Transient electromagnetic singularities in random three-dimensional optical fields.

We show that on all generic circular or elliptical paths in a random three-dimensional electromagnetic field, the electric and magnetic field vectors generate a cylinder throughout most of the optical cycle. At some point in the cycle, however, this cylinder transforms into a twisted ribbon with two sequential 180° twists. This ribbon exists for a short time and then unwinds, regenerating the cylinder. We discuss how and why these structures form and what determines their lifetimes.

Twisted ribbon carousels in random, three-dimensional optical fields.

The instantaneous electric vector in a random three-dimensional optical field is shown to generate twisted ribbon carousels that spin about their axes. The ribbons can be right or left handed and can unwind and rewind in time, changing their handedness during an optical cycle. Analytical formulas describing this behavior are presented.

Polarization Möbius strips on elliptical paths in three-dimensional optical fields.

Optical singularities in nonparaxial structured light are currently of special interest. Here we study polarization Möbius strips on elliptical paths surrounding lines of circular polarization in fully three-dimensional fields. We find that as the eccentricity, azimuthal orientation, or centering of the path changes, right-handed Möbius strips can change into left-handed ones, and vice versa, and that Möbius strips with one half twist can change into strips with three half twists, and vice versa. These transformations are shown to occur in a possibly unexpected way, not observed previously, that is universal for all two-component singularities.

Topological events on the lines of circular polarization in nonparaxial vector optical fields.

In nonparaxial vector optical fields, the following topological events are shown to occur in apparent violation of charge conservation: as one translates the observation plane along a line of circular polarization (a C line), the points on the line (C points) are seen to change not only the signs of their topological charges, but also their handedness, and, at turning points on the line, paired C points with the same topological charge and opposite handedness are seen to nucleate. These counter-intuitive events cannot occur in paraxial fields.

Singularity screening in generic optical fields.

We revisit the widely studied subject of screening in optical fields by topological charges and show that screening does not depend on charge ordering. Instead, for an array of N charges, screening requires that the variance of the charge fluctuations be small compared to N. We show by means of explicit examples that, when this requirement is met, screening can be complete, even for a spatially random arrangement of charges. We derive a minimal screening constraint on the charge correlation function and show that it is this constraint that is met in practice, rather than the more stringent constraints previously assumed.

Observer-dependent sign inversions of polarization singularities.

We describe observer-dependent sign inversions of the topological charges of vector field polarization singularities: C points (points of circular polarization), L points (points of linear polarization), and two virtually unknown singularities we call γ(C) and α(L) points. In all cases, the sign of the charge seen by an observer can change as she changes the direction from which she views the singularity. Analytic formulas are given for all C and all L point sign inversions.

Optical Möbius strips and twisted ribbon cloaks.

Optical Möbius strips that surround points of circular polarization, C points, in a generic three-dimensional optical field are cloaked by lines of twisted ribbons attached to the C points. When cloaking occurs, the observable signed twist index that counts the number of half-twists (one or three), and also measures the handedness (right or left), of a generic Möbius strip is determined by the twisted ribbon cloaks. Although some cloaks can be detached, they can never all be removed.

Ordinary polarization singularities in three-dimensional optical fields.

In generic three-dimensional optical fields the canonical point polarization singularities are points of circular polarization, C points on C lines, and points of linear polarization, L points on L lines. These special points are surrounded by a sea of ordinary points. In planes oriented normal to the principle axes of the polarization ellipse at the point, every ordinary point is also a singularity, here an ordinary polarization singularity, or O point. Interactions between O points, between O points and C points, and between O points and L points are described that highlight the fact that a consistent description of optical fields containing C and L lines must include O points.

Optical Möbius strips, twisted ribbons, and the index theorem.

The twist numbers of circular optical Möbius strips and twisted ribbons are shown to obey the index theorem under rotation of the plane of observation, and under change in the radius of the path.

Multitwist optical Möbius strips.

Circularly polarized Gauss-Laguerre GL00 and GL01 laser beams that cross at their waists at a small angle are shown to generate a quasi-paraxial field that contains a line of circular polarization, a C line, surrounded by polarization ellipses whose major and minor axes generate multitwist Möbius strips with twist numbers that increase with the distance from the C point.

Singularities in speckled speckle: screening.

We study screening of optical singularities in random optical fields with two widely different length scales. We call the speckle patterns generated by such fields speckled speckle, because the major speckle spots in the pattern are themselves highly speckled. We study combinations of fields whose components exhibit short- and long-range correlations and find unusual forms of screening.

Experimental measurements of topological singularity screening in random paraxial scalar and vector optical fields.

There exists a substantial body of theory that predicts mutual screening of signed topological singularities (topological charges) in random optical fields (speckle patterns). Such screening appears to be rather mysterious because there are neither energetic nor entropic reasons for its existence. We present the first experimental confirmation of mutual screening by the stationary points of the intensity, the canonical optical scalar field, and of mutual screening by C points in elliptically polarized light, the generic optical vector field. We also elucidate specific aspects of the geometry and topology of these fields that we argue give rise to screening.

Singularities in speckled speckle.

Speckle patterns produced by random optical fields with two (or more) widely different correlation lengths exhibit speckle spots that are themselves highly speckled. Using computer simulations and analytic theory we present results for the point singularities of speckled speckle fields, namely, optical vortices in scalar (one polarization component) fields and C points in vector (two polarization components) fields. In single correlation length fields both types of singularities tend to be more or less uniformly distributed. In contrast, the singularity structure of speckled speckle is anomalous; for some sets of source parameters vortices and C points tend to form widely separated giant clusters, for other parameter sets these singularities tend to form chains that surround large empty regions. The critical point statistics of speckled speckle is also anomalous. In scalar (vector) single correlation length fields phase (azimuthal) extrema are always outnumbered by vortices (C points). In contrast, in speckled speckle fields, phase extrema can outnumber vortices and azimuthal extrema can outnumber C points by factors that can easily exceed 10(4) for experimentally realistic source parameters.

Umbilic point screening in random optical fields.

Umbilic points--singular points of curvature characterized by a fractional topological charge q=+/-1/2--are the most numerous of all special points in the landscape of random optical fields (speckle patterns), outnumbering maxima, minima, saddle points, and optical vortices. To the best of our knowledge, we present the first experimental evidence that positive and negative umbilic points screen one another. Theory predicts that in the absence of screening the charge variance in a bounded region is proportional to the area of the region, whereas in the presence of screening the variance is drastically reduced and is proportional to the perimeter. Our data confirm this latter prediction and provide the first estimates of the screening lengths for umbilic points of the intensity and of the amplitude (field modulus).

Experimental umbilic diabolos in random optical fields.

The intensity of a random optical field consists of bright speckle spots (maxima) separated from dark areas (minima and optical vortices) by saddle points. We show that hidden in this complicated landscape are umbilic points--singular points at which the eigenvalues Lambda (+/-) of the Hessian matrix that measure the curvature of the landscape become degenerate. Although not observed previously in random optical fields, umbilic points are the most numerous of all special points, outnumbering maxima, minima, saddle points, and vortices. We show experimentally that the directions of principal curvature, the eigenvectors Psi (+/-), rotate about intensity umbilic points with positive or negative half-integer winding number, in accord with theory, and that Lambda (+) and Lambda (-) generate a double cone known as a diabolo. At optical vortices the curvature of the amplitude is singular, and we show from both theory and experiment that for this landscape Psi (+/-) rotate about vortex centers with a positive integer winding number. Diabolos can be classified as elliptic or hyperbolic, and we present initial results for the measured fractions of these two different types of umbilic diabolos.

Diabolo creation and annihilation.

A point of circular polarization embedded in a paraxial field of elliptical polarization is a polarization singularity called a C point. At such a point the major axis a and minor axis b of the ellipse become degenerate. Away from the C point this degeneracy is lifted such that surfaces a and b form nonanalytic cones that are joined at their apex (the C point) to produce a double cone called a diabolo. Typically, during propagation diabolo pairs are created or annihilated. We present rules based on geometry and topology that govern these events, provide initial experimental confirmation, and enumerate the allowed configurations in which diabolos can be created or annihilated.

Experimental optical diabolos.

The canonical point singularity of elliptically polarized light is an isolated point of circular polarization, a C point. As one recedes from such a point the surrounding polarization figures evolve into ellipses characterized by a major axis of length a, a minor axis of length b, and an azimuthal orientational angle alpha: at the C point itself, alpha is singular (undefined) and a and b are degenerate. The profound effects of the singularity in alpha on the orientation of the ellipses surrounding the C point have been extensively studied both theoretically and experimentally for over two decades. The equally profound effects of the degeneracy of a and b on the evolving shapes of the surrounding ellipses have only been described theoretically. As one recedes from a C point, a and b generate a surface that locally takes the form of a double cone (i.e., a diabolo). Contour lines of constant a and b are the classic conic sections, ellipses or hyperbolas depending on the shape of the diabolo and its orientation relative to the direction of propagation. We present measured contour maps, surfaces, cones, and diabolos of a and b for a random ellipse field (speckle pattern).

Polarization singularity proliferation in three-dimensional ellipse fields.

The classical singularities of elliptically polarized light are points of circular (linear) polarization, characterized by a half-integer (integer) topological index. On average, in any plane of a random ellipse field there is of the order of one each of these classical singularities per coherence area. It is shown that every ellipse in such a field is a multiple singularity characterized by nine different topological indices: Three indices characterize rotations of the principal axis system of the surrounding ellipses, and six indices characterize a one- or two-turn spiral precession of these axes. The nine indices can divide the field into 32,768 different volumes with different structures separated by singular surfaces (grain boundaries) on which an index becomes undefined. This unprecedented proliferation of singularities and structures can occur in other three-dimensional systems in which individual elements are described by unique principal axis systems, for example, liquid crystals, and should be sought in such systems.

Polarization singularity democracy: WYSIWYG.

The canonical point singularity of elliptically polarized light is a C point, an isolated point of circular polarization surrounded by a field of polarization ellipses. The defining singular property of a C point is that the surrounding ellipses rotate about the point. It is shown that this rotation is seen only for a particular line of sight (LOS) and, conversely, that there exists a unique LOS for every ellipse along which the ellipse is seen as a singularity. It is also shown that changes in LOS can turn singularities into stationary points and vice versa. The democratic behavior of polarization singularities and stationary points is a consequence of the fundamental "what you see is what you get" property of ellipse fields. Simple experiments are proposed for observing this unusual property of elliptically polarized light.

Polarization singularities in optical lattices.

Polarization singularities are shown to be unavoidable features of three-dimensional optical lattices. These singularities take the form of lines of circular polarization, C lines, and lines of linear polarization, L lines. The polarization figures surrounding a C line (L line) rotate about the line with winding number +/-1/2 (+/-1). C and L lines permeate the lattice, meander throughout the unit cell, and form closed loops. Surprisingly, every point in a linearly polarized optical lattice is found to be a singularity about which the surrounding polarization vectors rotate with an integer winding number.