*Method and apparatus for producing stationary intense wave fields of arbitrary shape*by

**Erasmo Recami**,

**Michel Rached Zamboni**,

**Hugo Enrique Ernandez Figueros**,

**Valerio Abate**,

**Cesar Augusto Dartora**,

**Kleber Suza Nobrega**,

**Marco Mattiuzzi**:

Method for producing a stationary wave field of arbitrary shape comprising the steps of defining at least one volume being limited in the direction of the axis of propagation of a beam, of the type $0 \leq z \leq L$; defining an intensity pattern within the said region $0 \leq z \leq L$ by a function $F(z)$, describing the said localized and stationary intensity pattern, which is approximated by means of a Fourier expansion or by a similar expansion in terms of (trigonometric) orthogonal functions; providing a generic superposition of Bessel or other beams highly transversally confined; calculating the maximum number of superimposed Bessel beams the amplitudes, the phase velocities and the relative phases of each Bessel beam of the superposition, and the transverse and longitudinal wavenumbers of each Bessel beam of the superposition.The invention is based on the following theoretical papers:

Lu, J., & Greenleaf, J. (1992). Experimental verification of nondiffracting X waves IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 39 (3), 441-446 DOI: 10.1109/58.143178 (pdf)

The propagation of acoustic waves in isotropic/homogeneous media and electromagnetic waves in free space is governed by the isotropic/homogeneous (or free space) scalar wave equation. A zeroth-order acoustic X wave (axially symmetric) was experimentally produced with an acoustic annular array transducer. The generalized expression includes a term for the frequency response of the system and parameters for varying depth of field versus beam width of the resulting family of beams. Excellent agreement between theoretical predictions and experiment was obtained. An X wave of finite aperture driven with realizable (causal, finite energy) pulses is found to travel with a large depth of field (nondiffracting length).

Recami, E., Zamboni-Rached, M., Nobrega, K., Dartora, C., & Hernandez F., H. (2003). On the localized superluminal solutions to the maxwell equations IEEE Journal of Selected Topics in Quantum Electronics, 9 (1), 59-73 DOI: 10.1109/JSTQE.2002.808194 (arXiv)

The various experimental sectors of physics in which superluminal motions seem to appear are briefly mentioned. In particular, a bird's-eye view is presented of the experiments with evanescent waves (and/or tunneling photons) and with the "localized superluminal solutions" (SLS) to the wave equation, like the so-called X-shaped beams. The authors also present a series of new SLSs to the Maxwell equations, suitable for arbitrary frequencies and arbitrary bandwidths, some of them endowed with finite total energy. Among the others, the authors set forth an infinite family of generalizations of the classical X-shaped wave and show how to deal with the case of a dispersive medium. Results of this kind may find application in other fields in which an essential role is played by a wave equation (like acoustics, seismology, geophysics, gravitation, elementary particle physics, etc.).

Recami, E. (1998). On localized “X-shaped” Superluminal solutions to Maxwell equations Physica A: Statistical Mechanics and its Applications, 252 (3-4), 586-610 DOI: 10.1016/S0378-4371(97)00686-9 (arXiv)

In this paper we extend for the case of Maxwell equations the "X-shaped" solutions previously found in the case of scalar (e.g., acoustic) wave equations. Such solutions are localized in theory, i.e., diffraction-free and particle-like (wavelets), in that they maintain their shape as they propagate. In the electromagnetic case they are particularly interesting, since they are expected to be Superluminal. We address also the problem of their practical, approximate production by finite (dynamic) radiators. Finally, we discuss the appearance of the X-shaped solutions from the purely geometric point of view of the Special Relativity theory.

Zamboni-Rached, M., Recami, E., & Hernández-Figueroa, H. (2002). New localized Superluminal solutions to the wave equations with finite total energies and arbitrary frequencies The European Physical Journal D, 21 (2), 217-228 DOI: 10.1140/epjd/e2002-00198-7 (arXiv)

By a generalized bidirectional decomposition method, we obtain new Superluminal localized solutions to the wave equation (for the electromagnetic case, in particular) which are suitable for arbitrary frequency bands; several of them being endowed with finite total energy. We construct, among the others, an infinite family of generalizations of the so-called “X-shaped" waves. Results of this kind may find application in the other fields in which an essential role is played by a wave-equation (like acoustics, seismology, geophysics, gravitation, elementary particle physics, etc.).

Zamboni-Rached, M., Nóbrega, K., Hernández-Figueroa, H., & Recami, E. (2003). Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth Optics Communications, 226 (1-6), 15-23 DOI: 10.1016/j.optcom.2003.08.022 (arXiv)

In this paper we set forth new exact analytical Superluminal localized solutions to the wave equation for arbitrary frequencies and adjustable bandwidth. The formulation presented here is rather simple, and its results can be expressed in terms of the ordinary, so-called "X-shaped waves". Moeover, by the present formalism we obtain the first analytical localized Superluminal approximate solutions which represent beams propagating in dispersive media. Our solutions may find application in different fields, like optics, microwaves, radio waves, and so on.

Zamboni-Rached, M., Shaarawi, A., & Recami, E. (2004). Focused X-shaped pulses Journal of the Optical Society of America A, 21 (8) DOI: 10.1364/JOSAA.21.001564 (pdf)

The space–time focusing of a (continuous) succession of localized X-shaped pulses is obtained by suitably integrating over their speed, i.e., over their axicon angle, thus generalizing a previous (discrete) approach. New superluminal wave pulses are first constructed and then tailored so that they become temporally focused at a chosen spatial point, where the wave field can reach very high intensities for a short time. Results of this kind may find applications in many fields, besides electromagnetism and optics, including acoustics, gravitation, and elementary particle physics.

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