Structure of the non-diffracting waves, and some interesting applications

This work is Chapter 2 of the Book. -- Since the early works on the so-called nondiffracting waves (called also Localized Waves), a great deal of results has been published on this important subject, from both the theoretical and the experimental point of view. Initially, the theory was developed taking into account only free space; however, in recent years, it has been extended for more complex media exhibiting effects such as dispersion, nonlinearity, anisotropy and losses. Such extensions have been carried out along with the development of efficient methods for obtaining nondiffracting beams and pulses in the subluminal, luminal and superluminal regimes. This Chapter (only partly a review) addresses some theoretical methods related to nondiffracting solutions of the linear wave equation in unbounded homogeneous media, as well as to some interesting applications of such waves. In section II we analyze the general structure of the Localized Waves, develop the so called Generalized Bidirectional Decomposition, and use it to obtain several luminal and superluminal (especially X-shaped) nondiffracting solutions of the wave equation. In section III we develop a space-time focusing method by a continuous superposition of X-Shaped pulses of different velocities. Section IV addresses the properties of chirped optical X-Shaped pulses propagating in material media without boundaries. Finally, in Section V, we show how a suitable superposition of Bessel beams can be used to obtain stationary localized wave fields, with a static envelope and a high transverse localization, and whose longitudinal intensity pattern can assume any desired shape within a chosen interval of the propagation axis.