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.     

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.     

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.     

利用非传统螺旋相位调控高阶涡旋光束的拓扑结构

本文提出一种利用非传统螺旋相位调控高阶涡旋光束拓扑结构的方法.数值模拟并实验研究了具有不均匀旋转梯度的非传统螺旋相位对高阶涡旋光束的调控行为.结果表明, 携带有非传统螺旋相位的高阶涡旋光束在传输过程中, 将退化为沿一条直线排列的多个一阶相位奇点, 并且, 这种非传统螺旋相位对高阶涡旋光束的调控特性, 可抑制相位噪声等扰动所引起的拓扑结构随机退化现象.本文的结论为涡旋光束拓扑结构的调控提供了一种可行的新途径, 在基于涡旋光束的光学通信、光学操控等方面具有潜在应用.     

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).     

Experiments with twisted light

The generic – that is, stable under perturbations – nodes of the field in a monochromatic light beam are optical vortices. We describe here their connection to Chladni's nodal lines in the oscillations of metal plates, as well as a few experiments that have been performed with optical vortices. We will describe how optical vortices can be generated experimentally; how it can be shown that they possess orbital angular momentum; how individual photons can be sorted according to their vortex state; and how optical vortices can be used to demonstrate higher-dimensional quantum entanglement.     

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.     

散射光场中的光学涡旋

散射光场中的光学涡旋叶茂，王鹏业，戴建华，张洪钧（四川大学物理系成都６１００６４）（中国科学院物理研究所北京１０００８０）ＯｐｔｉｃａｌＶｏｒｔｉｃｅｓｉｎＡＴｈｅｒｍａｌＲａｙｌｅｉｇｈＳｃａｔｔｅｒｉｎｇＬｉｇｈｔＦｉｅｌｄ￥ＭａｏＹｅ；ｐｅｎｇ...     

Lateral diffusion of the topological charge density in stochastic optical fields

Stochastic (i.e. random and quasi-random) optical fields may contain distributions of optical vortices that are represented by non-uniform topological charge densities. Numerical simulations are used to investigate the evolution under free-space propagation of topological charge densities that are inhomogeneous along one dimension. It is shown that this evolution is described by a diffusion process that has a diffusion parameter which depends on the propagation distance.     

Experimental investigation of local properties and statistics of optical vortices in random wave fields

We present the first direct experimental evidence of the local properties of optical vortices in a random laser speckle field. We have observed the Berry anisotropy ellipse describing the anisotropic squeezing of phase lines close to vortex cores and quantitatively verified the Dennis angular momentum rule for its phase. Some statistics associated with vortices, such as density, anisotropy ellipse eccentricity, and its relation to zero crossings of real and imaginary parts of the random field, are also investigated by experiments.     

Phase singularities and coherence vortices in linear optical systems

It is demonstrated for a time-invariant linear optical system that there exists a definite connection between the optical vortices (phase singularities of the field amplitude) which appear when it is illuminated by spatially coherent light and the coherence vortices (phase singularities of the field correlation function) which appear when it is illuminated by partially coherent light. Optical vortices are shown to evolve into coherence vortices when the state of coherence of the field is decreased. Examples of the connection are given. Furthermore, the generic behavior of coherence vortices in linear optical systems is described.     

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.     

The structure of a nonparaxial Gaussian beam near the focus: II. Optical vortices

Exact analytical solutions of Maxwell’s equations describing the behavior of a nonparaxial optical vortex in the vicinity of a focal waist are obtained using the Whittaker method of scalar potentials, the point complex source method, and approximate Davis boundary conditions. It is shown that nonparaxial optical vortices in free space fall into three large groups: even and odd vortices with preferential circular polarization and azimuthally symmetric TE and TM vortices. The fields of even and odd nonparaxial vortices agree well with the fields of guided homogeneous and inhomogeneous vortices of a weakly guiding fiber. In the paraxial approximation, the expressions obtained for the fields are transformed to the fields of paraxial optical vortices. In the focal region, a nonparaxial beam experiences elliptic deformation of the cross section. This elliptic deformation is shown to result from the asymmetric location of regions with negative energy flows. The reversal of sign of the topological charge and the helicity of a combination of even and odd vortices causes both rotation of the dislocation axis through π/2 and longitudinal displacement of the focal spot, which are the transverse and the longitudinal optical Magnus effects.     

Shearograms of an optical phase singularity

Shearograms are known to represent phase gradients but when vortices are present in the optical field, these do not represent true phase gradients. Phase gradients of an optical phase singularity are presented. A lateral shear interferometer is used for obtaining shearograms of optical fields with vortices. A diffractive phase element is used to generate vortices. It is shown that shearograms can be used in the detection of optical vortices. Shearogram of speckle field is also presented.     

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.     

Partially incoherent optical vortices in self-focusing nonlinear media

We observe stable propagation of spatially localized single- and double-charge optical vortices in a self-focusing nonlinear medium. The vortices are created by self-trapping of partially incoherent light carrying a phase dislocation, and they are stabilized when the spatial incoherence of light exceeds a certain threshold. We confirm the vortex stabilization effect by numerical simulations and also show that the similar mechanism of stabilization applies to higher-order vortices.     

利用全息法在偶氮聚合物薄膜中记录涡旋光场

利用全息技术在偶氮聚合物薄膜中记录了拓扑荷数q=–1,1,2,4的涡旋光场,并将记录的原始叉形光栅与计算全息光栅进行对比,对不同拓扑荷数涡旋光的记录速率和偶氮材料的可重复擦写性能进行了测试;记录完成后,将复现涡旋光与高斯光束干涉,并与原始涡旋光和原始叉形光栅对比,分析了记录质量.实验结果表明:高阶涡旋光场的全息叉形光栅会在记录过程中发生劈裂,轻微劈裂的涡旋光束仍维持一个稳定的环状结构;全息记录过程中不同拓扑荷数的涡旋光束记录速率较为统一,偶氮材料可经过上百次的擦写而不出现疲劳;再现涡旋光与原始涡旋光在光强分布结构上保持高度一致,再现涡旋光的干涉条纹与原始涡旋全息光栅保持高度一致,涡旋光及其携带的拓扑荷信息可被有效记录和读取.     

Vortex lines in a ferromagnetic spin-triplet superconductor

Based on Duan's topological current theory, we show that in a ferromagnetic spin-triplet superconductor there is a topological defect of string structures which can be interpreted as vortex lines. Such defects are different from the Abrikosov vortices in one-component condensate systems. We investigate the inner topological structure of the vortex lines. The topological charge density, velocity, and topological current of the vortex lines can all be expressed in terms of δ function, which indicates that the vortices can only arise from the zero points of an order parameter field. The topological charges of vortex lines are quantized in terms of the Hopf indices and Brouwer degrees of φ-mapping. The divergence of the self-induced magnetic field can be rigorously determined by the corresponding order parameter fields and its expression also takes the form of a δ-like function. Finally, based on the implicit function theorem and the Taylor expansion, we conduct detailed studies on the bifurcation of vortex topological current and find different directions of the bifurcation.     

Parallel two-photon photopolymerization of microgear patterns

We report parallel two-photon photopolymerization of microgear patterns by exposing a photoresist to holographically generated optical vortices. The optical vortices are created by imparting a helical pitch onto the incident light using a programmable lithographic phase mask realized with a computer addressable phase-only spatial light modulator. By varying the phase levels of the spatial light modulator, the truncated helical phase of an optical vortex results in output intensity patterns that typifies that of microgears instead of perfect doughnut beams. Our experiments and simulations are in good agreement implying a more efficient and highly parallel two-photon photopolymerization scheme that can be subsequently used for non-scanning fabrication of microgears.