Optical method of caustics applied in viscoelastic fracture analysis

The optical method of caustics is developed here to study the fracture of viscoelastic materials. By adopting a distribution of viscoelastic stress fields near the crack tip, the method of caustics is used to determine the viscoelastic fracture parameters from the caustic patterns near the crack tip. Two viscoelastic materials are studied. These are PMMA and ternary composites of HDPE/POE-g-MA/CaCO₃. The transmitted and reflective methods of caustics are performed separately to investigate viscoelastic fracture behaviors. The stress intensity factors (SIFs) versus time is determined by a series of shadow spot patterns combined with viscoelastic parameters evaluated by creep tests. In order to understand the viscoelastic fracture mechanisms of HDPE/POE-g-MA/CaCO₃ composites, their fracture surfaces are observed by a Scanning Electron Microscope (SEM). The results indicate that the method of caustics can be used to characterize the fracture behaviors of viscoelastic materials and further to optimize the design of polymer composites.     

Nondiffracting random intensity patterns

We report a simple approach for generating nondiffracting random intensity patterns that do not change with propagation. Experimental results are shown where the generating patterns are encoded onto a phase-only liquid-crystal display. We expect that these results will be useful for encryption or surveillance applications.     

Time-Domain Mapping of Electromagnetic Ray Movement Inside Anisotropic Spherical Resonator

This paper presents the analytical proof of Time-Domain Mapping Method for the spherical resonator made up of uniaxial crystal. In this way, the main types of caustics inside the spherical resonator made up of uniaxial crystal, which were investigated numerically before, are confirmed analytically. It is engraved that the problem of the ray flow inside the spherical resonator can be reduced to the problem of the ray flow inside metal cavity shaped as spheroid.     

被动方式产生近似无衍射贝塞尔光技术

无衍射光束具有中心光斑小且不随传播距离变化、自再现、产生局域空心光等特点,由于这些特殊的光束特性而在许多领域,如计量学,经典光学,非线性光学和生命科学等领域中得到广泛的应用。无衍射光束的产生与变换是目前的一个研究热点。通过衍射理论、干涉理论和几何光学方法可对无衍射光进行描述。实现无衍射贝塞尔光束的方法可分为主动式和被动式。本文主要综述地介绍了几种被动式产生近似无衍射贝塞尔光束的最新技术,通过分析其优缺点,给出了这些技术的适用范围。     

Nondiffracting wave beams in non-Hermitian Glauber–Fock lattice

We theoretically study non-Hermitian Glauber–Fock lattice with nonuniform hopping. We show how to engineer this lattice to get nondiffracting wave beams and find an exact analytical solution to nondiffracting localized waves. The exceptional points in the energy spectrum are also analyzed.     

A variational principle for actions on symmetric symplectic spaces

We present a definition of generating functions of canonical relations, which are real functions on symmetric symplectic spaces, discussing some conditions for the presence of caustics. We show how the actions compose by a neat geometrical formula and are connected to the hamiltonians via a geometrically simple variational principle which determines the classical trajectories, discussing the temporal evolution of such “extended hamiltonians” in terms of Hamilton–Jacobi-type equations. Simplest spaces are treated explicitly.     

Extended Feynman formula for time-dependent forced harmonic oscillator

Horváthy's modification of Feynman's path integral formula is generalized to the time-dependent forced harmonic oscillator. The propagator at caustics is then obtained by using its modified semi-group property. Finally, with our new formula, the propagator for a charged particle in a time-dependent electromagnetic field is evaluated exactly beyond and at caustics.     

Theoretical investigations on optical caustics of spherical microresonators: Analytical expressions of caustics and their asymptotic behaviors,computational simulations

This paper introduces analytical expressions of optical caustics r₁ and r₂ in micro-resonators (respectively called inner and outer caustics in micro-optics) in terms of radial and azimutal modal numbers n and l mutually. Such a theoretical study focuses on spherical micro-resonators and is carried out according to two complementary ways: Electromagnetic (E.M) approaches based respectively on an eigenvector problem resolution called modal field in integrated optics and an analogy with Quantum Mechanics (Q.M) using the notions of potential and geometric eikonal principle. Thus, using quantization and phase considerations, fundamental new formulation of caustics, that depend on the two previous quantum modal numbers simultaneously, have been shaped and confronted with numerical and graphical simulations. Both ways lead to results in good agreement (correlation coefficient superior to 0.99) and allow us to specify the evolution of such caustics according to each modal number of the quantification; such an analytical caustics formulation involves a specific function f whose study is expanded for various values of n ranging from 1 to 500. Moreover a whole study of asymptotic behaviors of f-function and its inverse f^? 1 on specific n and l values, directly related to the previous caustics expressions, is conducted, allowing the expansion of new analytical expressions of caustics in various limit cases and asymptotic behaviors. Then, the evolution of the positions of caustics in terms of modal numbers is interpreted, and the physical meanings of optical properties such as quasi-resonance, position and confinement of field with resonance, and associated electromagnetic fields are discussed.     

Propagation of laser beams and caustics in crystals

We investigated the propagation and transformation of laser beams and the formation of caustics in particular directions, including optical axes of crystals. We found a direct connection between the particular directions and the curvature of wave surfaces that is determined by the symmetry and optical properties of the crystals. We demonstrated that caustics in particular directions are characterized by a degree of the curvature degeneracy and are described by particular solutions of the corresponding nonlinear equations. __________ Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 74, No. 4, pp. 491–498, July–August, 2007.     

The Pentagram Map: A Discrete Integrable System

The pentagram map is a projectively natural transformation defined on (twisted) polygons. A twisted polygon is a map from \mathbb Z{\mathbb Z} into \mathbbRP²{{\mathbb{RP}}^2} that is periodic modulo a projective transformation called the monodromy. We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation. A research announcement of this work appeared in [16].     

无衍射特殊光束的产生与三维表征

无衍射光束(如贝塞尔光束、艾里光束)因具有无衍射、自愈合的特性, 在很多领域都有广泛的应用. 本文提出使用纯相位型空间光调制器对光场的复振幅进行调控, 从而可以产生多种复杂模式的无衍射光束, 如强度可独立调控的多个零阶贝塞尔光束, 两个高阶贝塞尔光束干涉生成的花瓣状无衍射光束, 具有多个主瓣的加速光束等特殊的无衍射光束. 通过在待测焦场附近放置一个平面反射镜, 使其沿光轴快速扫描光场, 并由数字相机同步拍摄反射回来的一系列二维光场强度分布信息, 可实现对无衍射光束三维光场强度分布的快速测量和表征. 本实验方法和技术可以快速产生各种复杂的特殊光场并获得其精确的三维可视化重建效果, 在光学显微、光学俘获、光学微加工等领域有潜在的应用价值.     

Shear wave caustics and the shadows of surface breaking cracks

A refinement to the detection of defects in rods and pipes by means of their ultrasonic shadows is presented. It has been previously shown that in standard non-destructive test configurations ultrasonic waves inside cylindrical test objects form caustics, surfaces of high sound intensity. Here it is demonstrated that when a crack edge crosses a caustic the diffraction of sound around the defect is enhanced, which allows the point at which this intersection occurs to be detected. Because the caustic is a well-defined geometric entity the position of the crack edge is now known. This effect presents a novel method of sizing defects that extend in from the surface. Experiments were performed in water immersion using 5 MHz sound. The test specimens were 76 mm diameter aluminium cylinders into which slots were cut to act as artificial defects.     

基于焦散线方法的自加速光束设计

以艾里光束为代表的自加速光束是一类在自由空间中具有弯曲传播特性的新型特殊光束.这类光束因其具有无衍射、自加速和自修复等奇异特性引起了人们的广泛关注,有望应用于光学微粒操控、激光微加工、全光路由和超分辨成像等诸多领域.由于艾里光束只能沿着抛物线的轨迹传播,限制了其在实际应用中的灵活性,因而设计出能够沿着不同轨迹传播的自加速光束是这一研究领域的关键问题,而基于焦散线方法的自加速光束设计是解决该问题的有效途径之一.这一方法是将设计的传播轨迹与光学焦散线联系起来,通过分析形成该焦散线所需的光线簇构造出对应的初始场分布.基于该原理并经过不断发展,不同类型的自加速光束相继得以实现,并且借助维格纳函数还可以同时实现实空间和傅里叶空间的自加速光束设计,为自加速光束的应用提供了更多的可能性.本文对基于焦散线方法的自加速光束设计原理和进展进行全面介绍.     

Non-integrability vs. integrability in pentagram maps

We revisit recent results on integrable cases for higher-dimensional generalizations of the 2D pentagram map: short-diagonal, dented, deep-dented, and corrugated versions, and define a universal class of pentagram maps, which are proved to possess projective duality. We show that in many cases the pentagram map cannot be included into integrable flows as a time-one map, and discuss how the corresponding notion of discrete integrability can be extended to include jumps between invariant tori. We also present a numerical evidence that certain generalizations of the integrable 2D pentagram map are non-integrable and present a conjecture for a necessary condition of their discrete integrability.     

无衍射光束的特性和生成方法

介绍了无衍射光束的概念和传输性质，描述了应用常规的光学元件和计算机产生的全息图生成无衍射光束的方法及其潜在的应用价值。     

Nature’s optics and our understanding of light

Optical phenomena visible to everyone have been central to the development of, and abundantly illustrate, important concepts in science and mathematics. The phenomena considered from this viewpoint are rainbows, sparkling reflections on water, mirages, green flashes, earthlight on the moon, glories, daylight, crystals and the squint moon. And the concepts involved include refraction, caustics (focal singularities of ray optics), wave interference, numerical experiments, mathematical asymptotics, dispersion, complex angular momentum (Regge poles), polarisation singularities, Hamilton’s conical intersections of eigenvalues (‘Dirac points’), geometric phases and visual illusions.     

Accelerating Airy–Gauss–Kummer localized wave packets

A general approach to generating three-dimensional nondiffracting spatiotemporal solutions of the linear Schrödinger equation with an Airy-beam time-dependence is reported. A class of accelerating optical pulses with the structure of Airy–Gauss–Kummer vortex beams is obtained. Our results demonstrate that the optical field contributions to the Airy–Gauss–Kummer accelerating optical wave packets of the cylindrical symmetry can be characterized by the radial and angular mode numbers.     

Controlling the evolution of nondiffracting speckle by complex amplitude modulation on a phase-only spatial light modulator

In this work we investigate the structure of nondiffracting speckle fields, both experimentally and theoretically. We are able to produce very good agreement between the experimentally recorded and theoretically calculated fields by using complex amplitude modulation on a phase-only spatial light modulator to implement controlled ring-slit experiments for the generation of nondiffracting speckle fields. The structure of the nondiffracting speckle due to binary and continuous phase modulations for both a uniform and a normal distribution is investigated. We find that we are able to engineer whether the nondiffracting field will appear as speckle or a structured zero-order Bessel beam by adjusting the standard deviation in the distribution. Having the ability to control where in the spectrum, from fully-developed nondiffracting speckle to a symmetric zero-order Bessel beam, the nondiffracting field will exist can prove to be a useful resource in the non-destructive testing of materials.     

Feynman's formula for a harmonic oscillator

Feynman's original formula for the Green's function of a harmonic oscillator includes the Maslov correction. It also leads to the known formula for caustics for the time intervals that are equal to integer multiplets of the half period.     

Pentagrams and Paradoxes

Klyachko and coworkers consider an orthogonality graph in the form of a pentagram, and in this way derive a Kochen-Specker inequality for spin 1 systems. In some low-dimensional situations Hilbert spaces are naturally organised, by a magical choice of basis, into SO(N) orbits. Combining these ideas some very elegant results emerge. We give a careful discussion of the pentagram operator, and then show how the pentagram underlies a number of other quantum “paradoxes”, such as that of Hardy.