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Recognized for their non-diffracting properties, Bessel beams can be conveniently combined to generate the so-called Frozen Waves, which are monochromatic beams endowed with topological charge and whose longitudinal intensity pattern can be shaped according to a previously chosen function. Continuous superposition of Bessel beams is specially suitable for micrometer-scale domains, being highly relevant for applications in optical tweezers, particle trapping and atom guidance. Previous studies have successfully constructed micrometer Frozen Wave solutions with null topological charge; nevertheless, some challenges persist in obtaining exact solutions when dealing with higher topological charge values (i.e., higher-order micrometer Frozen Waves). Typically, a topological charge raising operator is used to elevate the order; however, such solutions face significant issues when the orders become excessively high. In this paper, based in continuous superposition of higher order Bessel beams, we develop a novel analytic and exact solution for higher-order micrometer Frozen Waves, so obtaining a method for modeling the intensity of beams with arbitrary topological charge within micrometer spatial domains. We also investigate the behavior of the electric field's longitudinal component in such highly non-paraxial regimes through a vectorial treatment., Comment: 11 pages, 3 figures

While in linear optics the subject of structured light has been a fruitful field of both theoretical and applied research, its development in the arena of nonlinear optics has been underexplored. In this paper, we construct Frozen-Wave-type structured optical beams in Kerr nonlinear media, emphasizing the self-defocusing case, and use them to guide and control Gaussian optical beams. The results presented in this study support the expectation that structured light in nonlinear media can open new venues of theoretical research and applications, particularly in the realms of light controlling light and for all-optical photonics., Comment: 15 pages, 3 figures

We present an analytical method to achieve highly non-paraxial, azimuthally polarized structured beams that, when propagating through an absorbing stratified media, can assume in the last semi-infinite layer approximately any desired longitudinal intensity pattern within spatial regions few times larger than the wavelength. The possibility of managing the properties of a highly non-paraxial beam under adverse conditions, such as multiple reflections in stratified structures and energy loss to the material media, may be of great importance in many different optical applications, like trapping and micro-manipulation, remote sensing, thin films, medical devices, medical therapies and so on., Comment: 13 pages and 10 figures

Some years after the appearance of the so-called non-diffracting beams, there was the development of methods capable of structuring them spatially, being the so called Frozen Waves method the first and, perhaps, the most efficient one. That method allowed modelling the longitudinal intensity pattern of non-diffracting beams, being, however, little efficient in controlling their transverse spatial pattern, granting only the possibility of choosing their transverse dimensions, which remain invariant throughout the propagation. In this work, we have extended the Frozen Wave method in such a way to control, in addition to the longitudinal pattern, the transverse beam structure along the propagation. The new transversally and longitudinally structured beams can have potential applications in areas such as photonics, optical manipulation, optical atom guidance, lithography, etc.., Comment: 13 pages, 2 figures

This work presents the first optical trapping experimental demonstration of micro-particles with Frozen Waves. Frozen Waves are an efficient method to model longitudinally the intensity of non-diffracting beams obtained by superposing co-propagating Bessel beams with the same frequency and order. The experimental setup of a holographic optical tweezers using spatial light modulators has been assembled and optimized. We investigate the optical force distribution acting on micro-particles of two types of Frozen Waves.The results show that it is possible to obtain greater stability for optical trapping using Frozen Waves. The significant enhancement in trapping geometry from this approach shows promising applications for optical tweezers, micro-manipulations over a broad range., Comment: 16 pages, 9 figures

In this paper, we shall provide an analytical solution describing a Frozen Wave beam after passing through a pair of convex lenses with different focal distances., Comment: 5 pages and 1 figure; Replaced with minor improvements about the authors'names and Institutions

In this paper we perform a fusion between two important theoretical methodologies, one related to the Frozen Wave beams, which are non-diffracting beams whose longitudinal intensity pattern can be chosen a priori in an medium (absorbing or not), and the other related to the Airy-Type pulses, which are pulses resistant to dispersion effects in dispersive materials. As a result, a new method emerges, capable of providing vortex pulses resistant to three concomitant effects, i.e.: diffraction, dispersion and attenuation; while concurrently the spatial variation of the wave intensity along its axis of propagation can be engineered at will. The new approach can be seen as a generalization of the Localized Waves theory in the paraxial regime and the new pulses can have potential applications in different fields such as optics communications, nonlinear optics, micromanipulation, and so on., Comment: 14 pages, 6 figures

In this paper it is developed a simple, analytical and very efficient method capable to provide control of optical beam's intensity over an arbitrary curvilinear (planar) trajectory. The same method also provides the possibility of managing multifurcations of the optical beam. The results presented here can have valuable applications in fields like optical tweezers, optical lithography, atom optical guiding, structured light, etc.., Comment: 17 pages, 9 figures

In this paper, we study the propagation of the Frozen Wave type beams through non-absorbing stratified media and develop a theoretical method capable to provide the desired spatially shaped diffraction resistant beam in the last material medium. In this context, we also develop a matrix method to deal with stratified media with large number of layers. Additionally, we undertake some discussion about minimizing reflection of the incident FW beam on the first material interface by using thin films. Our results show that it is indeed possible to obtain the control, on demand, of the longitudinal intensity pattern of a diffraction resistant beam even after it undergoes multiple reflections and transmissions at the layer interfaces. Remote sensing, medical and military applications, noninvasive optical measurements, etc., are some fields that can be benefited by the method here proposed., Comment: 17 pages, 13 figures

In this paper we present a theoretical method, together with its experimental confirmation, to obtain structures of light by connecting diffraction-resistant cylindrical beams of finite lengths and different radii. The resulting "Lego-beams" can assume, on demand, various unprecedented spatial configurations. We also experimentally generate some of them on using a computational holographic technique and a spatial light modulator. Our new, interesting method of linking together various "pieces of light" can find applications in all fields where structured light beams are needed: in particular, such as optical tweezers, e.g. for biological manipulations, optical guiding of atoms, light orbital angular momentum control, holography, lithography, non-linear-optics, interaction of electromagnetic radiation with Bose-Einstein condensates, and so on, besides the field in general of Localized Waves (non-diffracting beams and pulses)., Comment: 20 pages with 6 Figures. Paper submitted for pub. In this Second Version more attention is paid to the applications, and more references are added. Suitable, related modifications are inserted (even in the Abstract)

We report on the theory and experimental generation of a class of diffraction-attenuation-resistant beams with state of polarization (SoP) and intensity that can be controlled on demand along the propagation direction. This is achieved by a suitable superposition of Bessel beams, whose parameters are systematically chosen based on closed-form analytic expressions provided by the Frozen Waves (FWs) method. Using an amplitude-only spatial light modulator, we experimentally demonstrate three scenarios. In the first, the SoP of a horizontally polarized beam evolves to radial polarization and is then changed to vertical polarization, with the beam intensity held constant. In the second, we simultaneously control the SoP and the longitudinal intensity profile, which was chosen such that the beam's central ring can be switched-off over predefined space regions, thus generating multiple foci with different SoP and at different intensity levels along the propagation. Finally, the ability to control the SoP while overcoming attenuation inside lossy fluids is shown experimentally for the first time in the literature (to the best of our knowledge). Therefore, we envision our proposed method to be of great interest for many applications, such as optical tweezers, atom guiding, material processing, microscopy, and optical communications., Comment: 11 pages, 8 figures

Several applications, such as optical tweezers and atom guiding, benefit from techniques that allow the engineering of optical fields' spatial profiles, in particular their longitudinal intensity patterns. In cylindrical coordinates, methods such as Frozen Waves allow an advanced control of beams' characteristics, but in Cartesian coordinates there is no analogous technique. Since Cartesian beams may also be useful for applications, we develop here a method to modulate on-demand the longitudinal intensity pattern of any (initially) unidimensional Cartesian beam with concentrated wavevector spectrum, thus encompassing all paraxial unidimensional beams. To this end, we write the total beam as a product of two unidimensional beams and explore the degree of freedom provided by the additional Cartesian coordinate. While in the plane where this coordinate is zero the chosen unidimensional beam keeps its structure with the additional desired intensity modulation, a sinusoidal-like oscillation appears in the direction of this variable and creates a spot whose size is tunable. Examples with Gaussian and Airy beams are presented and their corresponding experimental demonstrations are performed to show the validity of the method., Comment: 24 pages, 10 figures; added analysis in lossy media and additional comments in the introduction

In this paper, we develop a theoretical analysis to efficiently handle superpositions of waves with concentrated wavevector and frequency spectra, allowing an easy analytical description of fields with interesting transverse profiles. First, we analyze an extension of the paraxial formalism that is more suitable for superposing these types of waves, as it does not rely on the use of coordinate rotations combined with paraxial assumptions. Second, and most importantly, we leverage the obtained results to describe azimuthally symmetric waves composed of superpositions of zero-order Bessel beams with close cone angles that can be as large as desired, unlike in the paraxial formalism. Throughout the paper, examples are presented, such as Airy beams with enhanced curvatures, nonparaxial Bessel-Gauss beams and Circular Parabolic-Gaussian beams (which are based on the Cartesian Parabolic-Gaussian beams), and experimental data illustrates interesting transverse patterns achieved by superpositions of beams propagating in different directions., Comment: 21 pages, 10 figures

Light beams with azimuthal phase dependence [$exp(i \ell\phi)$] carry orbital angular momentum (OAM) which differs fundamentally from spin angular momentum (SAM) associated with polarization. Striking difference between the two momenta is manifested in the allowable values: where SAM is limited to $\hbar k_0$ per photon, the OAM has unbounded value of $\ell\hbar$ per photon ($\ell$ is integer), thus dramatically exceeding the value of SAM \cite{Ref1,Ref2, Ref3}. OAM has thus been utilized in optical trapping \cite{Ref4}, imaging\cite{Ref2}, and material processing \cite{Ref5}. Furthermore, the unbounded degrees-of-freedom in OAM states have been deployed in data communications \cite{Ref6}. Here, we report an \textit{exceptional} behavior for a class of light beams---known as Frozen Waves (FWs)---whose intensity and azimuthal phase profiles can be controlled along the propagation direction, at will. Accordingly, we generate rotating light patterns that can change their sense of rotation and order of phase twist with propagation. Manipulating OAM along the beam axis can open new directions in optical science and its applications., Comment: 16 Pages, 5 Figures, Journal paper

This work presents, for the first time the optical generation of non-diffracting beams via photorefractive holography. Optical generation of non-diffracting beams using conventional optics components is difficult and, in some instances, unfeasible, as it is wave fields given by superposition of non-diffracting beams. It is known that computer generated holograms and spatial light modulators (SLMs) successfully generate such beams. With photorefractive holography technique, the hologram of a non-diffracting beam is constructed (recorded) and reconstructed (reading) optically in a nonlinear photorefractive medium. The experimental realization of a non-diffracting beam was made in a photorefractive holography setup using a photorefractive Bi12SiO20 (BSO) crystal as the holographic recording medium, where the non-diffracting beams, the Bessel beam arrays and superposition of co-propagating Bessel beams (Frozen Waves) were obtained experimentally. The experimental results are in agreement with the theoretically predicted results, presenting excellent prospects for implementation of this technique for dynamical systems at applications in optics and photonics., Comment: 11 pages, 9 figures

In this paper we develop a method capable of modeling the space-time focusing of nondiffracting pulses. The new pulses can possess arbitrary peak velocities and, in addition to being resistant to diffraction, can have their peak intensities and focusing positions chosen a priori. More specifically, we can choose multiple locations (spatial ranges) of space/time focalization; also, the pulse intensities can be chosen in advance. The pulsed wave solutions presented here can have very interesting applications in many different fields, such as free-space optical communications, remote sensing, medical apparatus, etc., Comment: 15 pages, 7 figures

In recent times, we experimentally realized a quite efficient modeling of the shape of diffraction-resistant optical beams; thus generating for the first time the so-called Frozen Waves (FW), whose longitudinal intensity pattern can be arbitrarily chosen, within a prefixed space interval of the propagation axis. Such waves possess a host of potential applications: in medicine, biomedical optics, optical tweezers, atom guiding, remote sensing, tractor beams, optical communications or metrology, and other topics in photonic areas. In this work, we extend our theory of FWs -- which led to beams endowed with a static envelope -- through a dynamic modeling of the FWs, whose shape is now allowed to evolve in time in a predetermined way. And we experimentally create such dynamic FWs in Optics, via a computational holographic technique and a spatial light modulator. Experimental results are here presented for two cases of dynamic FWs, one of the zeroth and the other of higher order, the last one being the most interesting, consisting in a cylindrical surface of light whose geometry changes in space and time., Comment: 8 pages, 4 figures and 2 multimedia files

In this work, we propose the generation of diffraction resistant beams by using a parabolic reflector and a source of spherical waves positioned at a point slightly displaced from its focus (away from the reflector). In our analysis, considering the reflector dimensions much greater than the wavelength, we describe the main characteristics of the resulting beams, showing their properties of resistance to the diffraction effects. Due to its simplicity, this method may be an interesting alternative for the generation of long range diffraction resistant waves., Comment: 22 pages, 9 figures, Applied Optics, 2015

In this paper, we describe analytically the propagation of Airy-type pulses truncated by a finite-time aperture when second and third order dispersion effects are considered. The mathematical method presented here, based on the superposition of exponentially truncated Airy pulses, is very effective, allowing us to avoid the use of time-consuming numerical simulations. We analyze the behavior of the time truncated Ideal-Airy pulse and also the interesting case of a time truncated Airy pulse with a "defect" in its initial profile, which reveals the self-healing property of this kind of pulse solution., Comment: 9 pages. 5 figures

This work deals with exact solutions to the wave equations. We start by introducing the Non-Diffracting Waves (NDW), and by a definition of NDWs. Afterwards we recall -besides ordinary waves (gaussian beams, gaussian pulses)- the simplest non diffracting waves (Bessel beams, X-shaped pulses,...). In Sec.2 we show how to eliminate any backward-traveling components, first in the case of ideal NDW pulses, and then, in Sec.3, for realistic finite-energy NDW pulses. In particular, in subsec.3.1 we forward a general functional expression for any totally-forward non-diffracting pulses. Then, in Sec.4 an efficient method is set forth for the analytic description of truncated beams, a byproduct of its being the elimination of any need of lengthy numerical calculations. In Sec.5 we explore the question of the subluminal NDWs, or bullets, in terms of two different methods, the second one allowing the analytic description of non-diffracting waves with a static envelope ("Frozen Waves", FW), in terms of continuous Bessel beam superpositions. The production of such Frozen Waves (experimentally generated in recent time for Optics) is theoretically developed in Sec.6 also for the case of absorbing media. Sec.7 discusses the role of Special Relativity and of Lorentz transformations, relevant for the physical comprehension of the NDW issue. In Sec.8 we present further analytic solutions to the wave equations, with use of higher-order Bessel beams. Next, Sec.9 deals in detail with an application of NDWs to Optical Tweezers. In Sec.10 we show that "soliton-like" solutions can be found also in the different case of the ordinary linear Schroedinger equation within standard Quantum Mechanics. Some complementary issues are just mentioned in the last Section. This work also constitutes a part of a much longer Review in preparation., Comment: review of 74 pages, including 23 Figures. Source file in plain LaTeX. from Chap.1 (introductory chapter, pages 1-67) of the Book "Non-Diffracting Waves" (J.Wiley-VCH; Berlin, 2014), ed. by H.E.Hernandez-Figueroa, E.Recami, and M.Zamboni-Rached [ISBN 978-3-527-41195-5]; with permission

We propose in detail Antennas for generating Non-Diffracting Beams of Microwaves, for instance with frequencies of the order of 10 GHz, obtaining fair results even when having recourse to realistic apertures endowed with reasonable diameters. Our first proposal refers mainly to sets of suitable annular slits, having in mind various possible applications, including remote sensing. Our second proposal --which constitutes one of the main aims of this paper-- refers to the alternative, rather simple, use of a Parabolic Reflector, illuminated by a spherical wave source located on the paraboloid axis but slightly displaced with respect to the Focus of the Paraboloid. Such a parabolic reflector yields "extended focus" (non-diffracting) beams. [OCIS codes: 999.9999; 070.7545; 050.1120; 280.0280; 050.1755; 070.0070; 200.0200. Keywords: Non-Diffracting Waves; Microwaves; Remote sensing; Annular Arrays; Bessel beams; Extended focus; Reflecting paraboloids; Parabolic reflectors; Parabolic antennas]., Comment: 23 pages with 20 Figures (source-file in plain LaTeX)

In this paper we implement experimentally the spatial shape modelling of nondiffracting optical beams via computer generated holograms. The results reported here are the experimental confirmation of the so called Frozen Wave method, developed few years ago. Optical beams of this type can possess potential applications in optical tweezers, medicine, atom guiding, remote sensing, etc.., Comment: 18 pages, 9 figures

In the present paper the analytical aspects related to the maximum invariance depth of propagation finite energy Airy-type beams are investigated, considering not only the usual exponentially dumped Airy beams but also the truncated ones. It is obtained an analytic expression which seems capable of finding accurately the maximum invariance distance of these waves. The results seem to be in line with those obtained from numerical simulations., Comment: 16 pages. 7 figures

Our aim in this paper is to recall some essential points of "Extended special Relativity", now more correctly called "Non-Restricted special Relativity" theory (NRR), and in particular of the extended Maxwell Equations; as well as to set forth some further comments on the basic differences between Cherenkov Radiation and the so-called X-shaped Waves, met within the more recent realm of the Non-diffracting Waves (also known as Localized Waves). The occasion is furnished by some very recent Seshadri's comments[1] on a previous article of ours, titled "Cherenkov radiation versus X-shaped localized waves" (see[2], and arXiv:0807.4301[physics.optics]), and not less on NRR itself. OCIS codes: 320.5550; 350.7420; 070.7345; 350.5500; 070.0070; 100.7410; 050.050; 000.1600; 000.2690; 000.6800; 250.5530; 260.0260. PACS nos.: 41.60.Bq; 03.50.De; 03.30.+p; 41.20;Jb; 04.30.Db; 42.25.-p; 42.25.Fx; 47.35.Rs. Keywords: Non-diffracing Waves; Localized Waves; Cherenkov radiation; X-shaped waves; Wave equations; Bessel beams; Superluminal pulses; Maxwell equations; Special Relativity; Non-restricted Special Relativity; Extended special Relativity; Lorentz transformations; Superluminal point-charges., Comment: Standard LaTeX file, of 19 pages with five figures

In this paper, we have developed an analytic method for describing Airy-Type beams truncated by finite apertures. This new approach is based on suitable superposition of exponentially decaying Airy beams. Regarding both theoretical and numerical aspects, the results here shown are interesting because they have been quickly evaluated through a simple analytic solution, whose characteristics of propagation has agreed with those already published in literature through the use of numerical methods. To demonstrate the method's potentiality, three different truncated beams have been analyzed: ideal Airy, Airy-Gauss and Airy-Exponential., Comment: 11 pages, 6 figures

In this paper we present a simple and effective method, based on appropriate superpositions of Bessel-Gauss beams, which in the Fresnel regime is able to describe in analytic form the 3D evolution of important waves as Bessel beams, plane waves, gaussian beams, Bessel-Gauss beams, when truncated by finite apertures. One of the byproducts of our mathematical method is that one can get in few seconds, or minutes, high-precision results which normally require quite long times of numerical simulation. The method works in Electromagnetism (Optics, Microwaves,...), as well as in Acoustics. OCIS codes: (999.9999) Non-diffracting waves; (260.1960) Diffraction theory; (070.7545) Wave propagation; (070.0070) Fourier optics and signal processing; (200.0200) Optics in computing; (050.1120) Apertures; (070.1060) Acousto-optical signal processing; (280.0280) Remote sensing and sensors; (050.1755) Computational electromagnetic methods., Comment: Paper of 21 pages with 13 Figures. Source file in LaTeX

In recent times it has been paid attention to the fact that (linear) wave equations admit of "soliton-like" solutions, known as Localized Waves or Non-diffracting Waves, which propagate without distortion in one direction. Such Localized Solutions (existing also for K-G and Dirac equations) are a priori suitable, more than Gaussian's, for describing elementary particle motion. In this paper we show that, mutatis mutandis, Localized Solutions exist even for the ordinary Schroedinger equation, within standard Quantum Mechanics; and we obtain both approximate and exact solutions, setting forth particular examples for them. In the ideal case such solutions bear infinite energy, as well as plane or spherical waves: we also demonstrate, therefore, how to obtain finite-energy solutions. At last, we briefly consider solutions for a particle moving in the presence of a potential. Some physical comments are added., Comment: 31 pages, including 9 Figures; with a LaTeX source-file. Replaced by adding (in sect.4.1) several plots, which show new particular "soliton-like" solutions, and some new brief comments

In this paper, we show how one can change the stable equilibrium of a particle trapped into an optical tweezer by varying the intensity of superposed Bessel beams with different orders. The gradient forces acting on particles of different radii are determined, and the theoretical results indicates that it is possible to combine Bessel beams in such a way as to manipulate the particle into or out the centre of the beam by exploiting their ring-shaped intensity patterns, without any mechanical displacement of the lasers., Comment: 4 pages, 4 figures

This paper studies the possibility of using axicons in Free Space Optics (FSO) systems. The behavior of the pseudo- Bessel beams generated by "logarithmic" and "linear" axicons, with or without stops, was analyzed through the Huygens-Fresnel integral of diffraction in cylindrical coordinates. We also show that GRIN (Gradient Index) axicons, when well designed, could be used in order to choose the intensity pattern along the propagation axis, which could be a new technique for the alignment equipment., Comment: 5 pages, 6 figures

Recently, a method for obtaining diffraction-attenuation resistant beams in absorbing media was developed through suitable superposition of ideal zero-order Bessel beams. In this work, we will show that such beams maintain their resistance to diffraction and absorption even when generated by finite apertures. Also, we shall extend the original method to allow a higher control over the transverse intensity profile of the beams. Although the method has been developed for scalar fields, it can be applied to paraxial vector wave fields as well. These new beams can possess potential applications, such as free space optics, medical apparatuses, remote sensing, optical tweezers, etc.., Comment: 24 pages, 6 figures

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., Comment: 121 pages, with 41 Figures (source file in LaTeX). It is a preliminary version of Chap.4 in vol.156 of AIEP (2009)

In this paper we use a unidirectional decomposition capable of furnishing localized wave pulses, with luminal and superluminal peak velocities, in exact form and totally free of backward components, which have been a chronic problem for such wave solutions. This decomposition is powerful enough for yielding not only ideal nondiffracting pulses but also their finite energy versions still in exact analytical closed form. Another advantage of the present approach is that, since the backward spectral components are absent, the frequency spectra of the pulses do not need to possess ultra-widebands, as it is required by the usual localized waves (LWs) solutions obtained by other methods. Finally, the present results bring the LW theory nearer to the real experimental possibilities of usual laboratories., Comment: 28 pages, 6 figures

The Localized Waves are nondiffracting ("soliton-like") and self-reconstructing solutions to the wave equations, and are known to exist with subluminal, luminal and superluminal peak-velocities. The most studied ones are the "X-shaped" superluminal waves; which are associated with a cone, so that some authors [e.g., Walker and Kuperman, PRL 99 (Dec.2007) 244802] have been tempted to link them with Cherenkov radiation. However, the "X-waves" belong to a different realm, and exist even in the vacuum, independently of any media, as verified in a number of papers [listed e.g. in "Localized Waves" (J.Wiley; Jan.2008)]. We want to clarify the whole question on the basis of rigorous formalism and clear physical considerations. In particular, by explicit calculations based on Maxwell equations only, we show that: (i) the X-waves exist also inside both the front and the rear part of their double cone (that has nothing to do with Cherenkov's); (ii) they are to be found not via ad hoc assumptions, but by use of strict mathematical (or experimental) procedures; (iii) the ideal X-waves possess infinite energy, but finite-energy X-waves can be easily constructed (even without space-time truncations): And we do construct exact finite-energy solutions (totally free from backward-traveling waves); (iv) an actual attempt at comparing Cherenkov radiation with X-waves would lead one to consider the very different situation of the (X-shaped, too) field generated by a superluminal point-charge, a question exploited in previous papers [Recami et al., PRE 69 (2004) 027602]: We show explicitly, here, that the point-charge would not lose energy in the vacuum, and its field would not need to be continuously feeded by incoming side-waves (as it is the case, indeed, for an ideal, ordinary X-wave)., Comment: Replaced with the addition of a third Figure (representing a finite-energy, superluminal, X-type Localized Solution, in exact form and totally free from noncausal components). LaTeX source-file: 16 pages, with three Figures

In this work it is shown how to obtain, in a simple way, localized (non- diffractive) subluminal pulses as exact analytic solutions to the wave equations. These new ideal subluminal solutions, which propagate without distortion in any homogeneous linear media, are herein obtained for arbitrarily chosen frequencies and bandwidths, avoiding in particular any recourse to the non-causal components so frequently plaguing the previously known localized waves. The new solutions are suitable superpositions of --zeroth-order, in general-- Bessel beams, which can be performed either by integrating with respect to (w.r.t.) the angular frequency, or by integrating w.r.t. the longitudinal wavenumber: Both methods are expounded in this paper. The first one appears to be powerful enough; we study the second method as well, however, since it allows dealing even with the limiting case of zero-speed solutions (and furnishes a new way, in terms of continuous spectra, for obtaining the so-called "Frozen Waves", so promising also from the point of view of applications). We briefly treat the case, moreover, of non-axially symmetric solutions, in terms of higher order Bessel beams. At last, particular attention is paid to the role of Special Relativity, and to the fact that the localized waves are expected to be transformed one into the other by suitable Lorentz Transformations. The analogous pulses with intrinsic finite energy, or merely truncated, will be constructed in another paper. In this work we fix our attention especially on electromagnetism and optics: but results of the present kind are valid whenever an essential role is played by a wave-equation (like in acoustics, seismology, geophysics, gravitation, elementary particle physics, etc.), Comment: 38 pages with 7 figures [to be processed by PDFlatex]. (Replaced correcting a few misprints)

In the first part of this article the various experimental sectors of physics in which Superluminal motions seem to appear are briefly mentioned, after a sketchy theoretical introduction. In particular, a panoramic 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 waves. In the second part of this paper we present a series of new SLSs to the Maxwell equations, suitable for arbitrary frequencies and arbitrary bandwidths: some of them being endowed with finite total energy. Among the others, we set forth an infinite family of generalizations of the classic 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.). This e-print, in large part a review, was prepared for the special issue on "Nontraditional Forms of Light" of the IEEE JSTQE (2003); and a preliminary version of it appeared as Report NSF-ITP-02-93 (KITP, UCSB; 2002). Further material can be found in the recent e-prints arXiv:0708.1655v2 [physics.gen-ph] and arXiv:0708.1209v1 [physics.gen-ph]. The case of the very interesting (and more orthodox, in a sense) subluminal Localized Waves, solutions to the wave equations, will be dealt with in a coming paper. [Keywords: Wave equation; Wave propagation; Localized solutions to Maxwell equations; Superluminal waves; Bessel beams; Limited-dispersion beams; Electromagnetic wavelets; X-shaped waves; Finite-energy beams; Optics; Electromagnetism; Microwaves; Special relativity], Comment: LaTeX paper of 37 pages, with 20 Figures in jpg [to be processed by PDFlatex]

In the first part of this paper (mainly a review) we present general and formal (simple) introductions to the ordinary gaussian waves and to the Bessel waves, by explicitly separating the cases of the beams from the cases of the pulses; and, finally, 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 and important properties of the latter w.r.t. the former ones: 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.). In the second part of this paper (namely, in its Appendix), we recall at first how, in the seventies and eighties, the geometrical methods of Special Relativity (SR) had predicted --in the sense below specified-- the existence of the most interesting LWs, i.e., of the X-shaped pulses. At last, in connection with the circumstance that the X-shaped waves are endowed with Superluminal group-velocities (as carefully discussed in the first part of this article), 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 tunneling photons), and with the "localized Superluminal solutions" to the wave equations., Comment: Article (mainly a review) of 55 pages with 27 Figures, replaced with the quality of a few Figures improved. It is a preliminary version of a paper that in final form will appear as the introductory chapter of a book ["Localized Waves" (J.Wiley; in press), ed. by H.E.Hernandez Figueroa, M.Zamboni Rached, and E.Recami]

Since the early works[1-4] 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[5-7], nonlinearity[8], anisotropy[9] and losses[10]. 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[11-18]. This paper (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., Comment: Article (mainly a review) with 37 pages and 8 Figures. An improved version of this paper will appear as Chpater 2 of the book "Localized Waves" (J.Wiley; in press), ed. by H.E.Hernandez-Figueroa, M.Zamboni-Rached, and E.Recami

In this paper, starting from some general and plausible assumptions based on geometrical optics and on a common feature of the truncated Bessel beams, a heuristic derivation is presented of very simple analytical expressions, capable of describing the longitudinal (on-axis) evolution of axially-symmetric nondiffracting pulses when truncated by finite apertures. We apply our analytical formulation to several situations involving subluminal, luminal or superluminal localized pulses and compare the results with those obtained by numerical simulations of the Rayleigh-Sommerfeld diffraction integrals. The results are in excellent agreement. The present approach can be very useful, because it can yield, in general, closed analytical expressions, avoiding the need of time-consuming numerical simulations, and also because such expressions provide a powerful tool for exploring several important properties of the truncated localized pulses, as their depth of fields, the longitudinal pulse behavior, the decaying rates, and so on., Comment: 27 pages, 7 figures

In this work, in terms of suitable superpositions of equal-frequency Bessel beams, we develop a theoretical method to obtain nondiffractive beams in absorbing media (weakly conductive) capable to resist the loss effects for long distances., Comment: 12 pages, 3 figures

In this work, starting by suitable superpositions of equal-frequency Bessel beams, we develop a theoretical and experimental methodology to obtain localized stationary wave fields, with high transverse localization, whose longitudinal intensity pattern can approximately assume any desired shape within a chosen interval 0 < z < L of the propagation axis z. Their intensity envelope remains static, i.e. with velocity v=0; so that we have named ``Frozen Waves" (FW) these new solutions to the wave equations (and, in particular, to the Maxwell equations). Inside the envelope of a FW only the carrier wave does propagate: And the longitudinal shape, within the interval 0 < z < L, can be chosen in such a way that no nonnegligible field exists outside the pre-determined region (consisting, e.g., in one or more high intensity peaks). Our solutions are noticeable also for the different and interesting applications they can have, especially in electromagnetism and acoustics, such as optical tweezers, atom guides, optical or acoustic bistouries, various important medical apparata, etc., Comment: 29 pages, 9 figures

In this paper it is shown how one can use Bessel beams to obtain a stationary localized wavefield with high transverse localization, and whose longitudinal intensity pattern can assume any desired shape within a chosen interval 0 < z < L of the propagation axis. This intensity envelope remains static, i.e., with velocity v=0; and because of this we call "Frozen Waves" such news solutions to the wave equations (and, in particular, to the Maxwell equations). These solutions can be used in many different and interesting applications, as optical tweezers, atom guides, optical or acoustic bistouries, various important medical purposes, etc., Comment: LaTeX file (10 pages, including 2 sets of two Figures)

In this paper we analyze the properties of chirped optical X-shaped pulses propagating in material media without boundaries. We show that such ("superluminal") pulses may recover their transverse and longitudinal shape after some propagation distance, while the ordinary chirped gaussian-pulses can recover their longitudinal shape only (since gaussian pulses suffer a progressive spreading during their propagation). We therefore propose the use of chirped optical X-type pulses to overcome the problems of both dispersion and diffraction during the pulse propagation., Comment: Replaced with a much larger and deepened version (the number of pages going on from 4 to 24; plus 4 Figures added)

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. First, new Superluminal wave pulses are constructed, and then tailored in such a wave to get them temporally focused at a chosen spatial point, where the wavefield can reach for a short time very high intensities. Results of this kind may find applications in many fields, besides electromagnetism and optics, including acoustics, gravitation, and elementary particle physics. PACS nos.: 41.20.Jb; 03.50.De; 03.30.+p; 84.40.Az; 42.82.Et; 83.50.Vr; 62.30.+d; 43.60.+d; 91.30.Fn; 04.30.Nk; 42.25.Bs; 46.40.Cd; 52.35.Lv. Keywords: Localized solutions to Maxwell equations; Superluminal waves; Bessel beams; Limited-diffraction pulses; Finite-energy waves; Electromagnetic wavelets; X-shaped waves; Electromagnetism; Microwaves; Optics; Special relativity; Localized acoustic waves; Seismic waves; Mechanical waves; Elementary particle physics; Gravitational waves, Comment: Latex file, with 6 Figures

In a previous paper [Phys. Rev. E64 (2001) 066603; e-print physics/0001039], we showed that localized Superluminal solutions to the Maxwell equations exist, which propagate down (non-evanescence) regions of a metallic cylindrical waveguide. In this paper we construct analogous non-dispersive waves propagating along coaxial cables. Such new solutions, in general, consist in trains of (undistorted) Superluminal "X-shaped" pulses. Particular attention is paid to the construction of finite total energy solutions. Any results of this kind may find application in the other fields in which an essential role is played by a wave-equation (like acoustics, geophysics, etc.). [PACS nos.: 03.50.De; 41.20;Jb; 83.50.Vr; 62.30.+d; 43.60.+d; 91.30.Fn; 04.30.Nk; 42.25.Bs; 46.40.Cd; 52.35.Lv. Keywords: Wave equations; Wave propagation; Localized beams; Superluminal waves; Coaxial cables; Bidirectional decomposition; Bessel beams; X-shaped waves; Maxwell equations; Microwaves; Optics; Special relativity; Coaxial metallic waveguides; Acoustics; Seismology; Mechanical waves; Elastic waves; Guided gravitational waves.], Comment: plain LaTeX file (22 pages), plus 15 figures; in press in Phys. Rev. E

In a previous paper of ours [Phys. Rev. E64 (2001) 066603, e-print physics/0001039] we have shown localized (non-evanescent) solutions to Maxwell equations to exist, which propagate without distortion with Superluminal speed along normal-sized waveguides, and consist in trains of "X-shaped" beams. Those solutions possessed therefore infinite energy. In this note we show how to obtain, by contrast, finite-energy solutions, with the same localization and Superluminality properties. [PACS nos.: 41.20.Jb; 03.50.De; 03.30.+p; 84.40.Az; 42.82.Et. Keywords: Wave-guides; Localized solutions to Maxwell equations; Superluminal waves; Bessel beams; Limited-dispersion beams; Finite-energy waves; Electromagnetic wavelets; X-shaped waves; Evanescent waves; Electromagnetism; Microwaves; Optics; Special relativity; Localized acoustic waves; Seismic waves; Mechanical waves; Elastic waves; Guided gravitational waves.], Comment: plain LaTeX file (12 pages), plus 10 figures

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. [PACS nos.: 03.50.De ; 41.20.Jb ; 83.50.Vr ; 62.30.+d ; 43.60.+d ; 91.30.Fn ; 04.30.Nk ; 42.25.Bs ; 46.40.Cd ; 52.35.Lv. Keywords: Wave equation; Wave propagation; Optics; Localized beams; Superluminal waves; Bessel beams; X-shaped waves; Acoustics; Mechanical waves; Dispersion compensation; Seismology; Geophysics; Gravitational Waves; Elementary particle physics]., Comment: plain LaTeX file (16 pages), plus 9 figures

We show that localized (non-evanescent) solutions to Maxwell equations exist, which propagate without distortion along normal waveguides with Superluminal speed., Comment: plain LaTeX file (11 pages), plus 3 figures in eps format