Localized Waves: A not-so-short review

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 "localized 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.