[论文解读] Localized Waves: A not-so-short review

Michel Zamboni Rached, Erasmo Recami|arXiv (Cornell University)|Feb 16, 2009

Orbital Angular Momentum in Optics参考文献 180被引用 28

一句话总结

本文系统综述了局域波（LWs），包括X型脉冲和冻结波，通过分析其在无限均匀介质中的数学结构与物理特性。研究表明，贝塞尔波束的叠加可产生满足波动方程的非衍射、超光速及亚光速解，其应用涵盖电磁学、光学、声学及量子场论。

ABSTRACT

In the First Part of this paper (which is mainly a review) we present simple, general and formal, introductions to the ordinary gaussian waves and to the Bessel waves, by explicitly separating the case of beams from the case of pulses; and, afterwards, an analogous introduction is presented for the Localized Waves (LW), pulses or beams. Always we stress the very different characteristics of the gaussian with respect to the Bessel waves and to the LWs, showing the numerous important properties of the latter: Properties that may find application in all fields in which an essential role is played by a wave-equation (like electromagnetism, optics, acoustics, seismology, geophysics, gravitation, elementary particle physics, etc.). The First Part of this review ends with an Appendix, wherein: (i) we recall how, in the seventies and eighties, the geometrical methods of Special Relativity (SR) predicted --in the sense below specified-- the existence of the most interesting LWs, i.e., of the X-shaped pulses; and (ii) in connection with the circumstance that the X-shaped waves are endowed with Superluminal group-velocities (as discussed in the first part of this paper), we briefly mention the various experimental sectors of physics in which Superluminal motions seem to appear; in particular, a bird's-eye view is presented of the experiments till now performed with evanescent waves (and/or tunnelling photons), and with the Superluminal solutions to the wave equations. In the Second Part of this work, we address in more detail various theoretical approaches leading to nondiffracting solutions of the linear wave equation in unbounded homogeneous media, as well as some interesting applications of these waves. After some more introductory remarks (Sec.VI), we analyse in Section VII the general structure of the Localized Waves, develop the so called Generalized Bidirectional Decomposition, and use it to obtain several luminal and Superluminal nondiffracting solutions of the wave equations. In Section VIII we present a method for getting a space-time focusing by a continuous superposition of X-Shaped pulses of different velocities. Section IX addresses the properties of chirped optical X-Shaped pulses propagating in material media without boundaries. Finally, in the Third Part of this paper we complete our review by investigating also the not less interesting) case of subluminal Localized Solutions to the wave equations, which, among the others, allow us to emphasize the remarkable role of SR, in its extended, or rather non-restricted, formulation. [For instance, the various Superluminal and subluminal LWs are expected to be transformed one into the other by suitable Lorentz transformations]. We start by studying --by means of various approaches-- the very peculiar topic of zero-speed waves: Namely, of the localized fields with a static envelope; consisting, for instance, in light at rest. Actually, in Section X we show how a suitable superposition of Bessel beams can be used to construct stationary localized wave fields with high transverse localization, and with a longitudinal intensity pattern that assumes any desired shape within a chosen interval 0< z<L of the propagation axis. We have called Frozen Waves such solutions: As we shall see, they can have a lot of noticeable applications. In between, we do not forget to briefly treat the case of not axially-symmetric solutions, in terms of higher order Bessel beams. In this review we have fixed our attention especially on electromagnetism and optics: but results of the present kind are valid, let us repeat, whenever an essential role is played by a wave-equation.

研究动机与目标

• 系统回顾局域波（LWs）的理论基础，包括其与高斯波和贝塞尔波的区别。
• 通过相对论几何方法探讨X型脉冲及超光速群速度的产生机制。
• 提出广义双向分解方法，用于构建光速及超光速非衍射解。
• 研究通过连续叠加具有不同速度的X型脉冲实现时空聚焦的可能性。
• 引入并分析‘冻结波’——具有任意纵向强度分布的静态局域场——通过贝塞尔波束叠加实现。

提出的方法

• 利用广义双向分解推导线性波动方程的精确解，其具有局域化和非衍射特性。
• 在广义、无限制的狭义相对论形式下应用洛伦兹变换，关联超光速与亚光速LW解。
• 通过特定振幅与相位分布的贝塞尔波束叠加构造冻结波，实现静态的横向与纵向强度分布。
• 利用具有频率依赖参数的波动方程解，分析材料介质中的啁啾光学X型脉冲。
• 通过不同群速度的X型脉冲的连续叠加，在四维时空实现时空聚焦。
• 通过高阶贝塞尔波束引入非轴对称解，扩展可能的局域波结构类别。

实验结果

研究问题

• RQ1 局域波在传播与衍射特性方面，与高斯波和贝塞尔波的根本区别是什么？
• RQ2 狭义相对论在预测与统一超光速与亚光速局域波解中起什么作用？
• RQ3 能否构造具有任意纵向强度分布的静态非衍射波场（即冻结波）？
• RQ4 如何通过叠加具有不同群速度的X型脉冲实现时空聚焦？
• RQ5 局域波中的超光速群速度对实验物理及材料介质中波传播有何影响？

主要发现

• X型脉冲自然地从相对论几何方法中产生，表现出超光速群速度，与波动方程解一致。
• 冻结波通过贝塞尔波束叠加构造，可在定义区间 0 < z < L 内保持具有任意纵向形状的静态强度包络。
• 广义双向分解可推导出波动方程的光速与超光速非衍射解。
• 通过连续叠加具有不同速度的X型脉冲，实现时空聚焦，从而在时空中精确控制波前。
• 亚光速局域解存在，并可通过洛伦兹变换与超光速解关联，体现狭义相对论的统一作用。
• 材料介质中的啁啾光学X型脉冲保持其非衍射特性，表明其在超快光学与信号传输中具有应用潜力。