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.     

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.     

Elliptic critical points: C-points, a-lines, and the sign rule

The critical points of generic paraxial ellipse fields consist of singular points of circular polarization, called C -points, and azimuthal stationary points, i.e., maxima, minima, and saddle points. We define these stationary points here and review their properties. The sign rule for ellipse fields requires that the sign of the singularity indices I(C)=+/-1/2 of the C -points on non-self-intersecting lines of constant azimuthal ellipse orientation (modulo pi/2), i.e., a -lines, alternate along the line. We verify this rule experimentally, using a newly developed interferometric technique to measure C -points and a -lines in an elliptically polarized random optical field.     

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 2D and 3D speckle fields

The 3D structure of randomly polarized light fields is exemplified by its polarization singularities: lines along which the polarization is purely circular (C lines) and surfaces on which the polarization is linear (L surfaces). We visualize these polarization singularities experimentally in vector laser speckle fields, and in numerical simulations of random wave superpositions. Our results confirm previous analytical predictions [M. R. Dennis, Opt. Commun. 213, 201 (2002)] regarding the statistical distribution of types of C points and relate their 2D properties to their 3D structure.     

Polychromatic polarization singularities

Elliptical polarization can appear in only monochromatic optical fields. In polychromatic vector fields the polarization is a Lissajous figure, but in only commensurate fields do the figures have well-defined shapes; in other fields the shapes are undefined. Nonetheless, I show that a given paraxial polychromatic vector field has a coherency ellipse field associated with it that contains polarization singularities and stationary points that are surrogates for the corresponding critical points of the parent optical field.     

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).     

Cones of optical axes in gyrotropic crystals with scalar permittivity

A theory is constructed for the degeneracy of electromagnetic waves in gyrotropic crystals near their isotropy points (in temperature and other parameters), in which permittivity ? becomes scalar due to dispersion and optical anisotropy is entirely determined by gyrotropy. It is shown that closed lines of intersection of sheets of the refraction surface, which correspond to the cones and, in particular, planes of the optical axes, are formed for isotropic ?. The polarization characteristics of wave fields experience a jump upon a transition through such lines. The conditions for the existence and geometry of the degeneracy cones under investigation are analyzed for crystals of all symmetry classes permitting optical activity. It is shown that the degeneracy lines disappear for a small deviation of ? from isotropy, while polarization anomalies persist. Isolated (one or two) degeneracy points may retain in place of the lines in the case when the optical axes of a perturbed crystal with “switched-off” gyrotropy belong to the initial cone.     

Vector singularities of superposition of mutually incoherent orthogonally polarized beams

It is shown that, at an incoherent superposition of orthogonally polarized laser beams, a special type of singularities are formed in the cross section of a combined beam in place of the well-known singularities, such as optical vortices (for scalar fields); C points, at which the polarization is circular; and L lines, along which the polarization is linear (for coherent vector fields). These new singularities are U lines, along which the degree of polarization is zero and the state of polarization is undetermined, and P points, at which the degree of polarization is equal to unity and the state of polarization is determined by the nonzero component of the combined beam. Conditions of topological stability of U and P singularities are discussed, as well as peculiarities of the spatial distribution of the degree of polarization of the field in the vicinity of such singularities. First experimental results on the reconstruction of a vector skeleton formed by U and P singularities in combined speckle fields are presented.     

Hyperboloid structures formed by polarization singularities in coherent vector fields with longitudinal-transverse coupling

The propagation-dependent polarization vector fields are experimentally created from an isotropic microchip laser with a longitudinal-transverse coupling and entanglement of the polarization states. The experimental three-dimensional coherent vector fields are analytically reconstructed with a coherent superposition of orthogonal circularly polarized vortex modes. Each polarized component is found to comprise two Laguerre-Gaussian modes with different topological charges. With the analytical representation, the polarization singularities, on which the electric polarization ellipse is purely circular (C lines) or purely linear (L surfaces), are explored. The C line singularities are found to form an intriguing hyperboloidal structure.     

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.     

Intraband dynamics and terahertz emission in biased semiconductor superlattices coupled to double far-infrared pulses

This paper studies both the intraband polarization and terahertz emission of a semiconductor superlattice in combined dc and ac electric fields by using the superposition of two identical time delayed and phase shifted optical pulses. By adjusting the delay between these two optical pulses, our results show that the intraband polarization is sensitive to the time delay. The peak values appear again for the terahertz emission intensity due to the superposition of two optical pulses. The emission lines of terahertz blueshift and redshift in different ac electric fields and dynamic localization appears. The emission lines of THz only appear to blueshift when the biased superlattice is driven by a single optical pulse. Due to excitonic dynamic localization, the terahertz emission intensity decays with time in different dc and ac electric fields. These are features of this superlattice which distinguish it from a superlattice generated by a single optical pulse to drive it.     

Topological structure in polarization resolved conoscopic patterns for nematic liquid crystal cells

We investigate the polarization structure of coherent light, produced by a convergent light beam transmitted through nematic liquid crystal (NLC) cells with different director configurations. Employing solutions to the transmission problem for the case when plane wave propagates through an anisotropic layer, we analyze the arrangement of the topological elements, such as polarization singularities (C points with circular polarization and L lines with linear polarization), saddle points and extrema of polarization azimuth. We observe transformations of the topological structure under the variation of the incident light ellipticity and represent it by corresponding trajectories of topological elements in three-dimensional space. For the cells with uniform and non-uniform director configuration we describe the processes of creation/annihilation of C point pairs, which can be controlled precisely in the case of the cell with non-uniform director. Our experimental measurements for the homeotropically oriented NLC cells are in good agreement with the theoretical predictions.     

Strong-field photoionization of intense laser fields by controlling optical singularities

The spatial features of a light field, such as in the form of the optical singularities, provide a new degree of freedom for the application of light fields in different areas of science and technology. However, although the exploration of structured light is growing rapidly, the investigation of strong-field photoionization using such light fields is noticeably lagging behind. Here, we present an experimental study that reveals the signatures of intense, structured light fields with controlled optical singularities in strong-field photoionization. The different types of optical singularities can be identified through photoionization observables,i.e., photoelectron momentum distributions(PMDs). By concurrently shifting the locations of the phase and polarization singularities, the focal electric field features can be designated, and subsequently, the photoionization appearances can be manipulated. In this process, the behaviors of the different intense optical singularities are clearly visualized by the PMDs. This work will advance both the strong-field science and singularity optics.     

Fermionic out-of-plane structure of polarization singularities

A new classification of circular polarization C points in three-dimensional polarization ellipse fields is proposed. The classification type depends on the out-of-plane variation of the polarization ellipse axis, in particular, whether the ellipse axes are in the plane of circular polarization one or three times. A minimal set of parameters for this classification is derived and discussed in the context of the familiar in-plane C point classification into lemon, star, and monstar types. This new geometric classification is related to the M?bius index of polarization singularities recently introduced by Freund.     

Entangled optical vortex links

Optical vortices are lines of phase singularity which percolate through all optical fields. We report the entanglement of linked optical vortex loops in the light produced by spontaneous parametric down-conversion. As measured by using a Bell inequality, this entanglement between topological features extends over macroscopic and finite volumes. The entanglement of photons in complex three-dimensional topological states suggests the possibility of entanglement of similar features in other quantum systems describable by complex scalar functions, such as superconductors, superfluids, and Bose-Einstein condensates.     

Yang-Lee zeros of the Potts model

The Yang-Lee zeros of the three-component ferromagnetic Potts model in one dimension in the complex plane of an applied field are determined. The phase diagram consists of a triple point where three phases coexist. Emerging from the triple point are three lines on which two phases coexist and which terminate at critical points (Yang-Lee edge singularity). The zeros do not all lie on the imaginary axis but along the three two-phase lines. The model can be generalized to give rise to a tricritical point which is a new type of Yang-Lee edge singularity. Gibbs phase rule is generalized to apply to coexisting phases in the complex plane.Supported in part by the National Science Foundation under Grant No. DMR-81-06151.     

Dynamic evolution of transverse energy flow in focused asymmetric optical vector-vortex beams

We present here controlled generation of asymmetric optical vector-vortex beams using a two-mode optical fiber and study the dynamic evolution of the transverse energy flow (TEF) when focused through a spherical lens. The dependence of the TEF on various factors such as the vortex charge, vortex anisotropy and polarization structure around the vortex core is explored. It is found that the TEF is directly proportional to the phase gradient and its direction is governed by the vortex charge. The presence of C-point polarization singularity in the beam and the polarization structure around it results in vibrational phase gradient which is the major factor deciding the TEF in vector-vortex beams.