Since the early works[1-4] on the so-called nondiffracting waves (called alsoLocalized Waves), a great deal of results has been published on this importantsubject, 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 mediaexhibiting effects such as dispersion[5-7], nonlinearity[8], anisotropy[9] andlosses[10]. Such extensions have been carried out along with the development ofefficient methods for obtaining nondiffracting beams and pulses in thesubluminal, luminal and superluminal regimes[11-18]. This paper (partly areview) addresses some theoretical methods related to nondiffracting solutionsof the linear wave equation in unbounded homogeneous media, as well as to someinteresting applications of such waves. In section II we analyze the generalstructure of the Localized Waves, develop the so called GeneralizedBidirectional Decomposition, and use it to obtain several luminal andsuperluminal (especially X-shaped) nondiffracting solutions of the waveequation. In section III we develop a space-time focusing method by acontinuous superposition of X-Shaped pulses of different velocities. Section IVaddresses the properties of chirped optical X-Shaped pulses propagating inmaterial media without boundaries. Finally, in Section V, we show how asuitable superposition of Bessel beams can be used to obtain stationarylocalized wave fields, with a static envelope and a high transverselocalization, and whose longitudinal intensity pattern can assume any desiredshape within a chosen interval of the propagation axis.
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