Some exact solutions for light beams

Abstract

We give an infinite class of exact analytical solutions for monochromatic light beams with strong focusing. As the solutions do not contain integrals, they are easy to explore compared with diffraction-theory results for strongly focused light. All monochromatic beams can be decomposed into two standing waves, each proportional to a Hilbert transform of the other. This means a beam can be built from any standing wave and our class is derived using this procedure. We give a visual overview of some of the beams, which reveals many interesting energy and field structures, including vortices in the energy flow, angular momentum in the propagation direction, and knotted field lines. We also show how the method can be used to design beams with an arbitrary focal shape.

Chirality of self-dual electromagnetic beams

“…which may be written as superpositions of Bessel beams [28][29][30]. By causal we mean without backward propagation far from the focal region.…”

“…The results (2.4) and (2.5) rest on some intricate manipulation of highly singular integrals over products of Bessel functions. This analysis has been checked by the author against known beam wavefunctions with = m 0, 1 based on the 'proto-beam' [29], discussed also in [30]. The proto-beam has recently been shown to be the most tightly focused of all possible beams, according to an intensity criterion [31].…”

A note on conserved quantities for electromagnetic waves

“…When magnetic helicity is conserved it thus provides an interesting topological constraint on field dynamics. For electromagnetic waves, conserved magnetic helicity is particularly interesting when it is accompanied by knotted lines of B. Electromagnetic waves with knotted field lines have been found in the case of pulses [55,56,61,62], standing waves [63] and monochromatic beams [64]. In the pulse and beam cases the magnetic helicity of knotted B fields is conserved.…”

Theoretical proposal for the experimental realisation of a monochromatic electromagnetic knot

“…Electromagnetic knots were first described by Rañada and collaborators in a series of works [1][2][3][4] based on earlier studies by Woltjer [5] and Moffatt [6], making use of the Bateman construction [7]. Several properties of these Rañada-Hopf electromagnetic knots have been worked out in theory, including their orbital angular momentum [8] and helicity [9][10][11][12][13][14], and similar solutions of Maxwell's equations have been found more recently in the form of propagating light beams [15][16][17]. Although analogous structures have been observed experimentally in liquid crystals [18] and in fluid dynamics [19], no feasible proposal has been given to date for the realisation of a Rañada-Hopf type electromagnetic knot, the closest work being a theoretical proposal involving plasma physics and self-organisation [20,21].…”