We investigate pattern revivals in specially designed optical structures that combine different transverse modes. In general, the resulting pattern is not preserved under free propagation and gets transformed due to nonsynchronized Gouy phases. However, it is possible to build structures in which the Gouy phases synchronize at specific fractional values, thus recovering the initial pattern at the corresponding longitudinal positions. This effect is illustrated with a radially structured light spot in which the beam energy can be addressed to different positions without the need of intermediate optical components, which can be useful for optical communications and optical tweezing with structured beams.

In this review, we discuss the present status of the description of confining flux tubes in SU(N) pure Yang–Mills theory in terms of ensembles of percolating center vortices. This is based on three main pillars: modeling in the continuum the ensemble components detected in the lattice, the derivation of effective field representations, and contrasting the associated properties with Monte Carlo lattice results. The integration of the present knowledge about these points is essential to get closer to a unified physical picture for confinement. Here, we shall emphasize the last advances, which point to the importance of including the non-oriented center-vortex component and non-Abelian degrees of freedom when modeling the center-vortex ensemble measure. These inputs are responsible for the emergence of topological solitons and the possibility of accommodating the asymptotic scaling properties of the confining string tension.

In this work, we propose a 3D ensemble measure for center-vortex worldlines and chains equipped with non-Abelian degrees of freedom. We derive an effective field description for the center-element average where the vortices get represented by N flavors of effective Higgs fields transforming in the fundamental representation. This field content is required to accommodate fusion rules where N vortices can be created out of the vacuum. The inclusion of the chain sector, formed by center-vortex worldlines attached to pointlike defects, leads to a discrete set of Z(N) vacua. This type of SSB pattern supports the formation of a stable domain wall between quarks, thus accommodating not only a linear potential but also the Luscher term. Moreover, after a detailed analysis of the associated field equations, the asymptotic string tension turns out to scale with the quadratic Casimir of the antisymmetric quark representation. These behaviors reproduce those derived from Monte Carlo simulations in SU(N) 3D Yang-Mills theory, which lacked understanding in the framework of confinement as due to percolating magnetic defects.

Inspired by the center-vortex dominance in the infrared sector of $SU(N)$ Yang-Mills theory observed on the lattice, we propose a vacuum wave functional localized on an ensemble of correlated center vortices endowed with stiffness and magnetic monopoles that change the orientation of the vortex flux. In the electric-field representation, this wave functional becomes an effective partition function for N complex scalar fields. The inclusion of both oriented and nonoriented vortices as well as so-called N-vortex matchings leads to an effective potential that has only a center symmetry left. In the center-vortex condensed phase, this symmetry is spontaneously broken. In this case, the Wilson loop average can be approximated by a solitonic saddle point localized around the minimal surface. The asymptotic string tension thus obtained displays Casimir scaling.

A scenario to understand the asymptotic properties of confinement between quark probes, based on a 4D mixed ensemble of percolating center-vortex world surfaces and chains, was initially proposed by one of us in a non-Abelian setting. More recently, the same physics was reobtained by means of a Schrödinger wave functional peaked at Abelian-projected configurations, which deals with center-vortex lines and pointlike monopoles in real space. In this work, we formulate the Abelian-projected ensemble and reassess the non-Abelian one within the Weingarten lattice representation for the sum over surfaces. In the phase where world surfaces are stabilized by contact interactions and percolate, lattice gauge fields emerge. This generalizes the Goldstone modes in an Abelian loop condensate to the case where non-Abelian degrees of freedom are present. In this language, the different natural matching properties of elementary center-vortex world surfaces and monopole worldlines can be easily characterized. In the lattice, the Abelian setting also implements the original idea that the mixed ensemble reconciles N -ality with the formation of a confining flux tube. In this picture, center vortices and chains explain why Abelian-projected variables capture this property at asymptotic distances while simultaneously allowing for a “dual superconductor” description of the fundamental string. Common features, differences in the continuum, and perspectives will also be addressed.