Airy加速光束的局域空间频率及其在光束设计中的应用

Airy加速光束是近年来备受关注的一种新型无衍射光束. 它所具有的自由加速、无衍射及自恢复特性使其在光学微操纵、非线性光学、 电子加速等诸多领域显示出重要的应用价值. 因此, 如何方便高效地生成加速光束成为近年来的一个热点研究内容. 本文对Airy加速光束复振幅分布的空间振荡特性进行了分析, 建立了利用局域空间频率描述其加速特性的理论. 提出了利用零点坐标计算加速光束局域空间频率的方法, 通过非线性拟合给出了可以精确描述Airy光束局域空间频率的解析公式; 确定了加速光束的局域空间频率函数与加速轨迹之间的定量关系, 给出了由给定加速轨迹计算相应的局域空间频率以及加速光束的纯相位函数的一种简单计算方法. 将上述分析结果用于设计产生具有给定加速轨迹的加速光束所需的相位函数, 成功求出了能够产生圆弧形加速轨迹的新型加速光束的纯相位函数的解析表达式. 基于该相位函数设计的纯相位衍射光学元件的模拟衍射结果证明了上述方法的可行性. 关键词： 衍射光学 加速光束 局域空间频率     

部分相干Airy光束及其传输的研究

利用高斯-谢尔模型的交叉谱密度公式和惠更斯衍射积分, 得到了部分相干Airy光束的表达式. 对不同相干度的部分相干Airy光束的无衍射和横向加速偏移的性质进行了理论分析并予以实验验证, 结果一致表明部分相干Airy光束也具有无衍射和横向加速度的特性, 而且相干度越低, 部分相干Airy光束的无衍射距离越短, 横向加速度越小. 关键词： 傅里叶光学 全息 部分相干 艾里光束     

用傅里叶分析法研究艾里光束远场传播特性

为了从理论上深入分析新型无衍射光束艾里光束在有限能量条件下的远场传播特性,首先,从决定光波在自由空间传播的一维旁轴波动方程入手,采用傅里叶分析法,结合艾里函数的特殊性质,并利用经过指数衰减的有限能量初始条件,完整给出了有限能量条件下用于精确描述一维艾里光束在自由空间传播特性的波动方程解析解.然后,利用所得到解析解分别对一维和二维艾里光束在自由空间的传播特性进行了研究,重点分析了不同参量条件对艾里光束进行无衍射传播和横向自加速的影响.研究表明:当任意横向尺度为100μm,衰减系数为0.03、0.05、0.07、0.1、0.2时,二维艾里光束无衍射传播距离分别为1 014、624、455、338、193mm;当横向尺度保持不变时,衰减系数越小,艾里光束保持无衍射传播的距离越大;当衰减系数保持不变时,横向尺度越小,艾里光束横向自加速越大.所采用的研究方法也可用于研究艾里光束在介质中的传播特性.     

有限宽光束在生物组织中传输的蒙特卡罗方法

基于光在生物组织中的传输理论,研究有限宽光束在生物组织中的传输特性.利用入射光强与格林函数进行卷积计算,推导了有限宽光束在圆柱坐标系下的光传输过程,给出了有限宽光束的光传输方程.以高斯光源和平圆光源为例,进一步给出了两种光源的有限宽光束在生物组织中的光传输方程.介绍了Monte Carlo方法原理及模拟光在多层生物组织中的传输过程.给出了典型三层生物组织的散射系数、吸收系数和折射率等光学参数.利用Monte Carlo方法模拟了有限宽光束在三层生物组织中的传输特性,对比研究了高斯光源和平圆光源有限宽光束在三层生物组织传输时光学参数漫透射率、漫反射率和光能流率的变化规律.这些参数的变化规律可为光在生物组织传输的实际应用提供理论指导.     

部分相干Airy光束在大气湍流中的光强演化

为了利用Airy光束的无衍射、自恢复和自弯曲特性抑制大气湍流效应,实现远距离无线光通信,对部分相Airy光束在大气湍流中传输时的光强演化进行了研究.利用高斯-谢尔模型的交叉谱密度函数、广义惠更斯-菲涅尔原理以及Rytov相位近似法,推导了部分相干Airy在湍流大气中平均光强的表达式.分别从传播距离、湍流强度等方面对光强分布的影响进行了模拟仿真,并对光束自身参数对光强分布的影响进行了相关实验验证.结果表明:随着传播距离的增加,部分相干Airy光束的旁瓣逐渐衰减,主瓣逐渐扩散;在传播足够远时,其旁瓣逐渐消失,主瓣逐渐演化为高斯分布.仿真和实验结果一致表明光束的截断因子越小、特征长度和相干长度越长,光束的光强分布保持越完整.     

非偏振贝塞尔高斯光束的球散射

无衍射光束球散射性质的研究目前一般采用贝塞尔光束,但是贝塞尔光束在物理上是不可实现的。贝塞尔高斯光束作为近似无衍射光束,是亥姆霍兹方程在傍轴条件下的解,并且可以用激光振荡器直接产生,但其光束宽是有限的。应用傅里叶变换,平面波谱展开和球面矢量波函数展开法,推导了非偏振贝塞尔高斯光束的球散射远场的无量纲散射函数。通过数值模拟,对非偏振的贝塞尔高斯光束与贝塞尔光束,高斯光束的球散射远场进行了比较,比较发现:当球散射体偏离光轴时,非偏振贝塞尔高斯光束跟贝塞尔光束散射远场的差异主要是散射强度的差异,但是散射极点所在的方向基本保持不变;贝塞尔高斯光束和贝塞尔光束的散射在光束圆锥角方向上比较显著,但高斯光束的前向散射比较显著。     

加速四艾里光束的坡印亭矢量及角动量研究

利用光束标量衍射理论,数值模拟了加速四艾里光束(AQABs)在不同位置的光强分布。在此基础上,通过研究AQABs的坡印亭矢量,给出了AQABs的电磁场能量流动方式,从而理论上给出了AQABs的产生过程并揭示了这种光束的加速机理。进一步研究了AQABs的角动量特性,结果表明,这种光束并非是4个经典艾里光束的简单叠加,它是4个有着完全不一样光学特性的艾里光束部分的有机整体。研究结果揭示了这种奇异AQABs的内在光学本性和传输特性。     

基于轴棱锥产生近似无衍射Mathieu光束的新方法

提出了一种基于轴棱锥产生零阶近似无衍射Mathieu光束的新方法,利用轴棱锥聚焦具有椭圆高斯振幅调制的平面波,得到近似零阶无衍射Mathieu光束.根据椭圆高斯平面波经轴棱锥衍射的衍射积分公式,对光强分布进行了数值模拟,依据几何光学模型计算了近似无衍射Mathieu光束的最大无衍射距离,并设计了实验对理论模拟的结果进行了验证.实验采用柱透镜和准直扩束系统变换圆高斯光束产生具有椭圆高斯振幅调制的平面波,用轴棱锥聚焦该平面波后得到近似无衍射Mathieu光束,实验结果与理论模拟和计算相符.     

Airy光纤:基于阵列波导耦合的光场调控方法

近年来,Airy光束作为一种无衍射光束,其特性引起了研究者的广泛关注,人们对它的理论研究、实验验证、实际应用多个方面都取得了长足的进步.而Airy光纤作为一种可生成Airy光束的波导器件,结合其光纤自身优点可适用于多种应用领域,因此开展新型Airy光纤的探索研究、拓展Airy光束的应用范围具有重要现实意义.本文从Airy光束的原理、光纤结构设计、光纤内部光束生成机理、生成光束波长响应特性以及Airy光纤研究现状和应用五个方面展开了较系统的讨论.     

高斯光束在受限空间滤波器中传输的快速分析

根据矩形函数可以展开为有限复高斯函数之和的方法,从柯林斯衍射积分公式出发,对高斯光束通过受限空间滤波器传输特性进行分析,得到了相应的近似解析式。利用Matlab对其解析式进行数值模拟,得出一组直观的结果。通过比较这些结果可以看出透镜的有限尺寸对光束传输的影响,以及空间滤波器的滤波小孔在传输中具有明显作用。     

Observation of accelerating Airy beams

We report the first observation of Airy optical beams. This intriguing class of wave packets, initially predicted by Berry and Balazs in 1979, has been realized in both one- and two-dimensional configurations. As demonstrated in our experiments, these Airy beams can exhibit unusual features such as the ability to remain diffraction-free over long distances while they tend to freely accelerate during propagation.     

Bohmian trajectories of Airy packets

The discovery of Berry and Balazs in 1979 that the free-particle Schrödinger equation allows a non-dispersive and accelerating Airy-packet solution has taken the folklore of quantum mechanics by surprise. Over the years, this intriguing class of wave packets has sparked enormous theoretical and experimental activities in related areas of optics and atom physics. Within the Bohmian mechanics framework, we present new features of Airy wave packet solutions to Schrödinger equation with time-dependent quadratic potentials. In particular, we provide some insights to the problem by calculating the corresponding Bohmian trajectories. It is shown that by using general space–time transformations, these trajectories can display a unique variety of cases depending upon the initial position of the individual particle in the Airy wave packet. Further, we report here a myriad of nontrivial Bohmian trajectories associated to the Airy wave packet. These new features are worth introducing to the subject’s theoretical folklore in light of the fact that the evolution of a quantum mechanical Airy wave packet governed by the Schrödinger equation is analogous to the propagation of a finite energy Airy beam satisfying the paraxial equation. Numerous experimental configurations of optics and atom physics have shown that the dynamics of Airy beams depends significantly on initial parameters and configurations of the experimental set-up.     

CONSERVATION LAWS FOR THE SCHRÖDINGER–NEWTON EQUATIONS

In this Letter a first-order Lagrangian for the Schrödinger–Newton equations is derived by modifying a second-order Lagrangian proposed by Christian [Exactly soluble sector of quantum gravity, Phys. Rev. D 56(8) (1997) 4844–4877]. Then Noether's theorem is applied to the Lie point symmetries determined by Robertshaw and Tod [Lie point symmetries and an approximate solution for the Schrödinger–Newton equations, Nonlinearity 19(7) (2006) 1507–1514] in order to find conservation laws of the Schrödinger–Newton equations.     

A generalized Schrödinger equation via a complex Lagrangian of electrodynamics

In this paper we give a generalized form of the Schrödinger equation in the relativistic case, which contains a generalization of the Klein-Gordon equation. By complex Legendre transformation, the complex Lagrangian of electrodynamics produces a complex relativistic Hamiltonian H of electrodynamics, on the holomorphic cotangent bundle T′* M. By a special quantization process, a relativistic time dependent Schrödinger equation, in the adapted frames of (T′* M, H) is obtained. This generalized Schrödinger equation can be expressed with respect to the Laplace operator of the complex Hamilton space (T′*M, H). Finally, under some additional conditions on the proper time s of the complex space-time M and the time parameter t along the quantum state, by the method of separation of variables, we obtain two classes of solutions for the Schrödinger equation, one for the weakly gravitational complex curved space M, and the second in the complex space-time with Schwarzschild metric.     

A Symmetry Connection Between Hyperbolic and Parabolic Equations

Abstract

We give ansatzes obtained from Lie symmetries of some hyperbolic equations which reduce these equations to the heat or Schrödinger equations. This enables us to construct new solutions of the hyperbolic equations using the Lie and conditional symmetries of the parabolic equations. Moreover, we note that any equation related to such a hyperbolic equation (for example the Dirac equation) also has solutions constructed from the heat and Schrödinger equations.     

Reflectionless Analytic Difference Operators II. Relations to Soliton Systems

Abstract

This is the second part of a series of papers dealing with an extensive class of analytic difference operators admitting reflectionless eigenfunctions. In the first part, the pertinent difference operators and their reflectionless eigenfunctions are constructed from given “spectral data”, in analogy with the IST for reflectionless Schrödinger and Jacobi operators. In the present paper, we introduce a suitable time dependence in the data, arriving at explicit solutions to a nonlocal evolution equation of Toda type, which may be viewed as an analog of the KdV and Toda lattice equations for the latter operators. As a corollary, we reobtain various known results concerning reflectionless Schrödinger and Jacobi operators. Exploiting a reparametrization in terms of relativistic Calogero–Moser systems, we also present a detailed study of N-soliton solutions to our nonlocal evolution equation.     

Economical phase-covariant cloning with multiclones

This paper presents a very simple method to derive the explicit transformations of the optimal economical 1 to M phase-covariant cloning. The fidelity of clones reaches the theoretic bound [D'Ariano G M and Macchiavello C 2003 Phys. Rev. A 67 042306]. The derived transformations cover the previous contributions [Delgado Y, Lamata L et al, 2007 Phys. Rev. Lett.98 150502] in which M must be odd.     

Representation of Canonical Commutation Relations in a Gauge Theory,the Aharonov-Bohm Effect,and the Dirac-Weyl Operator

Abstract

We consider a representation of canonical commutation relations (CCR) appearing in a (non-Abelian) gauge theory on a non-simply connected region in the two-dimensional Euclidean space. A necessary and sufficient condition for the representation to be equivalent to the Schrödinger representation of CCR is given in terms of Wilson loops. A representation inequivalent to the Schrödinger representation gives a mathematical expression for the (non-Abelian) Aharonov-Bohm effect. Some aspects of the Dirac-Weyl operator associated with the representation of CCR are discussed.     

Comments on “Barut-Girardello Coherent States for the Parabolic Cylinder Functions”

In Chenaghlou and Faizy (Int. J. Theor. Phys. 2008), the authors claim that they have constructed the Barut-Girardello coherent states for the parabolic cylinder functions. However, we point out here that by introducing these coherent states, Schrödinger was able to put forth the idea of “coherent states of the quantum harmonic oscillator” over eighty years ago. These coherent states are derived not only from the Barut-Girardello eigenvalue equation, but also from the Schrödinger and the Klauder-Perelomov approaches. Thus, contrary to their claim, the authors have not introduced new coherent states. In particular, a wide range of the parabolic cylinder functions do not form an orthonormal basis.     

On the AC Spectrum of One-dimensional Random Schrödinger Operators with Matrix-valued Potentials

We consider discrete one-dimensional random Schrödinger operators with decaying matrix-valued, independent potentials. We show that if the ?²-norm of this potential has finite expectation value with respect to the product measure then almost surely the Schrödinger operator has an interval of purely absolutely continuous (ac) spectrum. We apply this result to Schrödinger operators on a strip. This work provides a new proof and generalizes a result obtained by Delyon et al. (Ann. Inst. H. Poincaré Phys. Théor. 42(3):283–309, 1985).