[Paper Review] Localized Waves: A not-so-short review

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

Orbital Angular Momentum in Optics180 references28 citations

TL;DR

This paper provides a comprehensive review of localized waves (LWs), including X-shaped pulses and frozen waves, by analyzing their mathematical structure and physical properties in unbounded homogeneous media. It demonstrates how superpositions of Bessel beams yield nondiffracting, superluminal, and subluminal solutions to the wave equation, with applications across electromagnetism, optics, acoustics, and quantum field theory.

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.

Motivation & Objective

• To systematically review the theoretical foundations of localized waves (LWs), including their distinction from Gaussian and Bessel waves.
• To explore the emergence of X-shaped pulses and superluminal group velocities through relativistic geometrical methods.
• To develop a generalized bidirectional decomposition for constructing luminal and superluminal nondiffracting solutions.
• To investigate the possibility of space-time focusing via continuous superposition of X-shaped pulses with varying velocities.
• To introduce and analyze 'frozen waves'—stationary localized fields with arbitrary longitudinal intensity profiles—using Bessel beam superpositions.

Proposed method

• Utilizes the generalized bidirectional decomposition to derive exact solutions of the linear wave equation with localized, nondiffracting behavior.
• Applies Lorentz transformations in an extended, non-restricted formulation of special relativity to relate superluminal and subluminal LWs.
• Constructs frozen waves by superposing Bessel beams with specific amplitude and phase distributions to achieve static transverse and longitudinal intensity patterns.
• Analyzes chirped optical X-shaped pulses in material media using solutions of the wave equation with frequency-dependent parameters.
• Employs continuous superposition of X-shaped pulses with different group velocities to achieve space-time focusing in four-dimensional spacetime.
• Considers non-axially symmetric solutions through higher-order Bessel beams to extend the class of possible localized wave structures.

Experimental results

Research questions

• RQ1 How do localized waves differ fundamentally from Gaussian and Bessel waves in terms of propagation and diffraction properties?
• RQ2 What is the role of special relativity in predicting and unifying superluminal and subluminal localized wave solutions?
• RQ3 Can a stationary, nondiffracting wave field (frozen wave) be constructed with an arbitrary longitudinal intensity profile?
• RQ4 How can space-time focusing be achieved through the superposition of X-shaped pulses with varying group velocities?
• RQ5 What are the implications of superluminal group velocities in localized waves for experimental physics and wave propagation in material media?

Key findings

• X-shaped pulses emerge naturally from relativistic geometrical methods and exhibit superluminal group velocities, consistent with wave equation solutions.
• Frozen waves are constructed via superposition of Bessel beams and can maintain a static intensity envelope with arbitrary longitudinal shape within a defined interval 0 < z < L.
• The generalized bidirectional decomposition enables the derivation of both luminal and superluminal nondiffracting solutions to the wave equation.
• Space-time focusing is achieved by continuously superposing X-shaped pulses with different velocities, enabling precise control over wavefronts in spacetime.
• Subluminal localized solutions exist and are related to superluminal ones via Lorentz transformations, demonstrating the unifying role of special relativity.
• Chirped optical X-shaped pulses in material media retain their nondiffracting character, indicating potential for applications in ultrafast optics and signal transmission.