Localized Waves: A not-so-short Review

TL;DR

This comprehensive review explores the theory, properties, and experimental realizations of localized waves, including superluminal, luminal, and subluminal solutions, highlighting their applications across various physical domains and introducing novel wavefield shaping techniques.

Contribution

The paper introduces a generalized bidirectional decomposition method to derive new localized wave solutions, including frozen waves with customizable intensity patterns, advancing the understanding and control of wave propagation.

Findings

Derived several luminal and superluminal non-diffracting solutions.

Developed a method to construct frozen waves with arbitrary longitudinal profiles.

Reviewed experimental evidence of localized wave phenomena across disciplines.

Abstract

In the FIRST PART we present simple introductions to gaussian and Bessel waves, and to the Localized Waves (LW), pulses or beams, showing the important properties of the latter, and their applications whenever a role is played by a wave-equation (electromagnetism, optics, acoustics, seismology, geophysics, gravitation, elementary particle physics,...). The First Part ends with a historical APPENDIX, recalling how the geometrical methods of Special Relativity (SR) had predicted the most interesting LWs, i.e., the X-shaped pulses; and presenting a bird's-eye view of the experiments performed with evanescent waves (and/or tunnelling photons), and with the "localized Superluminal solutions". In the SECOND PART, after some more theoretical introduction, we develop a Generalized "Bidirectional Decomposition", and obtain several luminal and Superluminal non-diffracting solutions; we get a space-time focusing of X-Shaped pulses; and deal with chirped optical X-shaped pulses in material media. Finally, in the THIRD PART we investigate also the subluminal LWs, which, among the others, allow to emphasize the role of SR, in its extended, or rather non-restricted, formulation. We study in particular the topic of zero-speed waves, endowed with a static envelope: Namely, we show how localized wavefields can be constructed with high transverse localization, and with a longitudinal intensity pattern that assumes any desired shape within a chosen interval of the propagation axis. Such "Frozen Waves" promise to have even more applications. In between, we do not forget to briefly treat the case of not axially-symmetric solutions, in terms of higher order Bessel beams.