams.

Singular knot bundle in light

“…Finally, we note that our analysis here may be used to approach the problem described recently in [48] in the main text, which may be paraphrased in the present terminology as "how can we determine ψ such that ϕ is a desired knotted field?" Taking the desired ϕ to be (A11), we can see immediately from (A5) that, for e − polarization,…”

An intuitive approach to structuring the three electric field components of light

“…, , and the transverse electric field components E ⊥ =(E x , E y ). In what follows, we consider the wavelength λ to be a fixed scaling parameter to equation (1). Note that E represents the complex amplitude vector of the beam; the full electric field has an additional trivial time-dependence w -…”

“…To create such a vector beam, the respective components can be prescribed in the focal plane as with σ y =0.42λ and σ z =0.5λ. The formulas of equation (23) were obtained according to [27,30], with coefficients adapted to the non-paraxial situation [1]. We filter in transverse Fourier space with H k 0 and H 0 , where the latter is only necessary for E z f .…”

“…Unless the beam's transverse electric field components are divergence-free in the two-dimensional transverse plane [1], tightly focused light typically leads to a non-negligible longitudinal electric field component [2,3], where the terms longitudinal and transverse electric field components refer to the components of the electric field that are parallel or perpendicular, respectively, to the direction of the mean Poynting flux. Having a longitudinal electric field component does not add a new degree of freedom, in the sense that all components of the electric and magnetic fields are still fixed by prescribing two components in a plane.…”

“…This paper aims to provide such a picture, and presents a novel approach towards an intuitive understanding of tightly focused beams by making analogies with fluid dynamics and with two-dimensional magneto-and electrostatics. For this, a two-dimensional Helmholtz decomposition of the transverse electric field components in the transverse plane [1] is key. The Helmholtz decomposition allows the generalization of the notion of 'radial' and 'azimuthal' polarization in the following way: radial polarization corresponds to an electric field that is 'curl-free' in the transverse plane-which as we shall see can be interpreted as a flow solely due to sinks and sources without vorticity-and is the part of the field that gives rise to the longitudinal polarization.…”