Localized Waves: A Review

Transcription

1 CHAPTER 4 Localized Waves: A Review Erasmo Recami* and Michel Zamboni-Rached Contents 1 Localized Waves: A Scientific and Historical Introduction Introduction and Preliminary Remarks A More Detailed Introduction The Localized Solutions Complementary Material: A Historical Perspective (Theoretical and Experimental) Introduction Historical Recollections Theory A Glance at the Experimental State of the Art Structure of Nondiffracting Waves and Some Interesting Applications Foreword Spectral Structure of Localized Waves and the Generalized Bidirectional Decomposition Space-Time Focusing of X-Shaped Pulses Chirped Optical X-Type Pulses in Material Media Frozen Waves and Subluminal Wave Bullets Modeling the Shape of Stationary Wave Fields: Frozen Waves Subluminal Localized Waves (or Bullets) A First Method for Constructing Physically Acceptable Subluminal Localized Pulses A Second Method for Constructing Subluminal Localized Pulses Stationary Solutions with Zero-Speed Envelopes The Role of Special Relativity and Lorentz Transformations Nonaxially Symmetric Solutions: Higher-Order Bessel Beams 34 Acknowledgments 344 References 344 * Faculty of Engineering, University of Bergamo, Bergamo, Italy; and INFN, Sezione di Milano, Milan, Italy Center of Natural and Human Sciences, Federal University of ABC, Santo André, SP, Brazil Advances in Imaging and Electron Physics, Volume 156, ISSN , DOI: 1.116/S (8) Copyright 29 Elsevier Inc. All rights reserved. 235

2 236 Erasmo Recami and Michel Zamboni-Rached 1. LOCALIZED WAVES: A SCIENTIFIC AND HISTORICAL INTRODUCTION 1.1. Introduction and Preliminary Remarks Diffraction and dispersion have long been known as phenomena that limit the applications of (optical, for instance) beams or pulses. Diffraction is always present and affects any waves that propagate in two-dimensional (2D) or three-dimensional (3D) unbounded media, even when homogeneous. Pulses and beams are composed of waves traveling along different directions, which produce a gradual spatial broadening (Born and Wolf, 1998). This effect is a limiting factor whenever a pulse is needed that maintains its transverse localization for example, in free space communications (Willebrand and Ghuman, 21), image forming (Goodman, 1996), optical lithography (Ito and Okazaki, 2; Okazaki, 1991), electromagnetic tweezers (Ashkin et al., 1986; Curtis, Koss, and Grier, 22), and so on. Dispersion acts on pulses propagating in material media and causes mainly a temporal broadening. This effect is known to be due to the variation of the refraction index with the frequency, so that each spectral component of the pulse possesses a different phase velocity. This entails a gradual temporal widening, which constitutes a limiting factor when a pulse is needed that maintains its time width for example, in communication systems (Agrawal, 1995). Therefore, it is important to develop techniques capable of reducing such phenomena. The localized waves (LWs), also known as nondiffracting waves, can indeed resist diffraction for a long distance in free space. Such solutions to the wave equations (and, in particular, to the Maxwell equations, under weak hypotheses) were theoretically predicted years ago (Barut, Maccarrone, and Recami, 1982; Bateman, 1915; Caldirola, Maccarrone, and Recami, 198; Courant and Hilbert, 1966; Recami and Maccarrone, 198, 1983; Maccarrone, Pavsic, and Recami, 1983; Stratton, 1941) (also compare with Recami, Zamboni-Rached, and Dartora (24), as well as Section 3 below), were mathematically constructed in more recent times (Lu and Greenleaf, 1992a; Lu, Greenleaf, and Recami, 1996; Recami, 1998; Shaarawi and Besieris, 2a; Ziolkowski, Besieris, and Shaarawi, 2), and soon thereafter were experimentally produced (Lu and Greenleaf, 1992b; Mugnai, Ranfagni, and Ruggeri, 2; Saari and Reivelt, 1997). Today localized waves are well established theoretically and experimentally and have innovative applications not only in vacuum but also in material (linear or nonlinear) media, and are able to resist also dispersion. Their potential applications are being intensively explored, always with surprising results, in fields such as acoustics, microwaves, and optics, with great promise also in mechanics, geophysics, and even gravitational

3 Localized Waves: A Review 237 waves and elementary particle physics. Of note, the so-called frozen waves presented in the fifth section of this chapter are already being applied in high-resolution ultrasound scanning of moving organs in the human body (Lu, Zou, and Greenleaf, 1993, 1994). To restrict our discussion to electromagnetism, recall the present-day studies on electromagnetic tweezers (Arlt et al., 21; Garcés-Chavez et al., 22; MacDonald et al., 22; McGloin, Garcés-Chavez, and Dholakia, 23), optical (or acoustic) scalpels, optical guiding of atoms or (charged or neutral) corpuscles (Arlt, Hitomi, and Dholakia, 2; Fan, Parra, and Milchberg, 2; Rhodes et al., 22), optical litography (Erdélyi et al., 1997; Garcés-Chavez et al., 22), optical (or acoustic) images (Herman and Wiggins, 1991), communications in free space (Lu and Greenleaf, 1992a; Lu and Shiping, 1999; Ziolkowski, 1989, 1991), remote optical alignment (Salo et al., 199; Salo, Friberg, and Salomaa, 21; Vasara, Turunen, and Friberg, 1989), optical acceleration of charged particles, and so on. The following two subsections provide a brief introduction to the theory and applications of localized beams and localized pulses, respectively (Recami, Zamboni-Rached, and Hernández-Figueroa, 28). Before proceeding, as in any review article (for obvious reasons of space), we had to select a few main topics, and such a choice can only be a personal one Localized (Nondiffracting) Beams The term beam refers to a monochromatic solution to the considered wave equation with a transverse localization of its field. Herein we explicitly refer to the optical case, but our considerations hold true for any wave equation (vectorial, spinorial, scalar, and in particular, for the acoustic case). The most common type of optical beam is the Gaussian one, whose transverse behavior is described by a Gaussian function. However all common beams are affected by diffraction, which spoils the transverse shape of their field, widening it gradually during propagation. As an example, the transverse width of a Gaussian beam doubles when it travels a distance z dif = 3π ρ 2 /λ, where ρ is the beam initial width and λ is its wavelength. A Gaussian beam with an initial transverse aperture of the order of its wavelength will already double its width after traveling just a few wavelengths. It was generally believed that the only wave devoid of diffraction was the plane wave, which does not incur any transverse changes. Some authors have shown that it is not the only one. For instance, in 1941 Stratton developed a monochromatic solution to the wave equation whose transverse shape was concentrated in the vicinity of its propagation axis and represented by a Bessel function. Such a solution, now called a Bessel beam, was not subject to diffraction because no change in its transverse shape

4 238 Erasmo Recami and Michel Zamboni-Rached took place with time. Courant and Hilbert (1966) later demonstrated how a large class of equations (including wave equations) admit non-distorted progressing waves as solutions. By 1915, Bateman and subsequently researches such as Barut and Chandola (1993), showed the existence of soliton-like, wavelet-type solutions to the Maxwell equations. However, despite the literature reports, they did not garner the attention they deserved. In Stratton s case, this can be partially justified since that (Bessel) beam was endowed with infinite energy (as much as the plane waves or Gaussian beams, incidentally), and, moreover it was not square-integrable in the transverse direction. An interesting problem, therefore, is investigating what would happen to the ideal Bessel beam solution when truncated by a finite transverse aperture. It was not until 1987 that a heuristic answer came from the experiment by Durnin, Miceli, and Eberly when it was shown that a realistic Bessel beam, endowed with wavelength λ =.6328 μm and central spot 1 ρ = 59 μm, passing through an aperture with radius R = 3.5 mm can travel 85 cm with its transverse intensity shape approximately unchanged (in the region ρ<<r surrounding its central peak). In other words, it was shown that the transverse intensity peak, as well as the surrounding field, do not change appreciably in shape along a large depth of field. As a comparison, recall again that a Gaussian beam with the same wavelength and with the central spot 2 ρ = 59 μm, when passing through an aperture with the same radius (R = 3.5 mm) doubles its transverse width after 3 cm, and after 6 cm its intensity is already diminished by a factor of 1. Therefore, in the considered case, a Bessel beam can travel approximately without deformation a distance 28 times larger than a gaussian beam. This remarkable property is due to the fact that when the transverse intensity fields (whose value decreases with increasing ρ), associated with the rings that constitute the (transverse) structure of the Bessel beam, diffract, they reconstruct the beam itself, all along a large field depth. This property depends on the Bessel beam spectrum (wave number and frequency) (Durnin, 1987; Herman and Wiggins, 1991; Salo et al., 199; Salo, Friberg, and Salomaa, 21; Vasara, Turunen, and Friberg, 1989), as explained in detail in Zamboni-Rached, Recami, and Hernández-Figueroa (22). Let us stress that, given a Bessel and a Gaussian beam both with the same power flux, the same spot ρ, and passing through apertures with the same radius R in the plane z = the percentage of the total energy E contained inside the central peak region ( ρ ρ ) is smaller 1 Let us define the central spot of a Bessel beam as the distance, along the transverse direction ρ, at which the first zero occurs of the Bessel function characterizing its transverse shape. 2 In the case of a Gaussian beam, let us define its central spot as the distance, along the transverse direction ρ, at which its transverse intensity has decayed by the factor 1/e.

5 Localized Waves: A Review 239 for a Bessel than for a Gaussian beam. This differing energy distribution on the transverse plane is responsible for the reconstruction of the Besselbeam central peak even at large distances from the source (and even after an obstacle, provided that its size is smaller than the aperture (Bouchal, Wagner, and Chlup, 1998; Grunwald et al., 23, 24, 25; Zamboni- Rached, 1998); this nice property is also possessed by the localized pulses as examined below (Zamboni-Rached, 1998)). Most experiments in this area have been performed rapidly and often with use of rather simple apparatus. For example, for the generation of a Bessel beam, Durnin, Miceli, and Eberly (1987a) had recourse to a laser source, an annular slit, and a lens (Figure 1). In a sense, such an apparatus produces what can be regarded as the cylindrically symmetric generalization of a couple of plane waves emitted at angles θ and θ, with respect to (w.r.t.) the z-direction, respectively (in which case the plane wave intersection moves along z with the speed c/ cos θ). These nondiffracting beams can be generated also by a conic lens (axicon) (see Herman and Wiggins (1991)), or by other means such as holographic elements (MacDonald et al., 1983; Salo, Friberg, and Salomaa, 21). We stress that many interesting applications of nondiffracting beams are currently being investigated in addition to those of Lu et al. in acoustics. In the optical sector, Bessel beams are used as optical tweezers to confine or move around small particles. In such theoretical and application areas, noticeable contributions include those presented by Zamboni- Rached (24, 26); Zamboni-Rached, Recami, and Hernández-Figueroa (25), wherein, by suitable superpositions of Bessel beams endowed with the same frequency but different longitudinal wave numbers, stationary envelopes have been mathematically constructed in closed form. These fields possess a high transverse localization and, more important, Annular aperture Lens R f Bessel beam a a a << f R FIGURE 1 The simple experimental setup used by Durnin, Miceli, and Eberly (1987a) to generate a Bessel beam.

6 24 Erasmo Recami and Michel Zamboni-Rached a longitudinal intensity shape that can be freely chosen inside a predetermined space interval z L. For instance, a high-intensity field with a static envelope can be created within a tiny region, with negligible intensity elsewhere. Section 3 addresses such frozen waves Localized (Nondiffracting) Pulses As noted in the previous subsection, the existence of nondiffractive (or localized) pulses was predicted years ago: (see Barut, Maccarrone, and Recami, 1982; Bateman, 1915; Caldirola, Maccarrone, and Recami, 198; Courant and Hilbert, 1966; Maccarrone, Pavsic, and Recami, 1983; Recami and Maccarrone, 198, 1983; Recami, Zamboni-Rached, and Dartora, 24, and more recent articles such as Barut and Bracken (1992); Barut and Grant (199)). Modern studies of nondiffractive pulses followed a development rather independent of those on nondiffracting beams, even if both phenomena are part of the same sector of physics: that of localized waves. In 1983, Brittingham set forth a luminal (V = c) solution to the wave equation (more particularly, to the Maxwell equations) that travels rigidly, i.e., without diffraction. The solution proposed by Brittingham, however, had infinite energy, and once again the issue arose of overcoming such a problem. As far as we know, a solution was first obtained by Sezginer (1985), who showed how to construct finite-energy luminal pulses, which do not propagate without distortion for an infinite distance: As expected, Sezginer s pulses travel with constant speed and approximately without deforming for a certain (long) depth of field much longer, in this case, too, than ordinary pulses like the Gaussian ones. In a series of subsequent papers (Besieris, Shaarawi, and Ziolkowski, 1989; Donnelly and Ziolkowski, 1993; Shaarawi, Besieris, and Ziolkowski, 199; Ziolkowski, 1989, 1991; Ziolkowski, Besieris, and Shaarawi, 1993), a simple theoretical method was developed, termed bidirectional decomposition, for constructing a new series of nondiffracting luminal pulses. Eventually, at the beginning of the 199s, Lu and Greenleaf (1992a,b) constructed, both mathematically and experimentally, new solutions to the wave equation in free space: namely, an X-shaped localized pulse, with the form predicted by the so-called extended special relativity (Barut, Maccarrone, and Recami, 1982; Caldirola, Maccarrone, and Recami, 198; Maccarrone, Pavsic, and Recami, 1983; Recami, 1986; Recami and Maccarrone, 198, 1983); for the connection between what Lu et al. called X-waves and extended relativity, see, for example, Recami, Zamboni- Rached, and Dartora (24), while brief excerpts of the latter theory can be found in Garavaglia (1998); Lu, Greenleaf, and Recami (1996); Maccarrone and Recami (198); Pavsic and Recami (1982); Recami (1985, 1987, 1998, 21); Recami, Fontana, and Garavaglia (2); Recami et al. (23);

7 Localized Waves: A Review 241 see also Shaarawi and Besieris (2a); Ziolkowski, Besieris, and Shaarawi (2). Lu et al. s solutions (which can be called the classic ones) were continuous superpositions of Bessel beams with the same phase velocity (i.e., with the same axicon angle α) (Friberg, Fagerholm, and Salomaa (1997); Lu and Greenleaf (1992a); Lu, Greenleaf, and Recami (1996); Recami (1986, 1998); Shaarawi and Besieris (2a); Ziolkowski, Besieris, and Shaarawi (2)), so that they could maintain their shape for long distances. Such X-shaped waves resulted in interesting and flexible localized solutions and have since been studied in a number of papers, even if their velocity V is supersonic or superluminal (V > c). Actually, when the phase velocity does not depend on the frequency, it is known that such a phase velocity becomes the group velocity. Remembering how a superposition of Bessel beams is generated (for example, by a discrete or continuous set of annular slits or transducers), the energy forming the LWs, coming from those rings is transported at the ordinary speed c of the plane waves in the considered medium (here c, representing the velocity of the plane waves in the medium, is the speed of sound in the acoustic case, and the speed of light in the electromagnetic case, and so on). Nevertheless, the peak of the LWs is faster than c. (Let us note that, when using a lens after an aperture located at its back-focus, as in Figure 2, then a classic X- shaped pulse can be generated even by a single annular slit, or transducer, illuminated however by a light, or sound, pulse; but the previous considerations about the actual transportation speed of the energy forming the X-shaped wave remain unaffected. The experimental setup in Figure 2, with various annular slits, is actually needed only for generating (e.g., X-shaped) pulses more complex than the classic one, namely, depending Annular slits Lens a N21 a N f R Bessel beam superposition a 2N a a << f R FIGURE 2 One of the simplest experimental setups for generating various Bessel beam superpositions.

8 242 Erasmo Recami and Michel Zamboni-Rached on the coordinates z and t not only through the quantity ζ z Vt; see the following). It is indeed possible to generate (in addition to the classic X-wave produced by Lu et al. in 1992) infinite sets of new X-shaped waves, with their energy increasingly concentrated in a spot corresponding to the vertex region (Zamboni-Rached, Recami, and Hernández-Figueroa, 22). It may therefore appear rather intriguing that such a spot (even if no violations of special relativity [SR] are obviously implied: all the results come from Maxwell equations, or from the wave equations (Barbero, Hernández- Figueroa, and Recami, 2; Brodowsky, Heitmann, and Nimtz, 1996; Shaaraawi and Besieris, 2)) travels superluminally when the waves are electromagnetic. For simplicity, we shall use the term superluminal for all X-shaped waves, even when the waves are acoustic. Figure 3, which refers to an X-wave possessing the velocity V > c, illustrates the fact that, if its vertex or central spot is located at P 1 at time t 1, it will reach the position P 2 at a time t + τ, where τ = P 2 P 1 /V < P 2 P 1 /c: All these points are discussed below. Soon after having mathematically and experimentally constructed their classic acoustic X-wave, Lu et al. started applying the waves to ultrasonic scanning, obtaining very high-quality images. Subsequently, in a 1996 e-print and report, Recami et al. (see Lu, Greenleaf, and Recami, 1996; Recami, 1998) published the analogous X-shaped solutions to the Maxwell equations by constructing scalar superluminal localized solutions for each component of the Hertz potential. That approach showed that the localized solutions to the scalar equation can also be used, under very weak conditions, to obtain localized solutions to Maxwell s equations. (Actually P 1 R 5 v P 2 R c FIGURE 3 This figure shows an X-shaped wave, that is, a localized superluminal pulse. It refers to an X-wave, possessing the velocity V > c, and illustrates the fact that, if its vertex or central spot is located at P 1 at time t, it will reach the position P 2 at a time t + τ, where τ = P 2 P 1 /V < P 2 P 1 /c: This is somewhat different from the illusory scissor effect, even if the feeding energy, coming from the regions R, has traveled with the ordinary speed c (which is the speed of light in the electromagnetic case, or the speed of sound in acoustics, and so on).

9 Localized Waves: A Review 243 Ziolkowski, Besieris, and Shaarawi (1993) had found something similar, which they called slingshot pulses, for the simple scalar case, but their solution went almost unnoticed). In 1997, Saari and Reivelt announced the laboratory production of an X-shaped wave in the optical realm, thus proving experimentally the existence of superluminal electromagnetic pulses. Three years later, in 2, Mugnai, Ranfagni, and Ruggeri produced superluminal X-shaped waves in the microwave region (their paper aroused various criticisms, to which those authors responded). 2. A MORE DETAILED INTRODUCTION Let us refer to the differential equation known as the homogeneous wave equation: simple, but so important in acoustics, electromagnetism (microwaves, optics), geophysics, and even, as we said, gravitational waves and elementary particle physics: ( ) 2 x y z c 2 t 2 ψ(x, y, z; t) =. (1) Let us write it in the cylindrical coordinates (ρ, φ, z) and, for simplicity s sake, confine ourselves to axially symmetric solutions ψ(ρ, z; t). Then, Eq. (1) becomes ( ) 2 ρ ρ ρ + 2 z c 2 t 2 ψ(ρ, z; t) =. (2) In free space, the solution ψ(ρ, z; t) can be written in terms of a Bessel Fourier transform w.r.t. the variable ρ, and two Fourier transforms w.r.t. variables z and t, as follows: ψ(ρ, z, t) = k ρ J (k ρ ρ) e ik zz e iωt ψ(k ρ, k z, ω) dk ρ dk z dω, (3) where J (.) is an ordinary zero-order Bessel function and ψ(k ρ, k z, ω) is the transform of ψ(ρ, z, t). Substituting Eq. (3) into Eq. (2), one obtains that the relation, among ω, k ρ, and k z, ω 2 c 2 = k2 ρ + k2 z (4)

10 244 Erasmo Recami and Michel Zamboni-Rached must be satisfied. As a consequence, by using condition (4) in Eq. (3), any solution to the wave equation [Eq. (2)] can be written ψ(ρ, z, t) = ω/c k ρ J (k ρ ρ) e i ω 2 /c 2 k 2 ρ z e iωt S(k ρ, ω) dk ρ dω, (5) where S(k ρ, ω) is the chosen spectral function, when k z > (and we disregard evenescent waves). The general integral solution in Eq. (5) yields, for instance, the (nonlocalized) Gaussian beams and pulses, to which we shall refer to illustrate the differences of the localized waves w.r.t. them. The Gaussian Beam A very common (nonlocalized) beam is the Gaussian beam (Newell and Molone, 1992), corresponding to the spectrum S(k ρ, ω) = 2a 2 e a2 k 2 ρ δ(ω ω ). (6) In Eq. (6), a is a positive constant, which will be shown to depend on the transverse aperture of the initial pulse. Figure 4 illustrates the interpretation of the integral solution in Eq. (5), with spectral function (6), as a superposition of plane waves. Namely, from Figure 4 one can easily realize that this case corresponds to plane waves propagating in all directions (always with k z ), the most intense ones being those directed along (positive) z. Note that, in the plane wave case, k z is the longitudinal component of the wave vector, k = k ρ + k z, where k ρ = k x + k y. k ê k z S(k, ) k ê z FIGURE 4 Visual interpretation of the integral solution [Eq. (5)], with spectral function [Eq. (6)], in terms of a superposition of plane waves.

11 Localized Waves: A Review 245 On substituting Eq. (6) into Eq. (5) and adopting the paraxial approximation, one meets the Gaussian beam ψ gauss (ρ, z, t) = ( ) 2a 2 ρ 2 exp 4(a 2 + iz/2k ) 2(a 2 + iz/2k ) e ik (z ct), (7) where k = ω /c. We can verify that such a beam, which suffers transverse diffraction, doubles its initial width ρ = 2a after having traveled the distance z dif = 3 k ρ 2 /2, called diffraction length. The more concentrated a Gaussian beam happens to be, the more rapidly it gets spoiled. The Gaussian Pulse The most common (nonlocalized) pulse is the Gaussian pulse, which is obtained from Eq. (5) by using the spectrum (Zamboni-Rached, Hernández-Figueroa, and Recami, 24) S(k ρ, ω) = 2ba2 π e a2 k 2 ρe b2 (ω ω ) 2, (8) where a and b are positive constants. Indeed, such a pulse is a superposition of Gaussian beams of different frequency. On substituting Eq. (8) into Eq. (5), and again adopting the paraxial approximation, one obtains the Gaussian pulse: ψ(ρ, z, t) = ( ) ( ) a 2 ρ 2 (z ct) 2 exp 4(a 2 exp + iz/2k ) 4c 2 b 2 a 2, (9) + iz/2k endowed with speed c and temporal width t = 2b, and suffering a progressing enlargement of its transverse width, so that its initial value is already doubled at position z dif = 3 k ρ 2 /2, with ρ = 2a The Localized Solutions We now proceed to the construction of the two most renowned localized waves: the Bessel beam and the ordinary X-shaped pulse.

12 246 Erasmo Recami and Michel Zamboni-Rached First, it is interesting to observe that, when superposing (axially symmetric) solutions of the wave equation in vacuum, three spectral parameters, (ω, k ρ, k z ), come into play, which must satisfy the constraint (4), deriving from the wave equation itself. Consequently, only two of them are independent, and we choose 3 here ω and k ρ. The possibility of choosing ω and k ρ already was apparent in the spectral functions generating Gaussian beams and pulses, which consist of the product of two functions one depending only on ω and the other on k ρ. We will see that further particular relations between ω and k ρ (or, analogously, between ω and k z ) can be enforced to produce interesting and unexpected results, such as the LWs The Bessel Beam Let us start by imposing a linear coupling between ω and k ρ (it could be actually shown (Durnin, 1987) that it is the unique coupling leading to localized solutions). Namely, let us consider the spectral function S(k ρ, ω) = δ (k ρ ω ) c sin θ k ρ δ(ω ω ), (1) which implies that k ρ = (ω sin θ)/c, with θ π/2: a relation that can be regarded as a space-time coupling. This linear constraint between ω and k ρ, together with Eq. (4), yields k z = (ω cos θ)/c. This is an important fact because it has been shown elsewhere (Zamboni-Rached, 1999, 24; Zamboni-Rached, Recami, and Hernández-Figueroa, 22) that an ideal LW must contain a coupling of the type ω = Vk z + b, where V and b are arbitrary constants. The interpretation of the integral function (5), this time with spectrum (1), as a superposition of plane waves is visualized in Figure 5, which shows that an axially symmetric Bessel beam is nothing but the result of the superposition of plane waves whose wave vectors lie on the surface of a cone, with the propagation line as its symmetry axis and an opening angle equal to θ; such θ being called the axicon angle. By inserting Eq. (1) into Eq. (5), one gets the mathematical expression of the so-called Bessel beam: ( ω ) [ ψ(ρ, z, t) = J c sin θρ exp i ω ( c cos θ z c )] cos θ t. (11) 3 Elsewhere we chose ω and k z.

13 Localized Waves: A Review 247 x k x k y k y k z z FIGURE 5 The axially symmetric Bessel beam is created by the superposition of plane waves whose wave vectors lie on the surface of a cone with the propagation axis as its symmetry axis and angle equal to θ ( axicon angle ). This beam possesses phase velocity v ph = c/ cos θ, and field transverse shape represented by a Bessel function J (.) so that its field in concentrated in the surroundings of the propagation axis, z. Moreover, Eq. (11) tells us that the Bessel beam maintains its transverse shape (which is therefore invariant) while propagating, with central spot ρ = 2.45c/(ω sin θ). The ideal Bessel beam, however, is not a square-integrable function and therefore has an infinite energy; that is, it cannot be experimentally produced. However, we can have recourse to truncated Bessel beams, generated by finite apertures. In this case, the (truncated) Bessel beams can still travel a long distance while maintaining their transfer shape, as well as their speed, approximately unchanged (Durnin, Miceli, and Eberly, 1987a,b; Overfelt and Kenney, 1991); that is, they still possess a large depth of field. For instance, the field depth of a Bessel beam generated by a circular finite aperture with radius R is given by Z max = R tan θ, (12) where θ is the beam axicon angle. In the finite aperture case, the Bessel beam can no longer be represented by Eq. (11), and it must be calculated by the scalar diffraction theory: for example, by using Kirchhoff s or Rayleigh Sommerfeld s diffraction integrals. But up to the distance Z max, Eq. (11) can still be used to approximately describe the beam, at least in the vicinity of the axis ρ =, that is, for ρ<<r. To realize the extent to which a truncated Bessel beam succeeds in resisting diffraction, let us also consider a Gaussian beam with the same frequency and central spot, and compare their field depths. In particular, let us assume for both beams λ =.63 μm and initial central spot size ρ = 6 μm. The Bessel beam will possess axicon angle θ = arcsin[2.45c/(ω ρ )]=.4 rad.

14 248 Erasmo Recami and Michel Zamboni-Rached C Gauss 2 C Bessel (mm) Z(m) (mm) (a) (b) FIGURE 6 Comparison between a Gaussian (a) and a truncated Bessel beam (b). The Gaussian beam doubles its initial transverse width already after 3 cm, while after 6 cm its intensity decays of a factor 1. By contrast, the Bessel beam maintains its transverse shape up to the distance of 85 cm..6.8 Z (m) Figure 6 shows the behavior of the two beams for a circular aperture with radius 3.5 mm. We can verify how the Gaussian beam doubles its initial transverse width already after 3 cm, and after 6 cm its intensity has become an order of magnitude smaller. By contrast, the truncated Bessel beam maintains its transverse shape until the distance Z max = R/ tan θ = 85 cm. Thereafter, the Bessel beam rapidly decays as a consequence of the sharp cut performed on its aperture (such cut being responsible also for the intensity oscillations suffered by the beam along its propagation axis, and for the fact that eventually the feeding waves, coming from the aperture, at a certain point become faint). The zero-order (axially symmetric) Bessel beam is nothing but one example of a localized beam. Further examples are the higher-order (not cylindrically symmetric) Bessel beams ( ω ) ( ψ(ρ, φ, z; t) = J ν c sin θρ exp (iνφ) exp i ω ( c cos θ z c )) cos θ t, (13) or the Mathieu beams (Dartora et al., 23), and so on The Ordinary X-Shaped Pulse Following the same procedure adopted in the previous subsection, let us construct pulses by using spectral functions of the type S(k ρ, ω) = δ (k ρ ω ) c sin θ k ρ F(ω), (14)

15 Localized Waves: A Review 249 where here the Dirac delta function furnishes the spectral space-time coupling k ρ = (ω sin θ)/c. Function F(ω) is, of course, the frequency spectrum; it is left for the moment undetermined. On using Eq. (14) in Eq. (5), one obtains ψ(ρ, z, t) = F(ω) J ( ω c sin θρ ) ( ω ( exp c cos θ z c )) cos θ t dω. (15) It is easy to see that ψ will be a pulse of the type ψ = ψ(ρ, z Vt) (16) with a speed V = c/ cos θ independent of the frequency spectrum F(ω). Such solutions are known as X-shaped pulses, and are localized (nondiffractive) waves in the sense that they obviously do maintain their spatial shape during propagation (see, Lu and Greenleaf, 1992a; Lu, Greenleaf, and Recami, 1996; Recami, 1998; Shaarawi and Besieris, 2a; Zamboni-Rached, Recami, and Hernández-Figueroa, 22; Ziolkowski, Besieris, and Shaarawi, 2, and references therein; as well as the following). At this point, some remarkable observations can be stressed: (i) When a pulse consists of a superposition of waves (in this case, Bessel beams) all endowed with the same phase velocity V ph (in this case, with the same axicon angle) independent of their frequency, then it is known that the phase velocity (in this case V ph = c/ cos θ) becomes the group velocity V (Esposito et al., 23; Zamboni-Rached, Fontana, and Recami, 23; Zamboni-Rached, Recami, and Fontana, 21); that is, V = c/ cos θ>c. In this sense, the X-shaped waves are called Superluminal localized pulses. [For simplicity, the group velocity (Garavaglia, 1998; Olkhovsky, Recami, and Jakiel, 24; Pavsic and Recami, 1982; Recami, Fontana, and Garavaglia, 2; Zamboni-Rached, Recami, and Hernández-Figueroa, 22) can be regarded as the peak velocity. Here, we add only the observations: (a) that the group velocity for a pulse, in general, is well defined only when the pulse has a clear bump in space, but it can be calculated by the approximate, simple relation V dω/dk only when some extra conditions are satisfied (namely, when ω as a function of k is also clearly bumped); and (b) that the group velocity can a priori be evaluated through the aforementioned, customary derivation of ω with respect to the wave number for the infinite total energy solutions; whereas, for the finite total energy superluminal solutions, the group velocity cannot be calculated through such

16 25 Erasmo Recami and Michel Zamboni-Rached an elementary relation, since in those cases there does not exist a one-to-one function ω = ω(k z )]. (ii) Such pulses, even if their group velocity is superluminal, do not contradict standard physics, having been found in what precedes on the basis of the wave equations in particular, of Maxwell equations (Lu, Greenleaf, and Recami, 1996; Recami, 1998; Shaarawi and Besieris, 2a; Ziolkowski, 1989; Ziolkowski, Besieris, and Shaarawi, 2) only. Indeed, as we shall see better in Section 3, their existence can be understood within SR itself (Barut, Maccarrone, and Recami, 1982; Caldirola, Maccarrone, and Recami, 198; Garavaglia, 1998; Lu, Greenleaf, and Recami, 1996; Maccarrone, Pavsic, and Recami, 1983; Pavsic and Recami, 1982; Recami, 1998, 21; Recami, Fontana, and Garavaglia, 2; Recami and Maccarrone, 198, 1983; Recami, Zamboni-Rached, and Dartora, 24; Recami et al., 23; Shaarawi and Besieris, 2a; Ziolkowski, Besieris, and Shaarawi, 2), on the basis of its ordinary postulates (Recami, 1993). Let us repeat: They are fed by waves originating at the aperture and carrying energy with the standard speed c of the medium (the light velocity in the electromagnetic case, and the sound velocity in the acoustic case). We can become convinced about the possibility of realizing superluminal X- shaped pulses by imagining the simple ideal case of a negligibly sized superluminal source S endowed with speed V > c in vacuum, and emitting electromagnetic waves W (each one traveling with the invariant speed c). The electromagnetic waves will be internally tangent to an enveloping cone C with S as its vertex, and as its axis the propagation line z of the source (Recami, 1986; Recami, Zamboni-Rached, and Dartora, 24): This is completely analogous to what happens for an airplane that moves in air with constant supersonic speed. The waves W interfere mainly negatively inside the cone C and constructively on its surface. We can place a plane detector orthogonally to z and record magnitude and direction of the waves W that hit on it as (cylindrically symmetric) functions of position and of time. It suffices, then, to replace the plane detector with a plane antenna that emits instead of recording exactly the same (axially symmetric) space-time pattern of waves W, for constructing a cone-shaped electromagnetic wave C that will propagate with the superluminal speed V (of course, without a source any longer at its vertex), even if each wave W travels with the invariant speed c. Again, this is exactly what would happen in the case of a supersonic airplane (in which case c is the sound speed in air; for simplicity, assume the observer to be at rest with respect to the air); for further details, see the quoted references. By suitable superpositions and interference of speed-c waves, one can obtain pulses increasingly more localized in the vertex region (Zamboni-Rached, Recami, and Hernández-Figueroa, 22): that is, very localized field blobs that

17 Localized Waves: A Review 251 z FIGURE 7 The truncated X-waves considered in this chapter, as predicted by SR (all wave equations have an intrinsic relativistic structure!), must have a leading cone in addition to the rear cone, with such a leading cone playing a role for peak stability. For example, when producing a finite conic wave truncated both in space and in time, the theory of SR suggests recourse, in the simplest case, to a dynamic antenna emitting a radiation cylindrically symmetric in space and symmetric in time, for a better approximation to what Courant and Hilbert (1966) called an undistorted progressing wave. travel with superluminal group velocity. This apparently has nothing to do with the illusory scissors effect, since such blobs, along their field depth, are a priori able to get two successive (weak) detectors, located at a distance L, to click after a time smaller than L/c. Incidentally, an analysis of the above-mentioned case (of a supersonic plane or a superluminal charge) led, as expected (Recami, 1986), to the simplest type of X-shaped pulse (Recami, Zamboni-Rached, and Dartora, 24). It might be useful to recall that SR (even the wave equations have an internal relativistic structure!) implies considering also the forward cone (Figure 7). The truncated X-waves considered in this chapter, for instance, must have a leading cone in addition to the rear cone, with such a leading cone playing a role for the peak stability (Lu and Greenleaf, 1992a). For example, in the approximate case in which we produce a finite conic wave truncated both in space and in time, the theory of SR suggested the biconic shape (symmetrical in space with respect to the vertex S) as a better approximation to a rigidly traveling wave (so that SR suggests recourse to a dynamic antenna emitting a radiation cylindrically symmetric in space and symmetric in time, for a better approximation to an undistorted progressing wave ). (iii) Any solutions that depend on z and on t only through the quantity z Vt, like Eq. (15), will appear with a constant shape to an observer traveling along z with the same speed V. That is, such a solution will propagate rigidly with speed V. This further explains why our X-shaped pulses, after having been produced, travel almost rigidly at speed V (in this case, a faster-than-light group velocity), all along their depth of field. For greater clarity, let us consider a generic function,

18 252 Erasmo Recami and Michel Zamboni-Rached depending on z Vt with V > c, and show, by explicit calculations involving the Maxwell equations only, that it obeys the scalar wave equation. Following Franco Selleri (2), let us consider, for example, the wave function, (x, y, z, t) = a [b ic(z Vt)] 2 + (V 2 c 2 )(x 2 + y 2 ), (17) with a and b nonzero constants, c the ordinary speed of light, and V > c (incidentally, this wave function is nothing but the classic X-shaped wave in cartesian coordinates). Let us naively verify that it is a solution to the wave equation On putting R 2 (x, y, z, t) 1 c 2 2 (x, y, z, t) 2 t =. (18) [b ic(z Vt)] 2 + (V 2 c 2 )(x 2 + y 2 ), (19) one can write = a/r and evaluate the second derivatives wherefrom [ ] 1 2 a 2 z 1 2 c 2 2 t 1 2 a 2 z = c2 R 3 3c2 R 5 [b ic(z Vt)]2 ; 1 2 c 2 a 2 x = V2 R (V 2 c 2) 2 x 2 R 5 ; 1 2 c 2 a 2 y = V2 R (V 2 c 2) 2 y 2 R a 2 t = c2 V 2 R 3 3c2 V 2 R 5 [b ic(z Vt)] 2 ; = V2 c 2 R (V 2 c 2) 2 [b ic(z Vt)] 2 R 5, and [ ] 1 2 a 2 x = 2 V2 c 2 y R (V 2 c 2) 2 x 2 + y 2 R 5.

19 Localized Waves: A Review 253 From the last two equations, remembering the previous definition, one finally gets 1 a [ ] 2 2 z x y 1 2 c 2 2 =, t which is nothing but the (d Alembert) wave equation [Eq. (18)], q.e.d. In conclusion, function is a solution of the wave equation even if it does obviously represent a pulse (Selleri says a signal!) propagating with superluminal speed. At this point, readers should be however informated that all the subluminal LWs, solutions of the ordinary homogeneous wave equation, have until now appeared to present singularities whenever they depend on z and t only via the quantity ζ z Vt. This is still an open, interesting research topic, which is also related to analogous results met in gravitation physics. After the previous three important comments, let us return to our evaluations with regard to the X-type solutions to the wave equations. Let us now consider in Eq. (15), for instance, the particular frequency spectrum F(ω) given by F(ω) = H(ω) a V exp ( a V ω ), (2) where H(ω) is the Heaviside step function and a a positive constant. Then, Eq. (15) yields ψ(ρ, ζ) X = a (a iζ) 2 + ( ), (21) V 2 1 ρ c 2 2 with ζ z Vt. This solution [Eq. (21)] is the well-known ordinary or classic X-wave, which constitutes a simple example of X-shaped pulse (Lu and Greenleaf, 1992a; Lu, Greenleaf, and Recami, 1996; Recami, 1998; Shaarawi and Besieris, 2a; Ziolkowski, Besieris, and Shaarawi, 2). Notice that function (2) contains mainly low frequencies, so that the classic X-wave is suitable for low frequencies only. Figure 8 depicts (the real part of) an ordinary X-wave with V = 1.1 c and a = 3m. Solutions (15), and in particular the pulse [Eq. (21)], have an infinite field depth and infinite energy. Therefore, as was done in the Bessel beam case, one should pass to truncated pulses, originating from a finite aperture.

20 254 Erasmo Recami and Michel Zamboni-Rached Re( ) FIGURE 8 a = 3m (m) (m) Plot of the real part of the ordinary X-wave, evaluated for V = 1.1 c with Afterward, our truncated pulses will keep their spatial shape (and their speed) all along the depth of field Z = R tan θ, (22) where, as before, R is the aperture radius and θ the axicon angle Further Observations Let us put forth some further observations. It is not strictly correct to call nondiffractive the LWs, since diffraction affects, more or less, all waves obeying Eq. (1). However, all LWs (both beams and pulses) possess the remarkable self-reconstruction property: That is, the LWs, when diffracting during propagation, immediately rebuild their shape (Bouchal, Wagner, and Chlup, 1998; Grunwald et al., 23, 24, 25) (even after obstacles with size much larger than the characteristic wavelengths, provided it is smaller than the aperture size), due to their particular spectral structure [as shown in detail, in Wiley (28)]. In particular, the ideal LWs (with infinite energy and field depth) are able to rebuild themselves for an infinite time; whereas, the finite-energy (truncated) ones can do it, and thus resist the diffraction effects, only along a certain depth of field.

21 Localized Waves: A Review 255 The interest in LWs (especially from the point of view of applications) lies in the circumstance that they are almost nondiffractive, rather than in their group velocity: From this point of view, superluminal, luminal, and subluminal localized solutions are equally interesting and suited to important applications. In reality, LWs are not restricted to the (X-shaped, superluminal) ones corresponding to the integral solution (15) to the wave equation; and, as already stated, three classes of localized pulses exist: the superluminal (with speed V > c), the luminal (V = c), and the subluminal (V < c) ones all of them with, or without, axial symmetry, and corresponding in any case to a single, unified mathematical background. This issue will be discussed again in this review. Incidentally, we have addressed elsewhere topics such as (i) the construction of infinite families of generalizations of the classic X-shaped wave (with energy increasingly concentrated around the vertex; compare with Figure 9 from Zamboni-Rached, Recami, and Hernández-Figueroa (22)); (ii) the behavior of some finite total-energy superluminal localized solutions (SLS); (iii) the techniques for building new series of SLS to the Maxwell equations suitable for arbitrary frequencies and bandwidths; (iv) questions related to the case of dispersive (and even lossy) media; (v) the construction x x (m) (m) (m) 2.6 (m) (a) FIGURE 9 Panel (a) represents (in arbitrary units) the square magnitude of the classic, X-shaped superluminal localized solution (SLS) to the wave equation, with V = 5c and a =.1 m. Families of infinite SLSs, however, exist, which generalize the classic X-shaped solution for instance, a family of SLSs obtained by suitably differentiating the classic X-wave (Zamboni-Rached, Recami, and Hernández-Figueroa, 22): Panel (b) depicts the first of these waves (corresponding to the first differentiation) with the same parameters. As noted, the successsive solutions in such a family are increasingly localized around their vertex. Quantity ρ is the distance (in meters) from the propagation axis z, while quantity ζ is the V-cone variable (still in meters) ζ z Vt, with V c. Because all these solutions depend on z only via the variable ζ, they propagate rigidly ; that is, without distortion (and they are called localized, or nondispersive, for this reason). Here we are assuming propagation in vacuum (or in a homogeneous medium). (b).2.4.6

22 256 Erasmo Recami and Michel Zamboni-Rached of (infinite- or finite-energy) superluminal LWs propagating down waveguides or coaxial cables; (vi) determining localized solutions to equations different from the wave equation, such as Schrödinger s; and (vii) using the above techniques for constructing, in general relativity, new exact solutions for gravitational ways. We discuss some of these issues in the second section of this chapter. Let us add here that X-shaped waves also have been easily produced in nonlinear media (Conti et al., 23). A more technical introduction to the subject of LWs (particularly w.r.t. the superluminal X-shaped ones) is found in the second section of this review, and in papers such as Recami et al. (23). The Appendix presents a historical perspective. 3. COMPLEMENTARY MATERIAL: A HISTORICAL PERSPECTIVE (THEORETICAL AND EXPERIMENTAL) This mainly historical appendix, written as much as possible in a (partially) self-consistent form, first refers, from the theoretical point of view, to the most intriguing localized solutions to the wave equation: the superluminal wave solutions (SLS), and in particular the X-shaped pulses. As a start, we recall their geometrical interpretation within SR. Afterward, to help resolving possible doubts, we present a bird s-eye view of the various experimental sectors of physics in which superluminal motions seem to appear; in particular, in experiments with evanescent waves (and/or tunneling photons), and with the SLSs. In some parts of this appendix the propagation line is called x (and no longer z) without creating any interpretation problems Introduction The question of superluminal (V 2 > c 2 ) objects or waves has a long history. In pre-relativistic times, various relevant papers include those by J.J. Thomson and by A. Sommerfeld. It is well known, however, that with SR the conviction spread that the speed c of light in vacuum was the upper limit of any possible speed. For instance, in 1917 Tolman was believed to have shown by his paradox that the existence of particles endowed with speeds larger than c would have allowed sending information into the past. The problem was tackled again only in the 195s and 6s, particularly after the papers (e.g., Bilaniuk, Deshpande, and Sudarshan, 1962) by George Sudarshan et al. and, later, by one of the present authors (Recami, 1978; Recami, and Mignani, 1974) as well as those by Corben and others. The first experimental attempts were performed by T. Alväger et al.

23 Localized Waves: A Review 257 We wish to address the still unusual issue of the possible existence of superluminal wavelets, and objects within standard physics and SR, since at least four different experimental sectors of physics seem to support such a possibility (apparently confirming some long-standing theoretical predictions (Barut, Maccarrone, and Recami, 1982; Bilaniuk, Deshpande, and Sudarshan, 1962; Maccarrone, Pavsic, and Recami, 1983; Recami, 1978, 1986)). The experimental review is necessarily short, but we provide enough bibliographical information, limited for brevity s sake to only the last century updated to the year Historical Recollections Theory A simple theoretical framework was long ago proposed (Bilaniuk, Deshpande, and Sudarshan, 1962; Recami, 1986; Recami, and Mignani, 1974), based merely on the space-time geometrical methods of SR, that appears to incorporate Superluminal waves and objects, and in a sense predicts (Barut, Maccarrone, and Recami, 1982; Caldirola, Maccarrone, and Recami, 198; Maccarrone, Pavsic, and Recami, 1983; Recami and Maccarrone, 198, 1983), among others, the superluminal X-shaped waves, without violating the Relativity principles. A suitable choice of the postulates of SR (equivalent to the other, more common, choices) includes the following: (i) the standard Principle of Relativity, and (ii) space-time homogeneity and space isotropy. It follows that one and only one invariant speed exists, and experience shows that invariant speed to be the light speed, c, in vacuum the essential role of c in SR being just due to its invariance, and not to the fact that it be a maximal, or minimal, speed. No subluminal or superluminal objects or pulses can be endowed with an invariant speed; therefore, their speed cannot play in SR the same essential role played the vacuum light-speed, c. Indeed, the speed c turns out to be also a limiting speed; but any limit possesses two sides and can be approached a priori both from below and from above (Figure 1). As Sudarshan stated, from the fact E m c 2 2c c v FIGURE 1 Energy of a free object as a function of its speed. (Bilaniuk, Deshpande, and Sudarshan, 1962; Recami, 1986; Recami, and Mignani, 1974).

24 258 Erasmo Recami and Michel Zamboni-Rached that no one could climb over the Himalayan ranges, people of India cannot conclude that there are no people north of the Himalayas. Indeed, speed-c photons exist, which are born, live, and die just at the top of the mountain, with no need to perform the impossible task of accelerating from rest to the speed of light. (Actually, the ordinary formulation of SR has been too restricted. For instance, even leaving superluminal speeds aside, it can easily be widened to include antimatter (Garavaglia, 1998; Maccarrone and Recami, 198; Pavsic and Recami, 1982; Recami, 1985, 1986, 1987; Recami, Fontana, and Garavaglia, 2)). An immediate consequence is that the quadratic form c 2 dt 2 dx 2 dx μ dx μ, called ds 2, with dx 2 dx 2 + dy 2 + dz 2, is invariant, except for its sign. Quantity ds 2, let us recall, is the four-dimensional (4D) length-element square, along the space-time path of any object. In correspondence with the positive (negative) sign, one gets the subluminal (superluminal) Lorentz transformations (LTs). More specifically, the ordinary subluminal LTs are known to leave, for instance, the quadratic forms dx μ dx μ,dp μ dp μ, and dx μ dp μ exactly invariant, where p μ is the component of the energyimpulse four-vector, whereas the superluminal LTs, by contrast, should change (only) the sign of such quadratic forms. This is enough to deduce some important consequences, such as the one that a superluminal charge has to behave as a magnetic monopole, in the sense specified in Recami (1986) and the references therein. A more important consequence for us is that the simplest subluminal object namely, a spherical particle at rest (which appears as ellipsoidal [Figure 11], due to Lorentz contraction, at subluminal speeds v), will appear (Barut, Maccarrone, and Recami, 1982; Caldirola, Maccarrone, and Recami, 198; Maccarrone, Pavsic, and Recami, 1983; Recami, 1986, 1998; Recami and Maccarrone, 198, 1983) as occupying the cylindrically symmetrical region bounded by a two-sheeted rotation hyperboloid and an indefinite double cone (Figure 11d) for superluminal speeds V. In the limiting case of a pointlike particle, one obtains only a double cone. Such a result is obtained by writing down the equation of the world tube of a subluminal particle and transforming it simply by changing the sign of the quadratic forms entering that equation. Thus, in , it was predicted (Barut, Maccarrone, and Recami, 1982; Caldirola, Maccarrone, and Recami, 198; Recami and Maccarrone, 198, 1983) that the simplest superluminal object appears not as a particle, but as a field or rather as a wave: namely, as an X-shaped pulse, the cone semi-angle α being given (with c = 1) by cotg α = V 2 1. Such X-shaped pulses will move rigidly with speed V along their motion direction. In fact, any X-pulse can be regarded at each instant of time as the (superluminal) Lorentz transform of a spherical object, which of course moves in vacuum or in a homogeneous medium without any deformation as

25 Localized Waves: A Review 259 z z z (a) (b) (c) (d) FIGURE 11 An intrinsically spherical (or pointlike, at the limit) object appears in vacuum as an ellipsoid contracted along the motion direction when endowed with a speed v < c. By contrast, if endowed with a speed V > c (evenifthec-speed barrier can be crossed neither from the left nor from the right), it would appear (Barut, Maccarrone, and Recami, 1982; Caldirola, Maccarrone, and Recami, 198; Maccarrone, Pavsic, and Recami, 1983; Recami, 1986; Recami and Maccarrone, 198, 1983) no longer as a particle, but rather as an X-shaped wave traveling rigidly---namely, as occupying the region delimited by a double cone and a two-sheeted hyperboloid---or as a double cone, at the limit, and moving without distortion in the vacuum, or in a homogeneous medium, with superluminal speed V (the square cotangent of the cone semi-angle being (V/c) 2 1). For simplicity, a space axis is skipped. Source: Barut, Maccarrone, and Recami (1982); Caldirola, Maccarrone, and Recami (198); Recami and Maccarrone (198, 1983). time elapses. The 3D picture of Figure 11d is shown in Figure 12, where its annular intersections with a transverse plane are shown (see Barut, Maccarrone, and Recami, 1982; Caldirola, Maccarrone, and Recami, 198; Recami and Maccarrone, 198, 1983). The X-shaped waves considered here are merely the simplest ones. If one starts not from an intrinsically spherical or pointlike object, but from a nonspherically symmetric particle, from a pulsating (contracting and dilating) sphere, or from a particle oscillating back and forth along the motion direction, then their superluminal Lorentz transforms would be increasingly more complicated. The above X-waves, however, are typical for a superluminal object much as the spherical or pointlike shape is typical for a subluminal object. Incidentally, it has long been believed that superluminal objects would allow sending information into the past, but such problems with causality seem to be solvable within SR. Once SR is generalized to include superluminal objects or pulses, no signal traveling backward in time is apparently left. For a solution of those causal paradoxes, see Bilaniuk, Deshpande, and Sudarshan (1962); Maccarrone and Recami (198); Pavsic and Recami (1982); Recami, Fontana, and Garavaglia (2), and especially Recami (1985, 1987), and references therein. When addressing the problem within this elementary appendix of the production of an X-shaped pulse like the one depicted in Figure 12 (maybe truncated, in space and in time, by use of a finite antenna radiating for

26 26 Erasmo Recami and Michel Zamboni-Rached V 1 V 2 c Z V FIGURE 12 Here we show the intersections of the superluminal object T represented in Figure 11d with planes P orthogonal to its motion line (the z-axis). For simplicity, we assumed again the object is spherical in its rest frame, and the cone vertex C coincides with the origin O for t =. Such intersections evolve in time so that the same pattern appears on a second plane shifted by x after the time t = x/v. On each plane, as time elapses the intersection is therefore predicted by (extended) SR to be a circular ring which, for negative times, continues to shrink until it reduces to a circle and then to a point (for t = ); afterward, such a point again becomes a circle and then a circular ring that broadens (Barut, Maccarrone, and Recami, 1982; Caldirola, Maccarrone, and Recami, 198; Maccarrone, Pavsic, and Recami, 1983; Recami, 1986, 1998; Recami and Maccarrone, 198, 1983). (Notice that, if the object is not spherical when at rest [but, e.g., is ellipsoidal in its own rest frame], then the axis of T will no longer coincide with x, but its direction will depend on the speed V of the tachyon itself). For the case when the space extension of the superluminal object T is finite, see Recami and Maccarrone (1983). Source: This picture is from Barut, Maccarrone, and Recami (1982); Caldirola, Maccarrone, and Recami (198); Recami (1986); Recami and Maccarrone (198, 1983). a finite time), all the considerations expounded under point (ii) of the subsection The Ordinary X-shaped Pulse become in order; here, we simply refer to them. Those considerations, together with the present ones (e.g., related to Figure 12), suggest the simplest antenna to consist of a series of concentric annular slits, or transducers (as in Figure 2) that suitably radiate, following specific time patterns (see, Zamboni-Rached, 26, and references therein). Incidentally, the above procedure can lead to a very simple type of X-shaped wave, as investigated below.

27 Localized Waves: A Review 261 From the present point of view, it is rather interesting to note that, during the past 15 years, X-shaped waves have been actually found as solutions to the Maxwell and wave equations (we repeat that the form of any wave equation is intrinsically relativistic). To further clarify the connection existing between what is predicted by SR (see, Figures 11 and 12) and the localized X-waves mathematically, and experimentally, constructed in recent times, let us tackle in detail the problem of the (X-shaped) field created by a superluminal electric charge, by following a paper (Recami, Zamboni-Rached, and Dartora, 24) recently appeared in Physical Review E The Particular X-Shaped Field Associated With a Superluminal Charge It is now well known that Maxwell equations admit of wavelet-type solutions endowed with arbitrary group-velocities ( < v g < ). We again confine ourselves to the localized solutions, rigidly moving, and, more in particular, to the superluminal ones (SLSs), the most interesting of which are X-shaped. The SLSs have been produced in a number of experiments, always by suitable interference of ordinary-speed waves. In this subsection we show, by contrast, that even a superluminal charge creates an electromagnetic X-shaped wave, in agreement with what was predicted (Barut, Maccarrone, and Recami, 1982; Caldirola, Maccarrone, and Recami, 198; Recami, 1986; Recami and Maccarrone, 198) within SR. In fact, on the basis of Maxwell equations, the field associated with a superluminal charge can be evaluated (at least, under the rough approximation of pointlikeness). As announced in what precedes, it constitutes a very simple example of true X-wave. Indeed, the theory of SR, when based on the ordinary postulates but not restricted to subluminal waves and objects (i.e., in its extended version), predicted the simplest X-shaped wave to be the one corresponding to the electromagnetic field created by a superluminal charge (Recami, 1986a; Recami, Zamboni-Rached, and Dartora, 24). It seems to be important evaluating such a field, at least approximately, by following Recami, Zamboni-Rached, and Dartora (24). The toy model of a pointlike superluminal charge. Let us start by considering, formally, a pointlike superluminal charge, even if the hypothesis of pointlikeness (already unacceptable in the subluminal case) is totally inadequate in the superluminal case (Recami, 1986). Then, let us consider the ordinary vector-potential A μ and a current density j μ (,, j z ; j o ) flowing in the z-direction (notice that the motion line is still the axis z). On assuming the fields to be generated by the sources only, one has that A μ (,, A z ; φ),

28 262 Erasmo Recami and Michel Zamboni-Rached which, when adopting the Lorentz gauge, obeys the equation A μ = j μ. We can write such a nonhomogeneous wave equation in the cylindrical coordinates (ρ, θ, z; t); for axial symmetry (which requires a priori that A μ = A μ (ρ, z; t)), when choosing the V-cone variables ζ z Vt; η z + Vt, with V 2 > c 2, we arrive at the equation [ ρ ( ρ ) ] ρ ρ γ 2 ζ γ 2 η A μ (ρ, ζ, η) = j μ (ρ, ζ, η), ζ η (23) where μ assumes the two values μ = 3, only, so that A μ (,, A z ; φ) and γ 2 [V 2 1] 1. (Notice that, whenever convenient, we set c = 1.) Let us now suppose A μ to be actually independent of η, namely, A μ = A μ (ρ, ζ). According to Eq. (23), we shall have j μ = j μ (ρ, ζ) too; and therefore j z = Vj (from the continuity equation) and A z = Vφ/c (from the Lorentz gauge). Then, by calling ψ A z, we obtain two equations, which allow us to analyze the possibility and consequences of having a superluminal pointlike charge, e, traveling with constant speed V along the z-axis (ρ = ) in the positive direction, in which case j z = evδ(ρ)/ρ δ(ζ). Indeed, one of those two equations becomes the hyperbolic equation [ 1 ( ρ ) ] ρ ρ ρ γ 2 ζ 2 ψ = ev δ(ρ) δ(ζ) (24) ρ which can be solved (Recami, Zamboni-Rached, and Dartora, 24) in a few steps. First, by applying (with respect to the variable ρ) the Fourier Bessel (FB) transformation f (x) = f ( )J ( x) d, quantity J ( x) being the ordinary zero-order Bessel function. Second, by applying the ordinary Fourier transformation with respect to the variable ζ (going on, from ζ, to the variable ω). And, third, by finally performing the corresponding inverse Fourier and FB transformations. Afterward, it is sufficient to have recourse to formulas ( ) and ( ) of Gradshteyn and Ryzhik (1965), still with ζ z Vt, to be able to write the solution of Eq. (24) in the form ψ(ρ, ζ) = ψ(ρ, ζ) = e V ζ 2 ρ 2 (V 2 1) for <γ ζ < ρ for ρ<γ ζ. (25)

29 Localized Waves: A Review 263 (8 ) 1/2 Ve (1 mm21 ) ca z (mm) (mm) FIGURE 13 Behavior of the field ψ A z generated by a charge supposed to be superluminal, as a function of ρ and ζ z Vt, evaluated for γ = 1 (i.e., for V = c 2). We skipped the points at which ψ must diverge, namely, the vertex and the cone surface. Source: Recami, Zamboni-Rached, and Dartora (24). 1 Figure 13 shows our solution A z ψ, as a function of ρ and ζ, evaluated for γ = 1 (i.e., for V = c 2). We skipped the points at which A z must diverge, namely, the vertex and the cone surface. For comparison, recall that the classic X-shaped solution (Lu and Greenleaf, 1992a) of the homogeneous wave equation which is shown in Figures 8, 9, and 12 has the form (with a > ): X = V (a iζ) 2 + ρ 2 (V 2 1). (26) The second equation in Eqs. (25) includes expression (26), given by the spectral parameter (Zamboni-Rached et al., 23; Zamboni-Rached, Recami, and Hernández-Figueroa, 22) a =, which indeed corresponds to the nonhomogeneous case [the nonnegligible fact that for a = these equations differ for an imaginary unit (Recami, 1986; Recami and Mignani, 1976) is discussed elsewhere]. At this point, it is rather important to notice that such a solution, Eq. (25), does represent a wave existing only inside the (unlimited) double cone C generated by the rotation around the z-axis of the straight lines ρ =±γζ. This too is in full agreement with the predictions of the extended

30 264 Erasmo Recami and Michel Zamboni-Rached theory of SR. For the explicit evaluation of the electromagnetic fields generated by the superluminal charge (and of their boundary values and conditions) we confine ourselves here to merely quoting Recami, Zamboni-Rached, and Dartora, 24. Incidentally, the same results found by following the above procedure can be obtained by starting from the four-potential associated with a subluminal charge (e.g., an electric charge at rest), and then applying to it the suitable superluminal Lorentz transformation. Also of note, this double cone does not have much in common with the Cherenkov cone (Folman and Recami, 1995; Recami, 1986a; Zamboni-Rached and Recami, 28b), and a superluminal charge traveling at constant speed, in vacuum, does not lose energy (Figure 14). Outside the cone C, that is, for <γ ζ < ρ, we get (as expected) no field, so that one meets a field discontinuity when crossing the doublecone surface. Nevertheless, the boundary conditions imposed by Maxwell equations are satisfied by our solution in Eq. (25), since at each point of the cone surface the electric and the magnetic fields are both tangent to the cone; for a discussion of this point, see Recami, Zamboni-Rached, and Dartora (24). Emission Absorption Z Retarded Advanced FIGURE 14 The spherical equipotential surfaces of the electrostatic field created by a charge at rest are transformed into two-sheeted rotation hyperboloids, contained inside an unlimited double cone, when the charge travels at superluminal speed (compare, Recami (1986a); Recami, Zamboni-Rached, and Dartora (24)). This figures shows that a superluminal charge traveling at constant speed in a homogeneous medium like a vacuum does not lose energy (Folman and Recami, 1995; Recami, 1986a). We mention, incidentally, that this double cone has nothing to do with the Cherenkov cone (Zamboni-Rached and Recami, 28b). Source: Recami (1986), figure 27, page 8.

31 Localized Waves: A Review 265 Here we stress that, when V, and therefore γ, the electric field tends to vanish, while the magnetic field tends to the value H φ = πe/ρ 2. This agrees once more with what is expected from extended SR, which predicted superluminal charges to behave (in a sense) as magnetic monopoles. In the present paper we can only mention such a circumstance and refer readers to Recami (1978, 1986); Recami, and Mignani (1974); Recami and Mignani (1976), and papers quoted therein A Glance at the Experimental State of the Art Extended relativity can also allow a better understanding of many aspects of ordinary physics (Recami, 1986), even if superluminal objects (tachyons) did not exist in our cosmos as asymptotically free objects. Anyway, at least three or four different experimental sectors of physics seem to suggest the possible existence of faster-than-light motions, or, at least, of superluminal group velocities. The following sets forth some information about the experimental results obtained in two of those different physics sectors, with only mere mention of the others Neutrinos A long series of experiments, started in 1971, seems to show that the square m 2 of the mass m of muon neutrinos, and more recently also of electron neutrinos, is negative, which, if confirmed, would mean that (when using a naive language, commonly adopted) such neutrinos possess an imaginary mass and are therefore tachyonic, or mainly tachyonic (Baldo Ceolin, 1993; Giani, 1999; Giannetto et al., 1986; Otten, 1995; Recami, 1986). (In extended SR, however, the dispersion relation for a free superluminal object does become ω 2 k 2 = 2,orE 2 p 2 = m 2 o, and there is no need, at all, therefore, of imaginary masses.) Galactic Microquasars As to the apparent superluminal expansions observed in the core of quasars (Zensus and Pearson, 1987) and, recently, in the so-called galactic microquasars (Gisler, 1994; Mirabel and Rodriguez, 1994; Tingay et al., 1995), we do not address that problem, except by mentioning that for those astronomical observations there also exist orthodox interpretations, based on Cavaliere, Morrison, and Sartori (1971) and Rees (1966), that are still accepted by the majority of astrophysicists. For a theoretical discussion, see Recami et al. (1986). Here, let us only emphasize that simple geometrical considerations in Minkowski space show that a single superluminal source of light would appear (Recami, 1986; Recami et al., 1986) (i) initially, in the optical boom phase (analogous to the acoustic boom

32 266 Erasmo Recami and Michel Zamboni-Rached produced by an airplane traveling with constant supersonic speed), as an intense source that suddenly comes into view; and which, afterward, (ii) seems to split into two objects receding from each other with speed V > 2c (all of this is similar to what has been actually observed, according to Gisler (1994); Mirabel and Rodriguez (1994); Tingay et al. (1995)) Evanescent Waves and Tunneling Photons Within quantum mechanics (and precisely in the tunneling processes), it had been shown that the tunneling time first evaluated as a simple Wigner s phase time and later calculated through the analysis of the wave packet behavior does not depend (Hartman, 1962; MacColl, 1932; Milonni, 22; Olkhovsky and Recami, 1992; Olkhovsky, Recami, and Jakiel, 21; Olkhovsky et al., 1995) on the barrier width in the case of opaque barriers ( Hartman effect ). This implies superluminal and arbitrarily large group velocities V inside long enough barriers (Figure 15) pen (s) pen (s) x (Å) FIGURE 15 Behavior of the average penetration time (in seconds) spent by a tunneling wave packet as a function of the penetration depth (in angstroms) down a potential barrier (from Olkhovsky et al., 1995). According to the predictions of quantum mechanics, the wave packet speed inside the barrier increases in an unlimited way for opaque barriers, and the total tunneling time does not depend on the barrier width (Hartman, 1962; MacColl, 1932; Milonni, 22; Olkhovsky and Recami, 1992; Olkhovsky, Recami, and Jakiel, 21; Olkhovsky et al., 1995).

33 Localized Waves: A Review 267 Experiments that may verify this prediction by, say, electrons or neutrons, are difficult and rare (Olkhovsky, Recami, and Jakiel, 24; Olkhovsky, Recami, and Zaichenko, 25). Luckily, however, the Schrödinger equation in the presence of a potential barrier is mathematically identical to the Helmholtz equation for an electromagnetic wave propagating, for instance, down a metallic waveguide (along the z-axis): as shown, for example in Chiao, Kwiat, and Steinberg (1991); Japha and Kurizki (1996); Kurizki, Kozhekin, and Kofman (1998); Kurizki et al. (1999); Martin and Landauer (1992); Ranfagni et al. (1991); and a barrier height U bigger than the electron energy E corresponds (for a given wave frequency) to a waveguide of transverse size lower than a cutoff value. A segment of undersized guide to go on with our example does therefore behave as a barrier for the wave (photonic barrier), as well as any other photonic band-gap filters. The wave assumes therein like a particle inside a quantum barrier an imaginary momentum or wave number and, as a consequence, results exponentially damped along x (Figure 16). It becomes an evanescent wave (reverting to normal propagation, even if with reduced amplitude, when the narrowing ends and the guide returns to its initial transverse size). Thus, a tunneling experiment can be simulated by having recourse to evanescent waves (for which the concept of group velocity can be properly extended; see Recami, Fontana, and Garavaglia (2)). The fact that evanescent waves travel with superluminal speeds (Figure 17) has been verified in a series of famous experiments. Various FIGURE 16 The damping taking place inside a barrier (Garavaglia, 1998; Recami, Fontana, and Garavaglia, 2). Such damping reduces the amplitude of the tunneling wave packet, imposing a practical limit on the adoptable barrier length.

34 268 Erasmo Recami and Michel Zamboni-Rached (a) b 1 b 2 b 1 (b) n a n c2 (c) n c1 a t c Length a (mm) FIGURE 17 Simulation of tunneling by experiments with evanescent classical waves (see text), which were predicted to be superluminal also on the basis of extended SR (Mugnai et al., 1995; Recami, 1986). The figure shows one of the measurement results by (Brodowsky, Heitmann, and Nimtz, 1996; Nimtz and Enders, 1992, 1993; Nimtz, Spieker, and Brodowsky, 1994; Nimtz and Heitmann, 1997) the average beam speed while crossing the evanescent region ( = segment of undersized waveguide, or barrier ) as a function of its length a. As theoretically predicted (Hartman, 1962; MacColl, 1932; Milonni, 22; Mugnai et al., 1995; Olkhovsky and Recami, 1992; Olkhovsky, Recami, and Jakiel, 21; Olkhovsky et al., 1995; Recami, 1986a), such an average speed exceeds c for long enough barriers. Further results appeared in Longhi et al. (22), and are reported below; see Figures 2 and 21 in the following text. experiments performed since 1992 by Nimtz et al. in Cologne (Brodowsky, Heitmann, and Nimtz, 1994, 1996; Nimtz and Enders, 1992, 1993; Nimtz, Spieker, and Brodowsky, 1994), by Chiao, Kwiat, and Steinberg at Berkeley (Steinberg, Kwiat, and Chiao, 1993), by Ranfagni and colleagues in Florence (Mugnai, Ranfagni, and Ruggeri, 2), and by others in Vienna, Orsay, and Rennes (Balcou and Dutriaux, 1997; Laude and Tournois, 1999; Spielmann et al., 1994), verified that tunneling photons travel with superluminal group velocities (such experiments also raised a great deal of interest (Begley, 1995; Brown, 1995; Landauer, 1993), in the popular press, and were reported in Scientific American, Nature, New Scientist, and other publications). In addition, extended SR had predicted (Mugnai et al., 1995; Recami, 1986b) evanescent waves to be endowed with faster-than-c speeds; the whole matter appears to be therefore theoretically self-consistent. The debate in the current literature does not refer to the experimental results (which can be correctly reproduced even by numerical simulations (Barbero, Hernández-Figueroa, and Recami, 2; Brodowsky, Heitmann,

35 Localized Waves: A Review 269 Amplitude (V/m) Amplitude (V/m) Amplitude (1 2 V/m) (a) Time (ns) (b) Time (ns) Time (ns) FIGURE 18 The delay of a wave packet crossing a barrier (see Figure 17) is due to the initial discontinuity. We then performed suitable numerical simulations (Barbero, Hernández-Figueroa, and Recami, 2) by considering an (indefinite) undersized waveguide, and therefore eliminating any geometric discontinuity in its cross section. This figure shows the envelope of the initial signal. Inset (a) depicts in detail the initial part of this signal as a function of time, while inset (b) depicts the Gaussian pulse peak centered at t = 1 ns. and Nimtz, 1996; Shaaraawi and Besieris, 2) based on Maxwell equations only; compare Figures 18 and 19), but rather to the question of whether they allow, or do not allow, sending signals or information with superluminal speed (see Milonni, 22; Nimtz and Haibel, 22; Shaarawi and Besieris, 2b; Ziolkowski, 21). In the above-mentioned experiments one meets a substantial attenuation of the considered pulses (see Figure 16) during tunneling [or during propagation in an absorbing medium]. However, by using gain doublets, undistorted pulses have been observed, propagating with superluminal group- velocity with a small change in amplitude (see Wang, Kuzmich, and Dogariu, 2). We emphasize that some of the most interesting experiments of this series seem to be the ones with two or more barriers (e.g., with two gratings in an optical fiber (Longhi et al., 22), or with two segments of undersized waveguide separated by a piece of normal-sized waveguide (Enders and Nimtz, 1993; Nimtz, Enders, and Spieker, 1993, 1994): see Figure 2).

36 27 Erasmo Recami and Michel Zamboni-Rached Amplitude (1 2 V/m) Amplitude (1 2 V/m) Amplitude (1 2 V/m) (a) Time (ns) (b) Time (ns) Time (ns) FIGURE 19 Envelope of the signal in Figure 18 after traveling a distance L = mm through the undersized waveguide. Inset (a) shows the initial part (in time) of such arriving signal, while inset (b) shows the peak of the Gaussian pulse that had been initially modulated by centering it at t = 1 ns. Its propagation took zero time, so that the signal traveled with infinite speed. The numerical simulation has been based on Maxwell equations only. Proceeding from Figure 18 to Figure 19 verifies that the signal strongly lowered its amplitute. However, the width of each peak did not change (and this might have some relevance when thinking of a Morse alphabet transmission ; see text). FIGURE 2 Interesting experiments have been performed with two successive barriers evanescence regions. For example, with two gratings in an optical fiber. This figure (Garavaglia, 1998) refers to the interesting experiment (Enders and Nimtz, 1993; Nimtz, Enders, and Spieker, 1993) performed with microwaves traveling along a metallic waveguide: the waveguide being endowed with two classical barriers undersized guide segments. See text for details. For suitable frequency bands namely, for tunneling far from resonances we found that the total crossing time does not depend on the length of the intermediate (normal) guide; that is, that the beam speed along it is infinite (Aharonov, Erez, and Reznik, 22; Enders and

37 Localized Waves: A Review 271 Nimtz, 1993; Esposito, 23; Nimtz, Enders, and Spieker, 1993; Olkhovsky, Recami, and Jakiel, 24; Olkhovsky, Recami, and Salesi, 22; Recami, 24). This agrees with the quantum mechanics prediction for nonresonant tunneling through two successive opaque barriers (Olkhovsky, Recami, and Salesi, 22; Recami, 24) (Figure 21). Such a prediction was verified first theoretically by Olkhovsky, Recami, and Salesi (22), and then, a second time, experimentally by taking advantage of the circumstance that evanescence regions can consist of a variety of photonic band-gap materials or gratings (from multilayer dielectric mirrors, or semiconductors, to photonic crystals). Indeed, the best experimental confirmation has come by having recourse to two gratings in an optical fiber: Longhi et al. (22); see Figures 22 and 23; particularly the rather peculiar (and quite interesting) results represented by the latter. We cannot omit a further topic which, being delicate, probably should not appear in an overview like this because it is currently raising much interest (Wang, Kuzmich, and Dogariu, 2). Even if all the ordinary causal I V V 1 2 II III IV L L1 FIGURE 21 Scheme of the (nonresonant) tunneling process through two successive (opaque) quantum barriers. Far from resonances, the (total) phase time for tunneling through the two potential barriers does depend neither on the barrier widths nor on the distance between the barriers ( generalized Hartman effect ) (Olkhovsky, Recami, and Jakiel, 24; Olkhovsky, Recami, and Salesi, 22; Recami, 21, 24; Recami et al., 23). See text for details. V Incident pulse Transmitted pulse n(z )5n [112V (z )cos(2 z /L)] L L L z z 5 z 5 L 12L FIGURE 22 Realization of the quantum theoretical setup in Figure 21 using, as classical (photonic) barriers, two gratings in an optical fiber. The corresponding experiment has been performed by Longhi et al. (22).

38 272 Erasmo Recami and Michel Zamboni-Rached Tunneling time (ps) Barrier separation (mm) FIGURE 23 Off-resonance tunneling time versus barrier separation for the rectangular symmetric double-barrier Fiber Bragg grating (FBG) structure considered in Longhi et al. (22) (see Figure 22). The solid line is the theoretical prediction based on group delay calculations; the dots are the experimental points as obtained by time-delay measurements (the dashed curve is the expected transit time from input to output planes for a pulse tuned far away from the stopband of the FBGs). The experimental results (Longhi et al., 22), as well as the early ones in Enders and Nimtz (1993); Nimtz, Enders, and Spieker (1993), do confirm the theoretical prediction of a generalized Hartman effect, in particular, the independence of the total tunneling time from the distance between the two barriers. paradoxes seem to be solvable (Recami, 1985, 1986, 1987), nevertheless one must consider (whenever it is met an object, O, traveling with superluminal speed) the possibility of dealing with negative contributions to the tunneling times (Olkhovsky, Recami, and Jakiel, 24; Olkhovsky et al., 1995; Recami, 1986), which should not be regarded as unphysical. In fact, whenever an object (particle, electromagnetic pulse) O overcomes (Maccarrone and Recami, 198; Recami, 1985, 1986, 1987) the infinite speed with respect to a certain observer, it will afterward appear to the same observer as the anti-object O traveling in the opposite space direction (Bilaniuk, Deshpande, and Sudarshan, 1962; Maccarrone and Recami, 198; Recami, 1985, 1986, 1987). For instance, when going from the lab to a frame F moving in the same direction as the particles or waves entering the barrier region, the object O penetrating through the final part of the barrier (with almost infinite speed, as in Figure 15) will appear in the frame F as an antiobject O crossing that portion of the barrier in the opposite space direction. In the new frame F, therefore, such anti-object O would yield a negative contribution to the tunneling time, which could even result as negative (for clarifications, see the quoted references). We stress that even the appearance of such negative times had been predicted within SR itself (Olkhovsky et al., 1995), on the basis of its ordinary postulates, and recently has been confirmed by quantum theoretical evaluations too (Olkhovsky, Recami, and Jakiel, 24; Petrillo and Refaldi, 2, 23; Refaldi, 2). (In the case of

39 Localized Waves: A Review 273 a nonpolarized beam, the wave anti-packet coincides with the initial wave packet; however, if a photon is endowed with helicity λ =+1, the antiphoton will bear the opposite helicity λ = 1.) From the theoretical point of view, besides the above-quoted papers (in particular Hartman, 1962; MacColl, 1932; Milonni, 22; Olkhovsky and Recami, 1992; Olkhovsky, Recami, and Jakiel, 21, 24; Olkhovsky et al., 1995), see more specifically Bolda, Chiao, and Garrison, 1993; Chiao, Kozhekin, and Kurizki, 1996; Garret and McCumber, 197. On the (very interesting!) experimental side, see the intriguing papers by Chu and Wong (1982); Macke et al. (1987); Mitchell and Chiao (1997); Nimtz (1999); Segard and Macke (1985); Wang, Kuzmich, and Dogariu (2). Let us add here that, via quantum interference effects, it is possible to obtain dielectrics with refraction indices that rapidly vary as a function of frequency, also in three-level atomic systems, with almost complete absence of light absorption (i.e., with quantum-induced transparency) (Alzetta et al., 1976). The group velocity of a light pulse propagating in such a medium can decrease to very low values, either positive or negative, with no pulse distortion. Experiments performed both in atomic samples at room temperature and in Bose Einstein condensates have shown the possibility of reducing the speed of light to a few meters per second. Similar, but negative group velocities, implying a propagation with superluminal speeds thousands of time higher than those previously mentioned, recently have been predicted, in the presence of such an electromagnetically induced transparency, for light moving in a rubidium condensate (Artoni et al., 21). Finally, faster-than-c propagation of light pulses can be (and has been, in same cases) observed by taking advantage of the anomalous dispersion near an absorbing line, nonlinear and linear gain lines (as already seen), nondispersive dielectric media, or inverted twolevel media, as well as of some parametric processes in nonlinear optics (see Kurizki et al. s works) Superluminal Localized Solutions (SLS) to the Wave Equations The X-shaped waves. The fourth sector (to leave aside the others) is no less important. It came into fashion again when it was rediscovered in a series of remarkable works that any wave equation (to fix the ideas, let us think of the electromagnetic case) also admits solutions as much subluminal as superluminal (besides the luminal ones, having speed c/n). Starting with the pioneering works of (Bateman, 1915), it slowly became known that all wave equations admit soliton-like (or rather wavelet-type) solutions with subluminal group velocities. Subsequently, superluminal solutions also started to be written (in one case (Barut and Chandola, 1993) simply by the mere application of a superluminal Lorentz transformation (Recami, 1986)).

40 274 Erasmo Recami and Michel Zamboni-Rached A remarkable feature of some of these new solutions (which attracted much attention for their possible applications) is that they propagate as localized, nondispersive pulses because of their self-reconstruction property. It is easy to realize the practical importance, for instance, of a radio transmission carried out by localized beams (independently of their speed), but nondispersive wave packets can be of use even in theoretical physics for a reasonable representation of elementary particles, and so on. Incidentally, from the point of view of elementary particles, the fact that the wave equations possess pulse-type solutions that, in the subluminal case, are ball-like (Figure 24) can have a bearing on the corpuscle/wave duality problem met in quantum physics (besides agreeing with Figure 11, for example). Further comments on this point are found below. We emphasize once again that, within extended SR, since 198 it has been shown that while the simplest subluminal object conceivable is a small sphere, or a point in the limiting case the simplest superluminal objects by contrast appear as (see Barut, Maccarrone, and Recami, 1982), and our Figures 11 and 12) an X-shaped wave, or a double cone, as their limit, which moreover travels without deforming (i.e., rigidly) in a homogeneous medium. It is not without meaning that the most interesting localized solutions to the wave equations happened to be the superluminal ones, and with a shape of that kind. Moreover, since from Maxwell equations under simple hypotheses one proceeds to the usual scalar wave equation for each electric or magnetic field component, the same solutions were expected to exist also in the fields of acoustic waves, seismic waves, and gravitational waves. Indeed, this has been suggested in the literature for all such cases, and demonstrated in acoustics. Such pulses (as suitable superpositions of Bessel beams) were mathematically constructed for the FIGURE 24 The wave equations possess pulse-type solutions that, in the subluminal case, are ball-like (in agreement with Figure 11). For comments, see the text.

41 Localized Waves: A Review 275 first time by Lu et al. in acoustics and were then called X-waves or rather X-shaped waves. (One should not forget that, however, LWs can be constructed in exact form even for other equations, such as Schröedinger s and Einstein s). It is important that the X-shaped waves have been produced experimentally, with both acoustic and electromagnetic waves; that is, X-pulses were produced that, in their medium, travel undistorted with a speed faster than sound, in the first case, and faster than light, in the second case. In acoustics, the first experiment was performed by Lu et al. in 1992 at the Mayo Clinic (and their papers received the first Institute of Electrical and Electronics Engineer 1992 award). In the electromagnetic case, certainly more intriguing, superluminal localized X-shaped solutions were first mathematically constructed (Compare with Figure 25) in Lu, Greenleaf, and Recami (1996); Recami (1998), and later were experimentally produced by Saari et al. (Saari and Reivelt, 1997) in 1997 at Tartu by visible light (Figure 26), and more recently by Ranfagni et al. at Florence by microwaves (Mugnai, Ranfagni, and Ruggeri, 2). In the theoretical sector the activity has been no less intense, in order to derive, for example, analogous new solutions with finite total energy or more suitable for high frequencies, on the one hand, and localized solutions superluminally propagating 2 mm (a) R e {F XBB } 2 mm Max Min. 5. Max e4 Min e4 4 m 4 m 2 mm (b) R e {(EY XBB ) } 2 mm Max e6 Min e6 Max Min m 4 m (c) R e {(BY XBB ) p } (d) R e {(BY XBB ) z } FIGURE 25 Real Part of the Hertz potential and of the field components of the localized electromagnetic ( classic, axially symmetric) X-shaped wave predicted, and first mathematically constructed for the electromagnetic case in Lu, Greenleaf, and Recami (1996), and Recami (1998). For the meaning of the various panels, see the quoted references. The dimension of each panel is 4 m (in the radial direction) 2 mm (in the propagation direction). The values shown on the top-right corner of each panel represent the maxima and the minima of the images before normalization for display (IS units).

42 276 Erasmo Recami and Michel Zamboni-Rached Pos.1 Pos.2 Pos.3 V L1 L2 PH M L3 FIGURE 26 Scheme of the experiment by (Saari and Reivelt, 1997), who announced (Physical Review Letters, Nov. 24, 1997) the production in optics of the beams depicted in Figure 25. The present figure shows the experimental results. The X-shaped waves are superluminal; indeed, they, running after plane waves (the latter regularly traveling with speed c), do catch up with the considered plane waves. An analogous experiment was performed later with microwaves at Florence by (Mugnai, Ranfagni, and Ruggeri, 2) (Physical Review Letters of May 22, 2). z 3D FIGURE 27 This figure depicts two elements of the trains of X-shaped pulses, mathematically constructed in Zamboni-Rached et al. (22), which propagate down a coaxial guide (in the Transverse Magnetic case). This picture is taken from (Zamboni-Rached et al., 22), but analogous X-pulses exist (with infinite or finite total energy) for propagation along a cylindrical, normal-sized metallic waveguide. even along a normal waveguide (see Figure 27), on the other hand; and so on. Let us eventually recall the problem of producing an X-shaped superluminal wave like the one in Figure 12, but truncated in space and in

43 Localized Waves: A Review 277 time (by use of a finite antenna, radiating for a finite time). In such a situation, the wave is known to keep its localization and superluminality only until a certain depth of field (i.e., as long as it is fed by the waves arriving (with speed c) from the antenna), decaying often abruptly afterward (Durnin, Miceli, and Eberly, 1987a; Zamboni-Rached, Recami, and Hernández-Figueroa, 22). Various authors, taking account of the time needed to foster such superluminal waves, have concluded that these localized superluminal pulses are unable to transmit information faster than c. Many of these questions have been discussed in preceding text and references; for further details, see Lu, Greenleaf, and Recami, 1996; Recami, 1998; Shaarawi and Besieris, 2a; Ziolkowski, Besieris, and Shaarawi, 2. In any case, the existence of the X-shaped superluminal (or supersonic) pulses seems to constitute (e.g., together with the superluminality of evanescent waves), a confirmation of extended SR: a theory (Recami, 1986) based on the ordinary postulates of SR and that consequently does not appear to violate any of the fundamental principles of physics. It is curious that one of the first applications of such X-waves (that takes advantage of their propagation without deformation) has been accomplished in the field of medicine precisely, of ultrasound scanners (Lu, Zou, and Greenleaf, 1993, 1994), whereas the most important applications of the (subluminal!) Frozen Waves will very probably once again affect human health problems such as cancer. After the digression in this appendix, we pass to a second part of the work with a slightly more technical (Zamboni-Rached, Recami, and Hernández-Figueroa, 28) review about the physical and mathematical characteristics of LWs and some interesting applications. In the third part we shall deal with the ones endowed with zero speed (i.e., with a static envelope) and, more in general, with the subluminal LWs. 4. STRUCTURE OF NONDIFFRACTING WAVES AND SOME INTERESTING APPLICATIONS 4.1. Foreword Since the early works (Brittingham, 1983; Durnin, Miceli, and Eberly, 1987a; Zamboni-Rached, Recami, and Hernández-Figueroa, 28) on the so-called nondiffracting waves (or Localized Waves), many articles have been published on this important subject from both the theoretical and the experimental points 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

44 278 Erasmo Recami and Michel Zamboni-Rached (Lu and Greenleaf, 1992b; Sõnajalg and Saari, 1996; Zamboni-Rached et al., 23), nonlinearity, Conti et al. (23), anisotropy, Salo et al. (1999), and losses (Zamboni-Rached, 26). Such extensions have been carried out in addition to the development of efficient methods for obtaining nondiffracting beams and pulses in the subluminal, luminal, and superluminal regimes (Besieris, Shaarawi, and Ziolkowski, 1989; Longhi, 24; Zamboni-Rached, 1999, 24a,b, 26; Zamboni-Rached, Recami, and Hernández-Figueroa, 22, 25). This section addresses some theoretical methods related to nondiffracting solutions of the linear wave equation in unbounded homogeneous media, as well as some interesting applications of such solutions. Zamboni- Rached, Recami, and Hernández-Figueroa (28). The usual cylindrical coordinates (ρ, φ, z) are used herein. We already know that in these coordinates the linear wave equation is written as ( 1 ρ ) ρ ρ ρ ρ 2 φ z c 2 =. (27) t2 In Section 4.2 we analyze the general structure of LWs, develop the socalled Generalized Bidirectional Decomposition, and use it to obtain several luminal and superluminal nondiffracting wave solutions of Eq. (27). In Section 4.3 we develop a space-time focusing method by a continuous superposition of X-Shaped pulses of different velocities. Section 4.4 addresses the properties of chirped optical X-shaped pulses propagating in material media without boundaries. Subsequently, we show at the beginning of Section 5 how a suitable superposition of Bessel beams can be used to obtain stationary localized wave fields with high transverse localization, and whose longitudinal intensity pattern can assume any desired shape within a chosen interval z L of the propagation axis. Because of space constraints, we necessarily omit many interesting results. Let us briefly mention, for example, that rather simple analytic expressions, capable of describing the longitudinal (on-axis) evolution of axially symmetric nondiffracting pulses, have been recently completed by Zamboni-Rached (26) even for pulses truncated by finite apertures. Excellent agreement has been found by comparing what is easily provided by such expressions, for several situations (involving subluminal, luminal, or superluminal localized pulses) with the results obtained by numerical evaluations of the Rayleigh Sommerfeld diffraction integrals. Therefore, those new closed-form expressions dispense with the need for time-consuming numerical simulations (and provide an effective tool for determining the most important properties of the truncated localized pulses).

45 Localized Waves: A Review Spectral Structure of Localized Waves and the Generalized Bidirectional Decomposition An effective approach to understand the concept of the (ideal) nondiffracting waves is furnishing a precise mathematical definition of these solutions, so to extract the necessary spectral structure from them. Intuitively, an ideal nondiffracting wave (beam or pulse) can be defined as a wave capable of maintaining indefinitely its spatial form (apart from local variations) while propagating. We can express such a characteristic by saying that a localized wave has to possess the property (Besieris, Shaarawi, and Ziolkowski, 1989; Zamboni-Rached, 1999). ( (ρ, φ, z, t) = ρ, φ, z + z, t + z ), (28) V where z is a certain length and V is the pulse propagation speed that here can assume any value: V. In terms of a Fourier-Bessel (FB) expansion, we can write a function (ρ, φ, z, t) as (ρ, φ, z, t) = n= [ dk ρ dk z dω k ρ A n (k ρ, k z, ω)j n (k ρ ρ)e ik zz e iωt e inφ ]. (29) On using the translation property of the Fourier transforms T[ f (x + a)] =exp(ika)t[ f (x)], we have that A n (k ρ, k z, ω) and exp[i(k z z ω z / V)]A n (k ρ, k z, ω) are the FB transforms of the left-hand side and right-hand side functions in Eq. (28). From this same equation we can obtain (Besieris, Shaarawi, and Ziolkowski, 1989; Zamboni-Rached, 1999) the fundamental constraint linking the angular frequency ω and the longitudinal wave number k z : ω = Vk z + 2mπ V z, (3) with m an integer. Obviously, this constraint can be satisfied by means of the spectral functions A n (k ρ, k z, ω). Now, let us explicitly mention that constraint (3) does not imply any breakdown of the wave equation. In fact, when inserting expression (29)

46 28 Erasmo Recami and Michel Zamboni-Rached in the wave equation (27), we derive ω 2 c 2 = k2 z + k2 ρ. (31) So, to obtain a solution of the wave equation by expression (29), the spectrum A n (k ρ, k z, ω) must possess the form ( )] A n (k ρ, k z, ω) = A n (k z, ω) δ [k 2ρ ω 2 c 2 k2 z, (32) where δ(.) is the Dirac delta function. With this, we can write a solution of the wave equation as (ρ, φ, z, t) = n= ρ dω ω/c ω/c dk z A n (k z, ω)j n ω 2 c 2 k2 z e ikzz e iωt e inφ (33) where we have considered positive angular frequencies only. Equation (33) is a superposition of Bessel beams, and it is understood that the integrations in the (ω, k z ) plane are confined to the region ω and ω/c k z ω/c. Now, to obtain an ideal nondiffracting wave, the spectra A n (k z, ω) must obey the fundamental constraint in Eq. (3), and so we write A n (k z, ω) = m= S nm (ω)δ [ω (Vk z + b m )], (34) where b m are constants representing the terms 2mπV/ z in Eq. (3), and S nm (ω) are arbitrary frequency spectra. By inserting Eq. (34) into Eq. (33), we get a general integral form of the ideal nondiffracting wave in Eq. (28): (ρ, φ, z, t) = ψ nm (ρ, φ, z, t) (35) n= m=

47 Localized Waves: A Review 281 with (ωmax ) m ψ nm (ρ, φ, z, t) = e ib mz/v dω S nm (ω) (ω min ) m ( 1 J n ρ c 2 1 ) V 2 ω 2 + 2b V 2 ω b2 V 2 e i V ω (z Vt) e inφ, (36) where ω min and ω max depend on the values of V as specified below: for subluminal (V < c) localized waves: b m >, (ω min ) m = cb m /(c + V) and (ω max ) m = cb m /(c V); for luminal (V = c) localized waves: b m >, (ω min ) m = b m /2 and (ω max ) m = ; for superluminal (V > c) localized waves: b m, (ω min ) m = cb m /(c + V) and (ω max ) m =.Orb m <, (ω min ) m = cb m /(c V) and (ω max ) m =. It is important to notice that each ψ nm (ρ, φ, z, t) in the superposition in Eq. (35) is a truly nondiffracting wave (beam or pulse) and the superposition of them [Eq. (35)] is just the most general form to represent a nondiffracting wave defined by Eq. (28). Due to this fact, the search for methods capable of providing analytic solutions for ψ nm (ρ, φ, z, t), [Eq. (36)], becomes an important task. Recall that Eq. (36) is also a Bessel beam superposition, but with constraint (3) linking the angular frequencies and longitudinal wave numbers. Despite the fact that Eq. (36) represents ideal nondiffracting waves, it is difficult to obtain closed analytic solutions from it. Because of this, we develop a method capable of overcoming such a difficulty, providing several interesting LW solutions (luminal and superluminal) of arbitrary frequencies, including some solutions endowed with finite energy The Generalized Bidirectional Decomposition For reasons that will be clear soon, instead of dealing with the integral expression (35), our starting point is the general expression in Eq. (33). Here, for simplicity, we restrict ourselves to axially symmetric solutions, adopting the spectral functions where δ n is the Kronecker delta. A n (k z, ω) = δ n A(k z, ω), (37)

48 282 Erasmo Recami and Michel Zamboni-Rached In this way, we get the following general solution (considering positive angular frequencies only), which describes axially symmetric waves: (ρ, φ, z, t) = dω ω/c ω/c dk z A(k z, ω)j ρ ω 2 c 2 k2 z e ikzz e iωt. (38) As we have seen, we can obtain ideal nondiffracting waves, given that the spectrum A(k z, ω) satisfies the linear relationship in Eq. (3). Therefore, it is natural to choose new spectral parameters, in place of (ω, k z ), that make it easier to implement the mentioned constraint (Besieris, Shaarawi, and Ziolkowski, 1989; Zamboni-Rached, 1999). With this in mind, let us choose the new spectral parameters (α, β) α 1 2V (ω + Vk z); β 1 2V (ω Vk z). (39) Let us consider here only luminal (V = c) and superluminal (V > c) nondiffracting pulses. With the change of variables [Eq. (39)] in the integral solution [Eq. (38)], and considering (V c), the integration limits on α and β must satisfy the three inequalities below: <α+ β< α c V c + V β (4) α c + V c V β. Let us suppose that both α and β are positive [α, β ]. The first inequality in Eq. (4) is then satisfied, whereas the coefficients (c V)/(c + V) and (c + V)/(c V) entering relations Eq. (4) are both negative (since V c). As a consequence, the other two inequalities in Eq. (4) are automatically satisfied. In other words, the integration limits α and β are contained in the limits of (4) and are therefore acceptable. Indeed, they constitute a rather suitable choice for facilitating all the subsequent integrations. Therefore, instead of Eq. (38), we shall consider the (more easily integrable) Bessel beam superposition in the new variables [with V c]

49 Localized Waves: A Review 283 (ρ, ζ, η) = dα where we have defined dβa(α, β) J (V ) ( ) 2 V ρ c 2 1 (α 2 + β 2 2 ) + 2 c αβ e iαζ e iβη, (41) ζ z Vt; η z + Vt. (42) This procedure is a generalization of the so-called bidirectional decomposition technique (Besieris, Shaarawi, and Ziolkowski, 1989), which was devised in the past for V = c. From the new spectral parameters defined in transformation (39), it is easy to see that the constraint in Eq. (3), i.e., ω = Vk z + b, is implemented just by making A(k z, ω) A(α, β) = S(α)δ(β β ) (43) with β = b/2v. The delta function δ(β β ) in the spectrum in Eq. (43) means that we are integrating Bessel beams along the continuous line ω = Vk z + 2Vβ and, in this way, the function S(α) will give the frequency dependence of the spectrum: S(α) S(ω/V β ). This method constitutes a simple, natural way for obtaining pulses with field concentration on ρ = and at ζ = z = Vt. It is important to emphasize (Zamboni-Rached, Recami, and Hernández-Figueroa, 22) that, when β > in Eq. (43), superposition in Eq. (41) gets contributions from both backward and forward traveling Bessel beams, corresponding to the frequency intervals Vβ ω<2vβ (where k z < ) and 2Vβ ω (where k z ), respectively. Nevertheless, we can obtain physical solutions when rendering the contribution of the backward-traveling components negligible by choosing suitable weight functions S(α). It is also noteworthy that we adopted the new spectral parameters α and β just to obtain (closed-form) analytic localized wave solutions. The spectral characteristics of these new solutions can be brought into evidence by using transformations in Eq. (39) and writing the corresponding spectrum in terms of the usual ω and k z spectral parameters. The following subsections consider some cases with β = and β >.

50 284 Erasmo Recami and Michel Zamboni-Rached Closed analytic expressions describing some ideal nondiffracting pulses. Let us first consider, in Eq. (41), spectra of the type in Eq. (43) with β = : A(α, β) = av δ(β)e avα (44) A(α, β) = av δ(β)j (2d α)e avα (45) A(α, β) = δ(β) sin(dα) e avα, (46) α a > and d being constants. We can obtain from the above spectra the following superluminal LW solutions, respectively: from spectrum (44), we can use the identity ( ) in Gradshteyn and Ryzhik (1965), obtaining the well-known ordinary X-wave solution (also called X-shaped pulse) (ρ, ζ) X = av (av iζ) 2 + ( ) ; (47) V 2 1 ρ c 2 2 by using spectrum (45) and the identity (6.6444) of Gradshteyn and Ryzhik (1965), one gets (ρ, ζ) = XJ V 2 [ c 2 1 (av) 2 d 2 X 2 ρ exp (av iζ) (av) 2 d 2 X 2] ; (48) the superluminal nondiffracting pulse [ ( (ρ, ζ) = sin 1 2 d X av 2 + (d/av) 2 + 2ρd(aV) V 2 2 /c 2 1 (49) ) 1 + X 2 + (d/av) 2 2ρd(aV) V 2 2 /c 2 1 is obtained from spectrum (46) by using identity ( ) of Gradshteyn and Ryzhik (1965) for a > and d >.

51 Localized Waves: A Review 285 From the previous discussion, we learn that any solutions obtained from spectra of the type in Eq. (43) with β = are free of noncausal (backwardtraveling) components. In addition, when β =, the pulsed solutions depend on z and t through ζ = z Vt only, and so propagate rigidly (i.e., without distortion). Such pulses can be transversally localized only if V > c, because if V = c the function must obey the Laplace equation on the transverse planes (Besieris, Shaarawi, and Ziolkowski, 1989; Zamboni-Rached, 1999). Many others superluminal LWs can be easily constructed (Zamboni- Rached, Recami, and Hernández-Figueroa, 22) from the above solutions simply by taking the derivatives (of any order) with respect to ζ. It is also possible to show that the new solutions, obtained in this way, have their spectra shifted toward higher frequencies. Now, let us pass to consider, in Eq. (41), a spectrum of the type in Eq. (43) with β > : A(α, β) = avδ(β β ) e avα (5) with a a positive constant. The presence of the delta function, with the constant β >, implies that we are integrating (summing) Bessel beams along the continuous line ω = Vk z + 2Vβ. Now, the function S(α) = avexp( avω) entails that we are considering a frequency spectrum of the type S(ω) exp( aω), and therefore with a bandwidth given by ω = 1/a. Since β >, the interval Vβ ω<2vβ (or, equivalently, in this case, α<β ), corresponds to backward Bessel beams negative values of k z. However, we can obtain physical solutions by making the contribution of this frequency interval negligible. In our case, this can be obtained by making aβ V << 1, so that the exponential decay of the spectrum S with respect to ω is very slow, and the contribution of the interval ω 2Vβ (where k z ) largely overruns the Vβ ω<2vβ (where k z < ) contribution. Incidentally, once we ensure the causal behavior of the pulse by making avβ << 1 in Eq. (5), we have that α = 1/aV >> β, and one can therefore simplify the argument of the Bessel function, in the integrand of superposition in Eq. (41), by neglecting the term (V 2 /c 2 1)β 2. With this, the superposition in Eq. (41), with the spectrum (5), can be written as (V ) ( ) (ρ, ζ, η) ave iβ 2 η V dαj ρ c 2 1 α c αβ e iαζ e avα. (51)

52 286 Erasmo Recami and Michel Zamboni-Rached Now, we can use identity ( ) of Gradshteyn and Ryzhik (1965) and obtain the new localized superluminal solution called (Zamboni-Rached, Recami, and Hernández-Figueroa, 22) Superluminal Focus Wave Mode (SFWM): SFWM (ρ, ζ, η) = e iβ η X exp [ β (V 2 + c 2 ) ( V 2 c 2 (av iζ) avx 1)], (52) where, as before, X is the ordinary X-pulse in Eq. (47). The center of the SFWM is located on ρ = and ζ = (i.e., at z = Vt). The intensity, 2,of this pulse propagates rigidly, being a function of ρ and ζ only. However, the complex function SFWM (i.e., its real and imaginary parts) propagate with local variations, recovering their entire 3-D form after each space and time interval z = π/β and t = π/β V. The SFWM solution above, for V c +, reduces to the well-known Focus Wave Mode (FWM) solution (Besieris, Shaarawi, and Ziolkowski, 1989), traveling with speed c: [ ] FWM (ρ, ζ, η) = ac e iβ η ac iζ exp β ρ 2. (53) ac iζ We also emphasize that, since β >, Eq. (5) corresponds to angular frequencies ω Vβ. Thus, our new solution also can be used to construct high-frequency pulses. Finite-energy nondiffracting pulses. In this subsection, we show how to obtain finite-energy LW pulses that can propagate for long distances while maintaining their spatial resolution (i.e., that possess a large depth of field). Ideal nondiffracting waves can be constructed by superposing Bessel beams [see Eq. (38) for cylindrical symmetry] with a spectrum A(ω, k z ) that satisfies a linear relationship between ω and k z. In the general bidirectional decomposition method, this can be obtained by using spectra of the type in Eq. (43) in superposition (41). Solutions of that type possess an infinite depth of field; however, they are endowed with infinite energy. To overcome this problem, we can truncate an ideal nondiffracting wave by a finite aperture, and the resulting pulse will have finite energy and a finite field depth. Even so, such field depths may be very large compared with those of ordinary waves.

53 Localized Waves: A Review 287 The problem in the present case is that the resulting field must be calculated by diffraction integrals (such as the well-known Rayleigh Sommerfeld formula) and, in general, a closed analytic formula for the resulting pulse cannot be obtained. However, there is another way to construct localized pulses with finite energy (Zamboni-Rached, Recami, and Hernández-Figueroa, 22) namely, by using spectra A(ω, k z ) in Eq. (38), whose domains are not restricted exactly to the straight line ω = Vk z + b, but are defined in the surroundings of that line, wherein the spectra should have their main values concentrated (in other words, any spectrum must be well localized in the vicinity of that line). Similarly, in terms of the generalized bidirectional decomposition given in Eq. (41), finite-energy nondiffracting wave pulses can be constructed by adopting spectral functions A(α, β) well localized in the vicinity of the line β = β, quantity β being a constant. To exemplify this method, let us consider the following spectrum: aqve avα e q(β β ) for β β A(α, β) = (54) for β<β in superposition (41), quantities a and q being free positive constants and V the peak s pulse velocity (here, V c). It is easy to see that the above spectrum is zero in the region above the β = β line, while it decays in the region below (as well as along) such a line. We can concentrate this spectrum on β = β by choosing values of q in such a way that qβ >> 1. The faster the spectrum decay takes place in the region below the β = β line, the larger is the field depth of the corresponding pulse. Once we choose qβ >> 1 to obtain pulses with a large field depth, we also can minimize the contribution of the noncausal (backward) components by choosing avβ << 1, similar to the SFWM case. Again by analogy with the SFWM case, when we choose qβ >> 1 (i.e., a long field depth) and avβ << 1 (a minimal contribution of the backward components), we can simplify the argument of the Bessel function, in the integrand of superposition (41), by neglecting the term (V 2 /c 2 1)β 2. With the help of the observations above, one can write the superposition (41), with spectrum (54), as (V ) ( ) 2 V (ρ, ζ, η) aqv dβ dα J ρ c 2 1 α c αβ β e iβη e iαζ e q(β β ) e avα (55)

54 288 Erasmo Recami and Michel Zamboni-Rached and, using identity ( ) given in Gradshteyn and Ryzhik (1965), we get [ (ρ, ζ, η) qx dβe q(β β) e iβη exp β V2 +c 2 ( β V 2 c 2 av iζ avx 1)], (56) which can be viewed as a superposition of the SFWM pulses [see Eq. (52)]. The above integration can be performed easily and results (Zamboni- Rached, Recami, and Hernández-Figueroa, 22) in the so-called Superluminal Modified Power Spectrum (SMPS) pulse: SMPS (ρ, ζ, η) = qx exp[(y iη)β ], (57) q (Y iη) where X is the ordinary X-pulse [Eq. (47)] and Y is defined by Y V2 + c 2 V 2 c 2 [ (av iζ) avx 1]. (58) The SMPS pulse is a superluminal LW, with field concentration around ρ = and ζ = (i.e., z = Vt), and with finite total energy. We will show that the depth of field, Z, of this pulse is given by Z SMPS = q/2. An interesting property of the SMPS pulse is related to its transverse width (the transverse spot size at the pulse center). It can be shown from Eq. (57) that, for the cases where av << 1/β and qβ >> 1 (i.e., for the cases considered by us), the transverse spot size, ρ, of the pulse center (ζ = ) is determined by the exponential function in Eq. (57) and is given by av ρ = c β (V 2 + c 2 ) + V 2 c 2 4β 2(V2 + c 2 ) 2, (59) which clearly does not depend on z, and so remains constant during its propagation. In other words, even though the SMPS pulse incurs an intensity decrease during propagation, it preserves its transverse spot size. This interesting characteristic is not met in ordinary pulses (like the Gaussian ones) where the amplitude of the pulse decreases and the width increases by the same factor. Figure 28 shows the intensity of a SMPS pulse, with β = 33 m 1, V = 1.1c, a = 1 12 s, and q = 1 5 m, at two different moments, for t = and

55 Localized Waves: A Review C SMPS 2.9 C 4 SMPS x(mm) y(mm) C SMPS 2 C SMPS y(mm) x (mm) (mm) (mm) z(mm) (mm) (a) (b) FIGURE 28 Representation of a Superluminal Modified Power Spectrum Pulse [Eq. (57)]. Its total energy is finite (even without any truncation), and so it is deformed while propagating since its amplitude decreases with time. Panel (a) represents, for t =, the pulse corresponding to β = 33m 1, V = 1.1c, a = 1 12 s, and q = 1 5 m. Panel (b) shows the same pulse after traveling 5km after 5 km of propagation where the pulse becomes less intense (precisely, with half of its initial peak intensity). Despite the intensity decrease, the pulse maintains its transverse width, as seen in the 2D plots in Figure 28, which show the field intensities in the transverse sections at z = and z = q/2 = 5 km. Three other important well-known finite-energy nondiffracting solutions can be obtained directly from the SMPS pulse: The first one, obtained from Eq. (57) by making β =, is the so-called (Zamboni-Rached, Recami, and Hernández-Figueroa, 22) superluminal splash pulse (SSP) SSP (ρ, ζ, η) = qx q + iη Y. (6) The other two pulses are luminal. By taking the limit V c + in the SMPS pulse of Eq. (57), we get the well-known (Besieris, Shaarawi, and Ziolkowski, 1989) luminal modified power spectrum (MPS) pulse ( ) aqce iβ η MPS (ρ, ζ, η) = (q + iη)(ac iζ) + ρ 2 exp β ρ 2. (61) ac iζ Finally, by taking the limit V c + and making β = in the SMPS pulse [or, equivalently, by making β = in the MPS of pulse in Eq. (61), or, instead, by taking the limit V c + in the SSP of Eq. (6)], we obtain the well-known (Besieris, Shaarawi, and Ziolkowski, 1989) luminal splash

56 29 Erasmo Recami and Michel Zamboni-Rached pulse (SP) solution SP (ρ, ζ, η) = aqc (q + iη)(ac iζ) + ρ 2. (62) It is also interesting to notice that the X and SFWM pulses can be obtained from the SSP and SMPS pulses (respectively) by making q in Eqs. (6) and (57). The solutions SSP and SMPS can be viewed as the finite-energy versions of the X and SFWM pulses, respectively. Some characteristics of the SMPS pulse. Let us examine the on-axis (ρ = ) behavior of the SMPS pulse. On ρ =, we have SMPS (ρ =, ζ, η) = aqve iβ z [(av iζ)(q + iη)] 1. (63) From this expression, we can show that the longitudinal localization z, for t =, of the SMPS pulse square magnitude is z = 2aV. (64) If we now define the field depth Z as the distance over which the pulse s peak intensity is at least 5% of its initial value, 4 then we can obtain [from Eq. (63)] the depth of field Z SMPS = q 2, (65) which depends only on q, as we expected since q regulates the concentration of the spectrum around the line ω = Vk z + 2Vβ. Now, let us examine the maximum amplitude M of the real part of Eq. (63), which for z = Vt writes (ζ = and η = 2z): M SMPS Re[ SMPS (ρ =, z = Vt)] = cos(2β z) 2(z/q) sin(2β z) 1 + 4(z/q) 2. Initially, for z =, t =, one has M = 1 and can also infer that: (i) when z/q << 1, namely, when z << Z, Eq. (66) becomes (66) M SMPS cos(2β z) for z << Z (67) 4 We can expect that, while the pulse peak intensity is maintained, the pulse also keeps its spatial form.

57 Localized Waves: A Review 291 and the pulse s peak actually oscillates harmonically with wavelength z = π/β and period t = π/vβ, all along its field depth; (ii) when z/q >> 1, namely, z >> Z, Eq. (66) becomes M SMPS sin(2β z) 2z/q for z >> Z. (68) Therefore, beyond its depth of field, the pulse continues oscillating with the same z, but its maximum amplitude decays proportionally to z. The next two sections show applications of the LW pulses Space-Time Focusing of X-Shaped Pulses This section shows how any known superluminal solution can be used to obtain a large number of analytic expressions for space-time focused waves, endowed with a very strong intensity peak at the desired location. The method presented here is a natural extension of that developed by Shaarawi et al. (Shaarawi, Besieris, and Said, 23), where the space-time focusing was achieved by superimposing a discrete number of ordinary X-waves, characterized by different values θ of the axicon angle. In this section, based on (Zamboni-Rached, Shaarawi, and Recami, 24), we proceed to more efficient superpositions for varying velocities V, related to θ through the known (Brittingham, 1983; Durnin, Miceli, and Eberly, 1987a; Recami, 1986) relation V = c/cos θ. This enhanced focusing scheme has the advantage of yielding analytic (closed-form) expressions for spatiotemporally focused pulses. We begin by considering an axially symmetric ideal nondiffracting superluminal wave pulse ψ(ρ, z Vt) in a dispersionless medium, where V = c/cos θ>c is the pulse velocity and θ is the axicon angle. As shown previously, pulses like these can be obtained by a suitable frequency superposition of Bessel beams. Suppose that we have now N waves of the type ψ n (ρ, z V n (t t n )), with different velocities c < V 1 < V 2 <...< V N, and emitted at (different) times t n ; quantities t n being constants, while n = 1, 2,..., N. The center of each pulse is located at z = V n (t t n ). To obtain a highly focused wave, we need all the wave components ψ n (ρ, z V n (t t n )) to reach the given point, z = z f, at the same time t = t f. On choosing t 1 = for the slowest pulse ψ 1, it is easily seen that the peak of this pulse reaches the point z = z f at the time t f = z f /V 1. So we obtain that, for each ψ n, the instant of emission t n must be t n = ( 1 V 1 1 V n ) z f. (69)

58 292 Erasmo Recami and Michel Zamboni-Rached With this in mind, we can construct other exact solutions to the wave equation, given by (ρ, z, t) = Vmax V min ( ( ( 1 dva(v)ψ ρ, z V t 1 )) )z f, (7) V min V where V is the velocity of the wave ψ(ρ, z Vt) that enters the integrand of Eq. (7). While integrating, V is considered a continuous variable in the interval [V min, V max ]. In Eq. (7), function A(V) is the velocity distribution that specifies the contribution of each wave component (with velocity V)to the integration. The resulting wave (ρ, z, t) can have a more or less strong amplitude peak at z = z f, at time t f = z f /V min, depending on A(V) and on the difference V max V min. Notice that the resulting wave field will propagate with a superluminal peak velocity, also depending on A(V). When the velocity-distribution function is well concentrated around a certain velocity value, the wave in Eq. (7) can be effected to increase its magnitude and spatial localization while propagating. Finally, the pulse peak acquires its maximum amplitude and localization at the chosen point z = z f, and at time t = z f /V min, as we know. Afterward, the wave suffers a progressing spreading, and a decreasing of its amplitude Focusing Effects by Using Ordinary X-Waves Here, we present a specific example by integrating Eq. (7) over the standard, classic (Lu and Greenleaf, 1992a) X-waves, X = av[(av i(z Vt)) 2 + (V 2 /c 2 1)ρ 2 ] 1/2. When using this ordinary X-wave, the largest spectral amplitudes are obtained for low frequencies. For this reason, one may expect that the solutions considered below will be suitable mainly for low-frequency applications. Let us choose, then, the function ψ in the integrand of Eq. (7) to be ψ(ρ, z, t) X(ρ, z V(t (1/V min 1/V)z f )), viz.: ψ(ρ, z, t) X = av [ ( ( ( ) ))] 2 ( ). av i z V t 1 V min V 1 z f + V 2 1 ρ c 2 2 (71) After some manipulations, one obtains the analytic integral solution (ρ, z, t) = Vmax V min av A(V) dv (72) PV 2 + QV + R

59 Localized Waves: A Review 293 with P [ ( a + i( t ( Q 2 t )) 2 z f V min + ρ 2 ) z f V min ai (z z f ) c 2 ] (73) R [ (z z f ) 2 ρ 2]. In the following examples, we illustrate the behavior of some new spatiotemporally focused pulses, by taking into consideration a few different velocity distributions A(V). These new pulses are closed analytic exact solutions of the wave equation. Example 1. Let us consider our integral solution in Eq. (72) with A(V) = 1s/m. In this case, the contribution of the X-waves is the same for all velocities in the allowed range [V min, V max ]. By using identity listed in (Gradshteyn and Ryzhik, 1965), we get the particular solution (ρ, z, t) = a ( ) PVmax P 2 + QV max + R PVmin 2 + QV min + R + aq 2P 3/2 ln 2 P(PVmin 2 + QV min + R) + 2PV min + Q 2 P(PVmax 2 + QV, max + R) + 2PV max + Q (74) where P, Q, and R are given in Eq. (73). A 3D plot of this function is shown in Figure 29; where we have chosen a = 1 12 s, V min = 1.1 c, V max = 1.5 c, and z f = 2 cm. This solution exhibits a rather evident space-time focusing. An initially spread-out pulse (shown for t = ) becomes highly localized at t = t f = z f /V min = 6.66 ns, the pulse peak amplitude at z f being 4.82 times greater than the initial one. In addition, at the focusing time t f the field is much more localized than at any other time. The velocity of this pulse is approximately V = 1.3 c. Example 2. In this case, we choose A(V) = 1/V s/m, and, using the identity in (Gradshteyn and Ryzhik, 1965), Eq. (72) yields (ρ, z, t) = a ln 2 P(PVmax 2 + QV max + R) + 2PV max + Q. P 2 P(PVmin 2 + QV min + R) + 2PV min + Q (75)

60 294 Erasmo Recami and Michel Zamboni-Rached C 2 t 5 ns t ns t ns (m) z (m) (m) z(m) (m) z (m) C 2 C 2.1 C t ns t ns t ns (m) z (m) (m) z(m) (m) z (m) C 2 FIGURE 29 Space-time evolution of the superluminal pulse represented by Eq. (74); the chosen parameter values are a = 1 12 s, V min = 1.1 c, V max = 1.5 c, while the focusing point is at z f = 2 cm. This solution is associated with a rather good spatiotemporal focusing. The field amplitude at z = z f is 4.82 times larger than the initial one. The field amplitude is normalized at the space-time point ρ =, z = z f, t = t f. C 2 Other exact closed-form solutions can be obtained Zamboni-Rached, Shaarawi, and Recami (24) by considering, for instance, velocity distributions like A(V) = 1/V 2 and A(V) = 1/V 3. Again, we can construct many other spatiotemporally focused pulses from the above solutions simply by taking their time derivatives (of any order). It also is possible to show (Zamboni-Rached, Shaarawi, and Recami, 24) that the new solutions obtained in this way have their spectra shifted toward higher frequencies Chirped Optical X-Type Pulses in Material Media The theory of LWs was initially developed for free space (vacuum). In 1996, Sõnajalg et al. (Sõnajalg and Saari, 1996) showed that the LW theory can be extended to include (unbounded) dispersive media. This was obtained by making the axicon angle of the Bessel beams vary with the frequency (Lu and Greenleaf, 1992a; Sõnajalg and Saari, 1996; Zamboni- Rached et al., 23) in such a way that a suitable frequency superposition of these beams compensates for the material dispersion. Soon after this idea was reported, many interesting nondiffracting/nondispersive pulses were obtained theoretically (Lu and Greenleaf, 1992a; Sõnajalg and Saari, 1996; Zamboni-Rached et al., 23) and experimentally (Sõnajalg and Saari, 1996). Despite the remarkable importance of such an extended method working well in theory its experimental implementation is not so simple. 5 5 We refer readers to references (Lu and Greenleaf, 1992a; Sõnajalg and Saari, 1996; Zamboni-Rached et al., 23) for a description, theoretical and experimental, of that extended method.

61 Localized Waves: A Review 295 In 24 Zamboni-Rached et al. (Zamboni-Rached, Hernández-Figueroa, and Recami, 24) developed a simpler way to obtain pulses capable of recovering their spatial shape, both transversally and longitudinally, after some propagation. It consisted of using chirped optical X-typed pulses, while keeping the axicon angle fixed. Recall that, by contrast, chirped Gaussian pulses in unbounded material media may recover only their longitudinal shape, since they undergo progressive transverse spreading while propagating. The present section is devoted to this approach. She start with an axially symmetric Bessel beam in a material medium with refractive index n(ω): ψ(ρ, z, t) = J (k ρ ρ) exp(iβz) exp( iωt), (76) which must obey the condition kρ 2 = n2 (ω)ω 2 /c 2 β 2, which connects the transverse and longitudinal wave numbers k ρ and β, and the angular frequency ω. In addition, we impose that kρ 2 and ω/β to avoid a nonphysical behavior of the Bessel function J (.) and to confine ourselves to forward propagation only. Once the conditions above are satisfied, we have the liberty of writing the longitudinal wave number as β = (n(ω)ω cos θ)/c and, therefore, k ρ = (n(ω)ω sin θ)/c, where (as in the free space case) θ is the axicon angle of the Bessel beam. Now we can obtain an X-shaped pulse by performing a frequency superposition of these Bessel beams, with β and k ρ given by the previous relations: ( ) n(ω)ω (ρ, z, t) = S(ω) J sin θρ exp[iβ(ω)z] exp( iωt) dω, c (77) where S(ω) is the frequency spectrum and the axicon angle is kept constant. The phase velocity of each Bessel beam in our superposition [Eq. (77)] is different and given by V phase = c/(n(ω) cos θ). So, the pulse represented by Eq. (77) will suffer dispersion during its propagation. As we said the method developed by Sõnajalg et al. (Sõnajalg and Saari, 1996), and explored by others (Sõnajalg and Saari, 1996; Zamboni-Rached et al., 23), to overcome this problem consisted of regarding the axicon angle θ as a function of the frequency in order to obtain a linear relationship between β and ω. Here, however, we choose to work with a fixed axicon angle, and we need to find another way to avoid dispersion and diffraction all along a certain propagation distance. To do so, we might choose a chirped Gaussian spectrum S(ω) in Eq. (77). S(ω) = T 2π(1 + ic) exp[ q 2 (ω ω ) 2 ] with q 2 = T 2 2(1 + ic), (78)

62 296 Erasmo Recami and Michel Zamboni-Rached where ω is the central frequency of the spectrum, T is a constant related with the initial temporal width, and C is the chirp parameter (we chose as temporal width the half-width of the relevant Gaussian curve when its heigth equals 1/e times its full heigth). Unfortunately, there is no analytic solution to Eq. (77) with S(ω) given by Eq. (78), so that some approximations are to be made. Then, let us assume that the spectrum S(ω), surrounding of the carrier frequency ω, is narrow enough to guarantee that ω/ω << 1, to ensure that β(ω) can be approximated by the first three terms of its Taylor expansion in the vicinity of ω ; that is, β(ω) β(ω ) + β (ω) ω (ω ω ) + (1/2)β (ω) ω (ω ω ) 2 ; when, after using β = n(ω)ω cos θ/c, it results that β ω = cos θ [ n(ω) + ω n ] ; c ω [ ] 2 β ω 2 = cos θ 2 n c ω + ω 2 n ω 2. (79) As we know, β (ω) is related to the pulse group velocity by the relation V g = 1/β (ω). Here we can see the difference between the group velocity of the X-type pulse (with a fixed axicon angle) and that of a standard Gaussian pulse. Such a difference is due to the factor cos θ in Eq. (79). Because of it, the group velocity of our X-type pulse is always greater than that of the Gaussian. In other words, (V g ) X = (1/ cos θ)(v g ) gauss. We also know that the second derivative of β(ω) is related to the group velocity dispersion (GVD) β 2 by β 2 = β (ω). The GVD is responsible for the temporal (longitudinal) spreading of the pulse. Here the GVD of the X-type pulse is always smaller than that of the standard Gaussian pulses due to the factor cos θ in Eq. (79) namely; (β 2 ) X = cos θ(β 2 ) gauss. On using the above results, we can write (ρ, z, t) = T exp[iβ(ω )z] exp( iω t) 2π(1 + ic) { exp i (ω ω ) [ z Vg t ] } exp V g dω J ( n(ω)ω c { ) sin θρ [ ]} (ω ω ) 2 iβ2 2 z q2. (8) The integral in Eq. (8) cannot be evaluated analytically, but for us it is sufficient to obtain the pulse behavior. Let us analyze the pulse for ρ =. In this case, we get (ρ =, z, t) = T exp[iβ(ω )z] exp( iω t) T 2 iβ 2(1 + ic)z exp [ (z Vg t) 2 (1 + ic) 2V 2 g [T2 iβ 2(1 + ic)z] ]. (81)

63 Localized Waves: A Review 297 From Eq. (81) one can immediately see that the initial temporal width of the pulse intensity is T and that, after a propagation distance z, the time-width T 1 becomes T 1 T = ( 1 + Cβ 2z T 2 ) 2 ( ) 2 β 2 z + T 2 1/2. (82) Equation (82) describes the pulse spreading behavior. Such behavior depends on the sign (positive or negative) of the product β 2 C, as is well known to happen for standard Gaussian pulses (Agrawal, 26). In the case β 2 C >, the pulse will monotonically become broader and broader with the distance z. On the other hand, if β 2 C <, the pulse will, in a first stage, narrow and then (during the rest of its propagation) it will spread. Thus, there will be a certain propagation distance AT after which the pulse will recover its initial temporal width (T 1 = T ). From Eq. (82), we can find such a distance Z T1=T (considering β 2 C < ) to be Z T1 =T = 2CT2 β 2 (C 2 + 1). (83) The maximum distance at which our chirped pulse, with given T and β 2, may recover its initial temporal width can be easily evaluated from Eq. (83), and it is L disp = T 2 /β 2. We call such a maximum value L disp the dispersion length : It is the maximum distance the X-type pulse may travel while recovering its initial longitudinal shape. Obviously, if we want the pulse to reassume its longitudinal shape at some desired distance z < L disp, we need only to suitably choose a new value for the chirp parameter. The property of recovering its own initial temporal (or longitudinal) width may be verified to exist also in the case of chirped standard Gaussian pulses. However, the latter suffer progressive transverse spreading, which is not reversible. The distance at which a Gaussian pulse doubles its initial transverse width w is z diff = 3πw 2 /λ, where λ is the carrier wavelength. Thus, optical Gaussian pulses with great transverse localization are spoiled within a few centimeters or even less. We now show that it is possible to recover the transverse shape of the chirped X-type pulse intensity; actually, it is possible to recover its entire spatial shape after a distance Z T1 =T. To do so, let us return to the integral solution in Eq. (8) and perform the change of coordinates (z, t) ( z, t c = z c /V g ), with z = z c + z t = t c z c V g, (84)

64 298 Erasmo Recami and Michel Zamboni-Rached where z c is the center of the pulse ( z is the distance from such a point), and t c is the time at which the pulse center is located at z c. We compare our integral solution in Eq. (8), when z c = (initial pulse), with that when z c = Z T1 =T = 2CT 2 /(β 2(C 2 + 1)). In this way, solution (8) can be written, when z c =, as (ρ, z c =, z) = T [ ] exp(iβ z) T 2 dω J (k ρ (ω)ρ) exp (ω ω ) 2 2π(1 + ic) 2(1 + C 2 ) { [ (ω ω ) z exp i + (ω ω ) 2 β 2 z + (ω ω ) 2 T 2C ]} V g 2 2(1 + C 2, (85) ) where we have taken the value q given by Eq. (78). To verify that the pulse intensity recovers its entire original form at z c = Z T1 =T = 2CT 2/ [β 2 (C 2 + 1)], we can analyze our integral solution at that point, obtaining [ ]} T exp {iβ z c z cz c cos θ n(ω )V g (ρ, z c = Z T1 =T, z) = 2π(1 + ic) [ ] T 2 dω J (k ρ (ω)ρ) exp (ω ω ) 2 2(1 + C 2 ) { [ ]} exp i (ω ω ) z + (ω ω ) 2 β 2 z + (ω ω ) 2 T 2C V g 2 2(1 + C 2 ), (86) where we put z = z. In this way, one immediately sees that (ρ, z c =, z) 2 = (ρ, z c = Z T1 =T, z) 2. (87) Therefore, from Eq. (87) it is clear that the intensity of a chirped optical X-type pulse is able to recover its original 3D shape, just with a longitudinal inversion at the pulse center. This method thus is a simple and effective procedure for compensating diffraction and dispersion in an unbounded material medium, and it is a method simpler than varying the axicon angle with the frequency. We note again that one can determine the distance z = Z T1 =T L disp at which the pulse resumes its spatial shape by choosing a suitable value of the chirp parameter.

65 Localized Waves: A Review 299 We have shown that the chirped X-type pulse recovers its 3D shape after some distance, and we also have obtained an analytic description of the pulse longitudinal behavior (for ρ = ) during propagation, by means of Eq. (81). However, we have not yet obtained the same information about the pulse transverse behavior. We just learned that it will be recovered at z = Z T1 =T. To complete the picture, we should also determine the transverse behavior in the plane of the pulse center z = V g t. We would then obtain quantitative information about the evolution of the pulse shape during its entire propagation. However, we do not expound all the relevant mathematical details here; let us state only that the transverse behavior of the pulse (in the plane z = z c = V g t), during its entire propagation, can approximately be described by (ρ, z = z c, t = z c /V g ) T exp[iβ(ω )z] exp( iω t) 2π(1 + ic) [ ] tan 2 θρ 2 exp 8 Vg 2( iβ 2z c /2 + q 2 ) iβ2 z c /2 + q 2 [ ( ( ) n(ω )ω sin θρ tan Ɣ(1/2)J )I 2 θρ 2 c 8 Vg 2( iβ 2z c /2 + q 2 ) + 2 p=1 2 p ( ) Ɣ(p + 1/2)Ɣ(p + 1) n(ω )ω sin θρ J 2p I 2p Ɣ(2p + 1) c ( )] tan 2 θρ 2 8 Vg 2( iβ 2z c /2 + q 2, (88) ) where I p (.) is the modified Bessel function of the first kind of order p, quantity Ɣ(.) being the gamma function, and q being given by Eq. (78). [Readers can check ref. (Zamboni-Rached, Hernández-Figueroa, and Recami, 24) for details on how Eq. (88) is obtained from Eq. (8).] At first glance, this solution appears to be very complicated, but the series in its right-hand side gives a negligible contribution. This circumstance renders our solution in Eq. (88) of important practical interest, and we use later. [For additional information about the transverse pulse evolution (to be extracted from Eq. (88)), readers can consult again (Zamboni-Rached, Hernández-Figueroa, and Recami, 24). The same paper, analyzes how the generation by a finite aperture affects the chirped X-type pulses.]

66 3 Erasmo Recami and Michel Zamboni-Rached C 2 C 2 The valuable methods developed in (Zamboni-Rached, Hernández- Figueroa, and Recami, 24) and partially revisited in this section are of general interest, and work is in progress for applying them, e.g., also to the (different) case of the Schrödinger equation An Example: Chirped Optical X-Type Pulse in Bulk Fused Silica For a bulk fused silica, the refractive index n(ω) can be approximated by the Sellmeier equation Agrawal (26) n 2 (ω) = 1 + N j=1 B j ω 2 j ω 2 j ω 2, (89) where ω j are the resonance frequencies, B j the strength of the jth resonance, and N the total number of the material resonances that appear in the frequency range of interest. For our purposes, it is appropriate to choose N = 3, which yields, for bulk fused silica Agrawal (26), the values B 1 = , B 2 = , B 3 = , λ 1 = μm, λ 2 = μm, and λ 3 = μm. Let us consider in this medium a chirped X-type pulse, with λ =.2 μm, T =.4 ps, C = 1, and with axicon angle θ =.84 rad, that correspond to an initial central spot with ρ =.117 mm. By using Eqs. (81) and (88) we obtain the longitudinal and transverse pulse evolution, which are represented in Figure 3. From Figure 3a, we can observe that the pulse initially suffers a longitudinal narrowing with an increase of intensity until the position z = T 2 /2β 2 =.186 m. After that point, the pulse starts broadening and C (Z/V g 2 t)/t.2 24 (m) Z (m) (a) (b) FIGURE 3 (a) Longitudinal shape evolution of a chirped X-type pulse, propagating in fused silica with λ =.2 μm, T =.4 ps, C = 1, and axicon angle θ =.84 rad, which correspond to an initial transverse width of ρ =.117 mm. (b) Transverse shape evolution for the same pulse. 25 C Z 5 V g t (m)

67 Localized Waves: A Review 31 decreasing its intensity, while recovering its entire longitudinal shape (width and intensity) at the point z = T 2 /β 2 =.373 m, as predicted. At the same time, from Figure 3b, we notice that the pulse maintains its transverse width ρ = 2.4 c/(n(ω )ω sin θ) =.117 mm (because T ω >> 1) during its entire propagation. The same does not occur, however, with the pulse intensity: Initially, the pulse suffers an increase of intensity until position z c = T 2 /2β 2 =.186 m; after that point the intensity starts decreasing and the pulse recovers its entire transverse shape at point z c = T 2 /β 2 =.373 m, as expected. In the calculations we could skip the series in the right-hand side of Eq. (88), because, as already stated, it yields a negligible contribution. Summarizing, from Figure 3 we can see that the chirped X-type pulse totally recovers its longitudinal and transverse shape at position z = L disp = T 2 /β 2 =.373 m, as expected. Let us recall that a chirped Gaussian pulse may recover just its longitudinal width, but with an intensity decrease, at the position given by z = Z T1 =T = L disp = T 2 /β 2. Its transverse width, on the other hand, suffers progressive and irreversible spreading. The next text section completes our review by investigating the (no less interesting) case of the subluminal localized solutions to the wave equations, which, among others, will allow us to set forth remarkable considerations about the role of (extended) SR. For instance, the various superluminal and subluminal LWs are expected to be transformed one into the other by suitable LTs. We start by studying, in terms of various different approaches, the peculiar topic of zero-speed waves namely, the question of constructing localized fields with a static envelope, consisting, for example, in light at rest endowed with zero peak velocity. We called such solutions, Frozen Waves they can have many applications. 5. FROZEN WAVES AND SUBLUMINAL WAVE BULLETS 5.1. Modeling the Shape of Stationary Wave Fields: Frozen Waves We begin this section by studying the peculiar topic of zero-speed waves, namely, the question of constructing localized fields with a static envelope (e.g., consisting in light at rest endowed with null peak velocity). We called such solutions; Frozen Waves they permit a priori many applications. In this section we develop a simple first method (Longhi, 24; Salo et al., 1999; Zamboni-Rached, 24b), based on Section 3, by using superpositions of forward-propagating and equal-frequency Bessel beams, that allow controlling the longitudinal beam intensity shape within a chosen interval z L, where z is the propagation axis and L can be much

68 32 Erasmo Recami and Michel Zamboni-Rached greater than the wavelength λ of the monochromatic light (or sound) being used. Inside such a space interval, we succeed in constructing a stationary envelope whose longitudinal intensity pattern can approximately assume any desired shape, including, for instance, one or more high-intensity peaks (with distances between them much larger than λ), and which in addition results to be naturally endowed with a good transverse localization. Since the intensity envelopes remain static (i.e., with velocity V = ), we called such new solutions frozen waves (FW) (Longhi, 24; Salo et al., 1999; Zamboni-Rached, 24b) to the wave equations. Although we deal here with exact solutions of the scalar wave equation, vectorial solutions of the same kind for the electromagnetic field can be determined. Indeed, solutions to Maxwell s equations may be naturally inferred even from the scalar wave equation solutions (Agrawal, 26; Bouchal and Olivik, 1995; Recami, 1998). We present first the method referring to lossless media (Longhi, 24; Zamboni-Rached, 24b) while, in the second part of this section, we extend the method to absorbing media (Zamboni-Rached, 26) Stationary Wave Fields With Arbitrary Longitudinal Shape in Lossless Media, Obtained by Superposing Equal-Frequency Bessel Beams We start from the well-known axis-symmetric zero-order Bessel beam solution to the wave equation: with ψ(ρ, z, t) = J (k ρ ρ)e iβz e iωt (9) k 2 ρ = ω2 c 2 β2, (91) where ω, k ρ, and β are the angular frequency, the transverse wave number, and the longitudinal wave number, respectively. We also impose the conditions ω/β > and k 2 ρ (92) (which imply ω/β c) to ensure forward propagation only (with no evanescent waves), as well as a physical behavior of the Bessel function J. Now, let us make a superposition of 2N + 1 Bessel beams with the same frequency ω, but with different (and still unknown) longitudinal wave

69 Localized Waves: A Review 33 numbers β m : (ρ, z, t) = e i ω t N m= N A m J (k ρ m ρ) e i β m z, (93) where the m represent integer numbers and the A m are constant coefficients. For each m, the parameters ω, k ρ m, and β m must satisfy Eq. (91), and, because of conditions (92), when considering ω >, we must have β m ω c. (94) Let us now suppose that we wish (ρ, z, t) 2, given by Eq. (93), to assume on the axis ρ = the pattern represented by a function F(z) 2, inside the chosen interval z L. In this case, the function F(z) can be expanded, as usual, in a Fourier series as follows: F(z) = m= B m e i 2π L mz, where B m = 1 L L F(z) e i 2π L mz d z. More precisely, our goal now is finding the values of the longitudinal wave numbers β m and the coefficients A m of Eq. (93) to reproduce approximately, within the said interval z L (for ρ = ), the predetermined longitudinal intensity pattern F(z) 2. Namely, we wish to have N m= N A m e iβ mz 2 F(z) 2 with z L. (95) Looking at Eq. (95), one might be tempted to take β m = 2πm/L, thus obtaining a truncated Fourier series, expected to represent approximately the desired pattern F(z). Superpositions of Bessel beams with β m = 2πm/L have been used in some works to obtain a large set of transverse amplitude profiles (Bouchal, 22). However, for our purposes, this choice is not appropriate, for two principal reasons. First, it yields negative values for β m (when m < ), which implies backward-propagating components

70 34 Erasmo Recami and Michel Zamboni-Rached (since ω > ), and second, when L >> λ, which is our interest here, the main terms of the series correspond to very small values of β m, which results in a very short field depth of the corresponding Bessel beams (when generated by finite apertures), preventing the creation of the desired envelopes far from the source. Therefore, we need a better choice for the values of β m, which permits forward-propagation components only, and a good depth of field. This problem can be solved by putting β m = Q + 2 π L m, (96) where Q > is a value to be chosen (as we shall see) according to the given experimental situation and the desired degree of transverse field localization. Due to Eq. (94), one gets Q ± 2 π L N ω c. (97) Inequality (97) can be used to determine the maximum value of m, which we call N max, once Q, L, and ω have been chosen. As a consequence, for getting a longitudinal intensity pattern approximately equal to the desired one, F(z) 2, in the interval z L, Eq. (93) has to be rewritten as (ρ =, z, t) = e iω t e iqz N m= N A m e i 2π L mz, (98) with A m = 1 L L F(z) e i 2π L mz d z. (99) Obviously, this yields only an approximation of the desired longitudinal pattern, because the trigonometric series in Eq. (98) is necessarily truncated (N N max ). Its total number of terms let us repeat is fixed once the values of Q, L, and ω have been chosen. When ρ =, the wave field (ρ, z, t) becomes (ρ, z, t) = e iω t e iqz N m= N A m J (k ρ m ρ) e i 2π L mz, (1)

71 Localized Waves: A Review 35 with ( kρ 2 m = ω2 Q + 2π m ) 2. (11) L The coefficients A m will yield the amplitudes and the relative phases of each Bessel beam in the superposition. Because we are adding zero-order Bessel functions, we can expect a high field concentration around ρ =. Moreover, due to the known nondiffractive behavior of the Bessel beams, we expect that the resulting wave field will preserve its transverse pattern in the entire interval z L. The present methodology addresses the longitudinal intensity pattern control. Obviously, we cannot get a total 3D control because the field must obey the wave equation. However, we can use two ways to exert some control over the transverse behavior. The first is through the parameter Q of Eq. (96). We have some freedom in the choice of this parameter, and FWs representing the same longitudinal intensity pattern can possess different values of Q. The important point is that, in superposition (1), using a smaller value of Q makes the Bessel beams have a higher transverse concentration (because on decreasing the value of Q, one increases the value of the Bessel beams transverse wave numbers), and this is reflected in the resulting field, which presents a narrower central transverse spot. The second way to control the transverse intensity pattern is by using higher-order Bessel beams (see Section 6.1.1). We now present a few examples of our methodology. Example 1. Let us suppose that we want an optical wavefield with λ =.632 μm (i.e., with ω = Hz) whose longitudinal pattern (along its z-axis) in the range z L is given by the function 4 (z l 1)(z l 2 ) (l 2 l 1 ) 2 for l 1 z l 2 F(z) = 1 for l 3 z l 4 4 (z l 5)(z l 6 ) (l 6 l 5 ) 2 for l 5 z l 6 elsewhere, (12) where l 1 = L/5 z 12 and l 2 = L/5 + z 12 with z 12 = L/5; while l 3 = L/2 z 34 and l 4 = L/2 + z 34 with z 34 = L/1; and, at last, l 5 = 4L/5 z 56 and l 6 = 4L/5 + z 56 with z 56 = L/5. In other words, the desired longitudinal shape, in the range z L, is a parabolic function for

72 36 Erasmo Recami and Michel Zamboni-Rached l 1 z l 2, a unitary step function for l 3 z l 4, and again a parabola in the interval l 5 z l 6, being zero elsewhere (within the interval z L, as stated). In this example, let us put L =.2 m. We can then easily calculate the coefficients A m, which appear in superposition (1), by inserting Eq. (12) into Eq. (99). Let us choose, for instance, Q =.999 ω /c. This choice yields for m a maximum value N max = 316, as can be inferred from Eq. (97). We emphasize that one is not compelled to use just N = 316, but can adopt for N any values smaller than it more generally, any value smaller than that calculated via inequality (97). When using the maximum value allowed for N, a better result is achieved. In the present case, let us adopt the value N = 3. Figure 31a compares the intensity of the desired longitudinal function F(z) with that of the FW, (ρ =, z, t), obtained from Eq. (98) by adopting the mentioned value N = 3. One can verify that good agreement between the desired longitudinal behavior and our approximate FW is already obtained for N = 3. The use of higher values for N can only improve the approximation. Figure 31b shows the 3D intensity of our FW, given by Eq. (1). This field possesses the desired longitudinal pattern and has good transverse localization. Example 2. (Controlling the transverse shape too). We wish to take advantage of this example to address an important issue. We can expect that, for a desired longitudinal pattern of the field intensity, by choosing smaller values of the parameter Q one will get FWs with narrower transverse width [for the same number of terms in the series entering Eq. (1)], because C 2.6 C z (m) (a) (b) FIGURE 31 (a) Comparison between the intensity of the desired longitudinal function F(z) and that of the FW, (ρ =, z, t), obtained from Eq. (98). The solid line represents the function F(z), and the dotted one the FW. (b) 3D plot of the field intensity of the FW chosen in this case..5 (mm) z (m).15.2

73 Localized Waves: A Review 37 the Bessel beams in Eq. (1) will possess larger transverse wave numbers, and, consequently, higher transverse concentrations. We can verify this expectation by considering, for instance, inside the usual range z L, the longitudinal pattern represented by the function F(z) = 4 (z l 1)(z l 2 ) (l 2 l 1 ) 2 for l 1 z l 2 elsewhere, (13) with l 1 = L/2 z and l 2 = L/2 + z. Such a function has a parabolic shape, with its peak centered at L/2 and with longitudinal width 2 z/ 2. By adopting λ =.632 μm (that is, ω = Hz), let us use superposition (1) with two different values of Q. We obtain two different FWs that, despite having the same longitudinal intensity pattern, possess different transverse localizations. Let us consider L =.6 m and z = L/1, and the two values Q =.999 ω /c and Q =.995 ω /c. In both cases, the coefficients A m will be the same, calculated from Eq. (99) using this time the value N = 45 in superposition (1). The results are shown in Figure 32. Both FWs have the same longitudinal intensity pattern, but the one with the smaller Q has a narrower transverse width. In this way, we can exert some control on the transverse spot size through the parameter Q. Equation (1), which defines our FW, is actually a superposition of zero-order Bessel beams, and because of this the resulting field is expected to possess a transverse localization around ρ =. Each Bessel beam in superposition (1) is associated with a central spot with transverse size, or width, ρ m 2.4/k ρ m. On the basis of the expected convergence of the series (1), we can estimate the width of the transverse C 2.4 C ( m) 25 (a) 4 2 z (mm) 6 5 ( m) 25 (b) 4 2 z (mm) FIGURE 32 (a) The FW with Q =.999 ω /c and N = 45, approximately reproducing the chosen longitudinal pattern represented by Eq. (13). (b) A different FW, now with Q =.995 ω /c (but still with N = 45) forwarding the same longitudinal pattern. In this case (with a lower value for Q) a higher transverse localization is obtained. 6

74 38 Erasmo Recami and Michel Zamboni-Rached spot of the resulting beam as follows: ρ 2.4 k ρ, m= = 2.4 ω 2 /c2 Q 2, (14) which is the same value as that for the transverse spot of the Bessel beam with m = in superposition (1). Relation (14) can be useful: Once we have chosen the desired longitudinal intensity pattern, we can even choose the size of the transverse spot, and use relation (14) to evaluate the corresponding needed value of parameter Q. For a more detailed analysis concerning the spatial resolution and residual intensity of FWs, see ref. Zamboni-Rached, Recami, and Hernández-Figueroa (25). The FWs, corresponding to zero group velocity, are a particular case of the subluminal LWs. As in the superluminal case, the (more orthodox, in a sense) subluminal LWs can be obtained by suitable superpositions of Bessel beams. Until now they have been largely neglected, however, because of the mathematical difficulties in deriving analytic expressions for them, difficulties associated with the fact that the superposition integral runs now over a finite interval. In Zamboni-Rached and Recami (28a) we have shown, by contrast, that one can arrive at exact (analytic) solutions in the case of general subluminal LWs both in the case of integration over the Bessel beams angular frequency ω and in the case of integration over their longitudinal wave number k z. We return to this point in the following text. Increasing control on the transverse shape by using higher-order Bessel beams. Here, we argue that it is possible to increase even more control on the transverse shape by using higher-order Bessel beams in the fundamental superposition (1). This new approach can be understood and accepted on the basis of simple and intuitive arguments (not presented here but found in Zamboni-Rached, Recami, and Hernández-Figueroa (25)). A brief description of that approach follows. The basic idea is obtaining the desired longitudinal intensity pattern not along the axis ρ =, but on a cylindrical surface corresponding to ρ = ρ >. To do that, we first proceed as before. Once we have chosen the desired longitudinal intensity pattern F(z), within the interval z L, we calculate the coefficients A m as before; that is, A m = (1/L) L F(z) exp( i2πmz/l) dz, and k ρ m = ω 2 (Q + 2πm/L)2. Afterward, we simply replace the zero-order Bessel beams J (k ρ m ρ), in superposition (1), with higher-order Bessel beams, J μ (k ρ m ρ), toget (ρ, z, t) = e iω t e iqz N m= N A m J μ (k ρ m ρ) e i 2π L mz. (15)

75 Localized Waves: A Review 39 From this result, and on the basis of intuitive arguments, Zamboni- Rached, Recami, and Hernández-Figueroa (25), we can expect that the desired longitudinal intensity pattern, initially constructed for ρ =, will approximately shift to ρ = ρ, where ρ represents the position of the first maximum of the Bessel function (the first positive root of the equation d J μ (k ρ, m= ρ)/dρ) ρ = ). By using such a procedure, one can obtain interesting stationary configurations of field intensity, as donuts, cylindrical surfaces, and much more. In the following example, we show how to obtain, for example, a cylindrical surface of stationary light. To obtain it, within the interval z L, let us first select the longitudinal intensity pattern given by Eq. (13), with l 1 = L/2 z and l 2 = L/2 + z, and with z = L/3. Moreover, let us choose L =.5 m, Q =.998 ω /c, and use N = 15. Then, after calculating the coefficients A m by Eq. (99), we use superposition (15), choosing, in this case, μ = 4. According to the previous discussion, one can expect the desired longitudinal intensity pattern to appear shifted to ρ 5.318/k ρ, m= = 8.47 μm, where is the value of k ρ, m= ρ for which the Bessel function J 4 (k ρ, m= ρ) assumes its maximum value, with k ρ, m= = ω 2 Q2. Figure 33 shows the resulting intensity field. Figure 33a shows the transverse section of the resulting beam for z = L/2. The transverse peak intensity is located at ρ = 7.75 μm, with a 8.5% difference with respect to the predicted value of 8.47 μm. Figure 33b shows the orthogonal projection of the resulting field, which corresponds to nothing but a cylindrical surface of stationary light (or other fields). We can see that the desired longitudinal intensity pattern has been approximately obtained, shifted, as desired, from ρ = toρ = 7.75 μm; and the resulting field resembles a cylindrical surface of stationary light 25 C 2 / C 2 max C 2 / C 2 max x (mm).5 (mm) y (mm) z (mm) (a) (b) FIGURE 33 (a) Transverse section at z = L/2 of the considered higher-order FW. (b) Orthogonal projection of the 3D intensity pattern of the same higher-order FW.

76 31 Erasmo Recami and Michel Zamboni-Rached with radius 7.75 μm and length 238 μm. Donut-like configurations of light (or sound) are also possible Stationary Wave Fields With Arbitrary Longitudinal Shape in Absorbing Media: An Extension of the Method When propagating in a nonabsorbing medium, the so-called nondiffracting waves maintain their spatial shape for long distances. However, the situation is not the same with absorbing media. In such cases, both the ordinary and the nondiffracting beams (and pulses) suffer the same effect: an exponential attenuation along the propagation axis. We present an extension Zamboni-Rached (26a) of the method given above with the aim of showing that, through suitable superpositions of equal-frequency Bessel beams, it is possible to obtain even in absorbing media nondiffracting beams, whose longitudinal intensity pattern can assume any desired shape within a chosen interval z L of the propagation axis z. As a particular example, we are going to obtain new nondiffracting beams capable of resisting the loss effects, maintaining amplitude and spot size of their central core for long distances. It is important to stress that the energy absorption by the medium continues as normal, but the new beams have an initial transverse field distribution, so to reconstruct (notwithstanding the presence of absorption) their central cores for distances considerably longer than the penetration depths of ordinary (nondiffracting or diffracting) beams. In this sense, the new method can be regarded as extending, for absorbing media, the selfreconstruction properties (Grunwald et al., 24) that usual LWs are known to possess in lossless media. In the same way as for lossless media, we construct a Bessel beam with angular frequency ω and axicon angle θ in the absorbing materials by superposing plane waves, with the same angular frequency ω, and whose wave vectors lie on the surface of a cone with vertex angle θ. The refractive index of the medium can be written as n(ω) = n R (ω) + in I (ω), quantity n R being the real part of the complex refraction index and n I the imaginary one, responsible for the absorbtion effects. For a plane wave, the penetration depth δ for the frequency ω is given by δ = 1/α = c/2ωn I, where α is the absorption coefficient. Therefore, a zero-order Bessel beam in dissipative media can be written as ψ = J (k ρ ρ)exp(iβz)exp( iωt) with β = n(ω) ω cos θ/c = n R ω cos θ/c + in I ω cos θ/c β R + iβ I ; k ρ = n R ω sin θ/c + in I ω sin θ/c k ρr + ik ρi, and so k 2 ρ = n2 ω 2 /c 2 β 2. Thus, the result is ψ = J ((k ρr + ik ρi )ρ)exp(iβ R z)exp( iωt)exp( β I z), where β R, k ρr are the real parts of the longitudinal and transverse wave numbers, and β I, k ρi are the imaginary ones, while the absorption coefficient of a Bessel beam with axicon angle θ is given by α θ = 2β I = 2n I ω cos θ/c, its penetration depth being δ θ = 1/α θ = c/2ωn I cos θ.

77 Localized Waves: A Review 311 Because k ρ is complex, the amplitude of the Bessel function J (k ρ ρ) starts decreasing from ρ = until the transverse distance ρ = 1/2k ρi, and afterward it starts growing exponentially. This behavior is not physically acceptable, but it occurs only because an ideal Bessel beam needs an infinite aperture to be generated. However, in any real situation, when a Bessel beam is generated by finite apertures, that exponential growth in the transverse direction, starting after ρ = 1/2k ρi, will not occur indefinitely, stopping at a given value of ρ. Moreover, we emphasize that, when generated by a finite aperture of radius R, the truncated Bessel beam (Zamboni-Rached, Recami, and Hernández-Figueroa, 25) possesses a depth of field Z = R/ tan θ and can be approximately described by the solution in the previous paragraph, for ρ<r and z < Z. Experimentally, to guarantee that the mentioned exponential growth in the transverse direction does not even start, so as to meet only a decreasing transverse intensity, the radius R of the aperture used to generate the Bessel beam should be R 1/2k ρi. However, as noted by Durnin, Miceli, and Eberly (1987a) the same aperture also must satisfy the relation R 2π/k ρr. From these two conditions, one can infer that, in an absorbing medium, a Bessel beam with only a decreasing transverse intensity can be generated only when the absorption coefficient is α<2/λ; that is, if the penetration depth is δ>λ/2. The present method does refer to these cases it is always possible to choose a suitable finite aperture size in such a way that the truncated versions of all solutions, including the general one given by Eq. (111), do not develop any unphysical behavior. We now outline the method Zamboni-Rached and Recami (28a). Let us consider an absorbing medium with the complex refraction index n(ω) = n R (ω) + in I (ω), and the following superposition of 2N + 1 Bessel beams with the same frequency ω: (ρ, z, t) = N m= N A m J ( (kρrm + ik ρim )ρ ) e i β Rm z e iωt e β Im z, (16) where the m are integer numbers, the A m are constant coefficients (yet unknown), quantities β Rm and k ρrm (β Im and k ρim ) are the real (the imaginary) parts of the complex longitudinal and transverse wave numbers of the mth Bessel beam in superposition (16); the following relations being satisfied k 2 ρ m = n 2 ω2 c 2 β2 m (17) β Rm β Im = n R n I, (18) where β m = β Rm + iβ Im, k ρm = k ρrm + ik ρim, with k ρrm /k ρim = n R /n I.

78 312 Erasmo Recami and Michel Zamboni-Rached Our goal is to find the values of the longitudinal wave numbers β m and the coefficients A m to reproduce approximately, inside the interval z L (on the axis ρ = ), a freely chosen longitudinal intensity pattern that we call F(z) 2. The problem for the particular case of lossless media Longhi (24); Zamboni-Rached (24b) when n I = β Im = was solved in the previous subsection. For those cases, it was shown that the choice β = Q + 2πm/L, with A m = L F(z)exp( i2πmz/l)/l dz, can be used to provide approximately the desired longitudinal intensity pattern F(z) 2 in the interval z L, and, at the same time, to regulate the spot size of the resulting beam by means of the parameter Q. Such parameter, incidentally, also can be used to obtain large field depths and moreover to inforce the linear polarization approximation to the electric field for the Transverse Electric (TE) electromagnetic wave (see details in Longhi (24); Zamboni-Rached (24b)). However, when dealing with absorbing media, the procedure described in the last paragraph does not work, due to the presence of the functions exp( β Im z) in the superposition (16), because in this case that series does not became a Fourier series when ρ =. On attempting to overcome this limitation, let us write the real part of the longitudinal wave number, in superposition (16), as β Rm = Q + 2πm L (19) with Q + 2πm L n ω R c. (11) where inequality (11) guarantees forward propagation only, with no evanescent waves. In this way, the superposition (16) can be written (ρ, z, t) = e i ω t e iqz N m= N A m J ( (kρrm + ik ρim )ρ ) e i 2πm L z e β Im z, (111) where, by using Eq. (18), we have β Im = (Q + 2πm/L)n I /n R, and k ρm = k ρrm + ik ρim is given by Eq. (17). Obviously, the discrete superposition (111) could be written as a continuous one (i.e., as an integral over β Rm ) by taking L, but we prefer the discrete sum due to the difficulty of obtaining closed-form solutions to the integral form. Now, let us examine the imaginary part of the longitudinal wave numbers. The minimum and maximum values among the β Im are (β I ) min = (Q 2πN/L)n I /n R and (β I ) max = (Q + 2πN/L)n I /n R, the central one being

79 Localized Waves: A Review 313 given by β I (β I ) m= = Qn I /n R. With this in mind, let us evaluate the ratio =[(β I ) max (β I ) min ]/β I = 4πN/LQ. Thus, when <<1, there are no considerable differences among the various β Im, because β Im β I holds for allm.inthesameway,therearenoconsiderabledifferencesamongtheexponential attenuation factors, since exp( β Im z) exp( β I z). So, when ρ = the series in the right hand side, of Eq. (111) can be approximately considered a truncated Fourier series multiplied by the function exp( β I z), and, therefore, superposition (111) can be used to reproduce approximately the desired longitudinal intensity pattern F(z) 2 (on ρ = ), within z L, when the coefficients A m are given by A m = 1 L L F(z) e βiz 2πm i e L z dz, (112) the presence of the factor exp(β I z) in the integrand being necessary to compensate for the factors exp( β Im z) in superposition (111). Since we are adding zero-order Bessel functions, we can expect a good field concentration around ρ =. In short, we have shown in this section how to obtain, in an absorbing medium, a stationary wave field with a good transverse concentration, and whose longitudinal intensity pattern (on ρ = ) can approximately assume any desired shape F(z) 2 within the predetermined interval z L. This method is a generalization of a previous one, Longhi (24); Zamboni-Rached (24b), and consists of the superposition in Eq. (111) of Bessel beams whose longitudinal wave numbers are individuated by the real and imaginary parts given in Eqs. (19) and (18), respectively, while their complex transverse wave numbers are given by Eq. (17). Finally, the coefficients of the superposition are given by Eq. (112). The method is justified when 4πN/LQ << 1; fortunately, this condition is satisfied in many situations. With regard to the generation of these new beams, once we have an apparatus capable of generating a single Bessel beam, we can simply use an array of such apparatuses to generate a sum of Bessel beams, with the appropriate longitudinal wave numbers and amplitudes/phases (as specified by our method), thus producing the desired final beam. For instance, we can use, Longhi (24); Zamboni-Rached (24b), a laser illuminating an array of concentric annular apertures (located at the focus of a convergent lens), with the appropriate radii and transfer functions to yield both the required longitudinal wave numbers (once a value for Q has been chosen) and the coefficients A n of the fundamental superposition (111). Examples For generality s sake, let us consider a hypothetical medium in which a typical XeCl excimer laser (λ = 38nm ω = Hz) has a

80 314 Erasmo Recami and Michel Zamboni-Rached penetration depth of 5 cm that is, an absorption coefficient α = 2m 1, and therefore n I = In addition, let us suppose that the real part of the refraction index for this wavelength is n R = 1.5 and therefore n = n R + in I = i Note that the value of the real part of the refractive index is not as important for us because we are dealing with monochromatic wave fields. A Bessel beam with ω = Hz and with an axicon angle θ =.141 rad (thus, with a transverse spot of radius 8.4 μm), when generated by an aperture, say, of radius R = 3.5 mm, can travel in vacuum a distance equal to Z = R/ tan θ = 25 cm while resisting the diffraction effects. However, in the material medium considered here, the penetration depth of this Bessel beam would be only z p = 5 cm. We now consider two interesting applications of this method. Example 1. Almost undistorted beams in absorbing media. We can use the extended method to obtain, in the same medium and for the same wavelength, an almost undistorted beam capable of preserving its spot size and the intensity of its central core for a distance many times larger than the typical penetration depth of an ordinary beam (nondiffracting or not). To this purpose, let us suppose that, in the considered material medium, we want a beam (with ω = Hz) that maintains amplitude and spot size of its central core for a distance of 25 cm a distance five times greater than the penetration depth of an ordinary beam with the same frequency. We can model this beam by choosing the desired longitudinal intensity pattern F(z) 2 (on ρ = ), within z L, to be given by the function F(z) = { 1 for z Z elsewhere, (113) and by putting Z = 25 cm, with, for example, L = 33 cm. Now, we can use the Bessel beam superposition (111) to reproduce approximately the selected intensity pattern. Let us choose Q =.9999ω/c for the β Rm in Eq. (19), and N = 2 (notice that, according to inequality (11), N could assume a maximum value of 158.) After having chosen the values of Q, L, and N, the values of the complex longitudinal and transverse wave numbers of the Bessel beams happen to be defined by relations (19), (18), and (17). Eventually, we have recourse to Eq. (112) and determine the coefficients A m of the fundamental superposition (111), that defines the resulting stationary wave field. (Note that the condition 4πN/LQ << 1 is perfectly satisfied in this case.) Figure 34a shows the 3D field intensity of the resulting beam. The field possesses a good transverse localization (with a spot size smaller than 1 μm) and is capable of maintaining spot size and intensity of its central

81 Localized Waves: A Review 315 1log 1 [ C 2 / C max 2 ] [db] C (mm) (a) z (mm) (mm) z (mm) FIGURE 34 (a) Three-dimensional field intensity of the resulting beam. (b) The resulting beam, in an orthogonal projection and in logaritmic scale. (b) core up to the desired distance (a better result could be reached by using a higher value of N). It is interesting to note that at that distance (25 cm), an ordinary beam would have its initial field intensity attenuated 148 times. As noted, the energy absorption by the medium continues as normal; the difference is that these new beams have an initial transverse field distribution sophisticated enough to be able to reconstruct (even in the presence of absorption) their central cores up to a certain distance. For a better visualization of this field intensity distribution and of the energy flux, Figure 34b shows the resulting beam, in an orthogonal projection and in logarithmic scale. It appears clear that the energy comes from the lateral regions in order to reconstruct the central core of the beam. On the plane z =, within the region ρ R = 3.5 mm, there is an uncommon field intensity distribution; it is very dispersed instead of concentrated. This uncommon initial field intensity distribution is responsible for the construction of the central core of the resulting beam and for its reconstruction all along the distance z = 25 cm. Due to absorption, the beam (total) energy, flowing through different z planes, is not constant, but the energy flowing in the beam spot area, and the beam spot size itself, are conserved up to the distance (in this case) z = 25 cm. Example 2. Beams in absorbing media with a growing longitudinal field intensity. We consider again the previous hypothetical medium, in which an ordinary Bessel beam with θ =.141 rad and ω = Hz would have a penetration depth of 5 cm. We now want to construct a beam that, instead of possessing a constant core intensity up to the position z = 25 cm, presents, on the contrary, a (moderate) exponential growth of its intensity, up to that distance (z = 25 cm).

82 316 Erasmo Recami and Michel Zamboni-Rached Let us assume we wish to get the following longitudinal intensity pattern F(z) 2, in the interval < z < L: F(z) = { exp(z/z) for z Z elsewhere, (114) with Z = 25 cm and L = 33 cm. Using again Q =.9999 ω/c, and N = 2, we can proceed as in the first example, calculating the complex longitudinal and transverse wave numbers of the Bessel beams, and finally the coefficients A m of the fundamental superposition (111). In Figure 35 we can see the 3D field intensity of the resulting beam. The field presents the desired longitudinal intensity pattern with a good transverse localization (a spot size smaller than 1 μm). Obviously, the amount of energy necessary to construct these new beams is greater than that necessary to generate an ordinary beam in a nonabsorbing medium. It is also clear that there is a limitation on the depth of field of these new beams. In the first example, for distances larger than 1 times the penetration depth of an ordinary beam, the field intensity in the lateral regions would be higher than that at the core, and the field would lose 8 C (mm) z (mm) FIGURE 35 Three-dimensional field intensity of the resulting beam in an absorbing medium with a growing longitudinal field intensity. 3

83 Localized Waves: A Review 317 the usual characteristics of a beam (transverse field concentration), not to mention the greater energy demand Subluminal Localized Waves (or Bullets) In this subsection, we face the more general problem of obtaining, in a simple way, localized (nondiffractive) subluminal pulses as exact analytic solutions to the wave equations, Zamboni-Rached and Recami (28a). These new ideal subluminal solutions, which propagate without distortion in any homogeneous linear media, will be obtained here for arbitrarily chosen frequencies and bandwidths, avoiding in particular any recourse to the noncausal (backward-moving) components that so frequently plague the previously known LWs. The new solutions are suitable superpositions of zero-order, in general Bessel beams, which can be performed either by integrating with respect to the angular frequency ω, or by integrating with respect to the longitudinal wave number k z ; both methods are expounded in this review. The first appears to be powerful enough, but we also present the second method since it allows dealing once more (from a different starting point) also with the limiting case of zero-speed solutions (and furnishes a new way, in terms of continuous spectra, for obtaining our FWs, so promising also from the point of view of applications). Moreover, we also pay attention to the known role of SR, and to the fact that the LWs are expected to be transformed one into the other by suitable LTs. Finally, we briefly treat the case of non axially symmetric solutions in terms of higher-order Bessel beams. The analogous pulses with intrinsic finite energy, or merely truncated, are considered elsewhere. We devote our attention especially to electromagnetism and optics but we reiterate that results of the same kind are valid whenever an essential role is played by a wave equation (as in acoustics, seismology, geophysics, elementary particle physics [as we verified even in the slightly different case of the Schrödinger equation], and gravitation [for which we recently got stimulating new results], and so on) Foreword For self-consistency, let us repeat here the following considerations. For more than 1 years, the so-called (nondiffracting) LWs, which are new solutions to the wave equations (scalar, vectorial, spinorial etc.), have been in fashion, both in theory and in experiment. In particular, rather well known are those with luminal or superluminal peak velocity, Hernández-Figueroa, Zamboni-Rached, and Recami (28) like the socalled X-shaped waves (see Lu and Greenleaf (1992a); Zamboni-Rached, Recami, and Hernández-Figueroa (22) and references therein; for a

84 318 Erasmo Recami and Michel Zamboni-Rached review, see Recami et al. (23)), which are supersonic in acoustics, Lu and Greenleaf (1992b), and superluminal in electromagnetism (see Recami (1998) and references therein). Since the work of Bateman (1915) and later Courant and Hilbert (1966), it has been known, e.g., that luminal LWs exist, which are solutions to the wave equations. More recently, some attention also has been paid to the subluminal LWs. Let us recall that all LWs propagate without distortion and in a self-reconstructive way in a homogeneous linear medium (apart from local variations): in the sense that their square magnitude maintains its shape during propagation, while local variations are shown only by its real, or imaginary, part. As in the superluminal case, subluminal LWs can be obtained by suitable superpositions of Bessel beams, Zamboni-Rached and Recami (28a). Until recently, as we know, they have been almost neglected because of the mathematical difficulties in getting analytic expressions for them, difficulties associated with the fact that the superposition integral runs over a finite interval. Here we readdress the question of such subluminal LWs, showing, by contrast, that exact (analytic) solutions can indeed be determined both in the case of integration over the Bessel beams angular frequency ω and in the case of integration over their longitudinal wave number k z. As already claimed, this work is devoted to the exact, analytic solutions: that is to ideal solutions. The corresponding pulses with finite energy, or truncated, are presented elsewhere. In the past, not enough attention was paid to Brittingham s 1983 paper, Brittingham (1983), wherein he showed the possibility of obtaining pulsetype solutions to the Maxwell equations, which propagate in free space as a new kind of speed-c solitons. That lack of attention was partially due to the fact that Brittingham had been unable to either obtain correct finite-energy expressions for such wavelets and to make suggestions about their practical production. Two years later, however, Sezginer (1985) was able to obtain quasi-nondiffracting luminal pulses endowed with a finite energy. Finite-energy pulses no longer travel undistorted, as we know, for an infinite distance, but they can nevertheless propagate without deformation for a long field depth, much larger than the one achieved by ordinary pulses like the Gaussian ones (see Arlt et al. (21); Garcés- Chavez et al. (22); Lu, Zou, and Greenleaf (1993, 1994); MacDonald et al. (22); McGloin, Garcés-Chavez, and Dholakia (23), and references therein). Only after 1985 was the general theory of LWs extensively developed, Arlt, Hitomi, and Dholakia (2); Ashkin et al. (1986); Erdélyi et al. (1997); Fan, Parra, and Milchberg (2); Goodman (1996); Herman and Wiggins (1991); Rhodes et al. (22); Willebrand and Ghuman (21); Ziolkowski

85 Localized Waves: A Review 319 (1989, 1991), both in the case of beams and in the case of pulses. For reviews, see Besieris et al. (1998); Recami (21); Recami et al. (23); Recami, Zamboni-Rached, and Hernández-Figueroa (28) and citations therein. For the propagation of LWs in bounded regions (like wave guides), see Barbero, Hernández-Figueroa, and Recami (2); Zamboni-Rached, Fontana, and Recami (23); Zamboni-Rached, Recami, and Fontana (21); Zamboni-Rached et al. (22), and references therein. For the focusing of LWs, see the fourth section of this review (as well as Shaarawi, Besieris, and Said (23); Zamboni-Rached, Shaarawi, and Recami (24) and citations therein). With regard to the construction of general LWs propagating in dispersive media, see references Conti et al. (23); Hernández- Figueroa, Zamboni-Rached, and Recami (28); Longhi (23); Porras, Valiulis, and Di Trapani (23); Porras et al. (23); Salo et al. (1999); Sõnajalg and Saari (1996); Zamboni-Rached, Hernández-Figueroa, and Recami (24); Zamboni-Rached et al. (23), and for lossy media compare with Zamboni- Rached (26a) and references therein. Finally, for finite-energy, or truncated, solutions, see references Besieris et al. (1998); Zamboni-Rached (24, 26); Zamboni-Rached, Fontana, and Recami (23); Zamboni- Rached, Recami, and Hernández-Figueroa (22); Ziolkowski, Besieris, and Shaarawi (1993), and work in progress. By LWs have been experimentally produced (Lu and Greenleaf (1992b); Mugnai, Ranfagni, and Ruggeri (2); Saari and Reivelt (1997); Valtua, Reivelt, and Saari (27)), and are being applied in fields ranging from ultrasound scanning (Lu and Greenleaf (1995); Lu, Zou, and Greenleaf (1993, 1994)) to optics (e.g., for the production, of a new type of tweezers Zamboni-Rached et al. (25)). We now know that nondiffracting pulses can travel with an arbitrary peak speed v, that is, with < v <, whereas Brittingham and Sezginer had confined themselves to the luminal case (v = c) only. Superluminal and luminal LWs have been, and continue to be, intensively studied, but the subluminal ones have been neglected. The few papers dealing with them until now have been restricted to the paraxial (Newell and Molone, 1992) approximation (Besieris and Shaarawi, 24) or to numerical simulations (Salo and Salomaa, 21), because of the mathematical difficulty in obtaining exact analytic expressions for subluminal pulses. Indeed, only one analytic solution was known, Besieris and Shaarawi (22, 24); Donnelly and Ziolkowski (1993); Lu and Greenleaf (1995); Mackinnon (1978); Rodrigues, Vaz, and Recami (1994), biased by the physically unconvenient facts that (1) its frequency spectrum is very large, (2) it does not possess a well-defined central frequency, and (3) that backward-traveling components (ordinarily called non causal because they should be entering the antenna or generator) were needed to construct it. The next text subsections show that subluminal localized exact

86 32 Erasmo Recami and Michel Zamboni-Rached solutions can be constructed with any spectra, in any frequency bands, and for any bandwidths without using (Recami, Zamboni-Rached, and Hernández-Figueroa (28); Zamboni-Rached, Recami, and Hernández- Figueroa (22)) any backward-traveling components A First Method for Constructing Physically Acceptable Subluminal Localized Pulses Axially symmetric solutions to the scalar wave equation are known to be superpositions of zero-order Bessel beams over the angular frequency ω and the longitudinal wave number k z : That is, in cylindrical coordinates, (ρ, z, t) = dω ω/c ω/c dk z S(ω, k z )J ρ ω 2 c 2 k2 z e ikzz e iωt, (115) where kρ 2 ω2 /c 2 kz 2 is the transverse wave number. Quantity k2 ρ must be positive since evanescent waves cannot come into play. The condition characterizing a nondiffracting wave is the existence (Besieris et al. (1998); Zamboni-Rached and Hernàndez-Figueroa (2)) of a linear relation between longitudinal wave number k z and frequency ω for all the Bessel beams entering superposition (113); that is, the chosen spectrum must entail, Zamboni-Rached, Recami, and Hernández-Figueroa (22), for each Bessel beam a linear relationship of the type: 6 ω = vk z + b, (116) with b. Requirement (116) also can be regarded as a specific space-time coupling, implied by the chosen spectrum S. Equation (116) must be obeyed by the spectra of any one of the three possible types (subluminal, luminal, or superluminal) of nondiffracting pulses. With the choice in Eq. (116) the pulse regains its initial shape after the space interval z 1 = 2πv/b. But the more general case also can be considered, Zamboni-Rached, Recami, and Hernández-Figueroa (22, 28)) when b assumes any values b m = mb (with m an integer), and the periodicity space interval becomes z m = z 1 /m. We are referring to the real (or imaginary) part of the pulse, since its modulus is known to be endowed with rigid motion. 6 More generally, as shown in Zamboni-Rached, Recami, and Hernández-Figueroa (22), the chosen spectrum has to call into play, in the plane ω, k z, if not exactly the line (116), at least a region in the proximity of a straight line of that a type. It is interesting that in the latter case one obtains solutions endowed with finite energy, but possessing a finite depth of field, that is, nondiffracting only for a certain finite distance.

87 Localized Waves: A Review 321 In the subluminal case, of interest here, the only exact solution known until recently, just represented by Eq. (124) below, was the one found by Mackinnon (1978). Indeed, by taking into explicit account that the transverse wave number k ρ of each Bessel beam entering Eq. (115) must be real, it can be easily shown (as first noticed by Salo and Salomaa (21) for the analogous acoustic solutions) that in the subluminal case b cannot vanish, but must be larger than zero: b >. Then, on using conditions (116) and b >, the subluminal localized pulses can be expressed as integrals over the frequency only: [ [ (ρ, z, t) = exp ib z ] ω + dω S(ω) J (ρk ρ ) exp v ω iω ζ v ], (117) where now k ρ = 1 v 2bω b 2 (1 v 2 /c 2 )ω 2 (118) with ζ z vt (119) and with ω = ω + = b 1 + v/c b 1 v/c. (12) As anticipated, the Bessel beam superposition in the subluminal case proves to be an integration over a finite interval of ω, which clearly shows that the backward-traveling (noncausal) components correspond to the interval ω <ω<b. It could be noticed that Eq. (117) does not represent the most general exact solution, which on the contrary is a sum (Zamboni- Rached, Recami, and Hernández-Figueroa (28)) of such solutions for the various possible values of b mentioned above that is, for the values b m = mbwith spatial periodicity z m = z 1 /m. However, we can confine ourselves to solution (117) without any real loss of generality, since the actual problem is evaluating in analytic form the integral entering Eq. (117). For mathematical and physical details, see Zamboni-Rached, Recami, and Hernández-Figueroa (28). Now, if one adopts the change of variable ω b 1 v 2 /c 2 (1 + v s), (121) c

88 322 Erasmo Recami and Michel Zamboni-Rached Eq. (117) becomes, Salo and Salomaa (21), (ρ, z, t) = b c 1 1 v [ i 1 v 2 /c 2 exp bv ] [ z exp i b v ds S(s) J ( b c ) ρ 1 s 2 exp 1 v 2 /c 2 ] 1 1 v 2 /c 2 ζ [ i b c ] 1 1 v 2 /c 2 ζs. (122) In the following we adhere to some symbols standard in SR (since the topic of subluminal, luminal, and superluminal LWs is strictly connected (Recami (1998); Recami, Zamboni-Rached, and Dartora (24); Recami et al. (23)) with the principles and structure of SR [compare Barut, Maccarrone, and Recami (1982); Recami (1986) and references therein], as we shall mention also in the concluding remarks); namely: β v c ; γ 1 1 β 2. (123) As stated, Eq. (122) has until now yielded one analytic solution for S(s) = constant only (for instance, S(s) = 1), which means S(ω) = const.; in this case, one gets the Mackinnon solution, Donnelly and Ziolkowski (1993); Lu and Greenleaf (1995); Mackinnon (1978); Zamboni-Rached (26), (ρ, ζ, η) = 2 b [ c v γ2 exp i b ] c βγ2 η sinc b 2 c 2 γ2 ( ρ 2 + γ 2 ζ 2), (124) which, however, because of its previously mentioned drawbacks, is endowed with little physical and practical interest. In Eq. (124) the sinc function has the ordinary definition and sinc x (sin x)/x, η z Vt, with V c2 v, (125) where V and v are related by the de Broglie relation. (Note that in Eq. (124), and in the following ones, is eventually a function [besides of ρ] ofz, t via ζ and η, both functions of z and t.)

89 Localized Waves: A Review 323 However, we can use a simple method to construct new subluminal pulses corresponding to any spectrum, and devoid of backward-moving (i.e., entering ) components, by taking advantage of the fact that in Eq. (122) the integration interval is finite; that is, by transforming it into a good, instead of a harm. Let us first observe that Eq. (122) does not admit only the already known analytic solution corresponding to S(s) = constant, and more in general to S(ω) = constant, but it also yields an exact, analytic solution for any exponential spectra of the type [ ] i2nπω S(ω) = exp, (126) with n any integer number, which means for any spectra of the type S(s) = exp [inπ/β] exp [inπs], as can be easily seen by checking the product of the various exponentials entering the integrand. In Eq. (126) we have set ω + ω. The solution in this more general case is written as: [ (ρ, ζ, η) = 2bβγ 2 exp i b ] c βγ2 η [ exp in π ] b sinc 2 ( ) b 2 β c 2 γ2 ρ 2 + c γ2 ζ + nπ. (127) Notice that in Eq. (127) quantity η also is defined as in Eq. (125), where V and v obey the de Broglie relation vv = c 2, where the subluminal quantity v is the velocity of the pulse envelope, and V plays the role (in the envelope s interior) of a superluminal phase velocity. The next step, as anticipated, consists of taking advantage of the finiteness of the integration limits for expanding any arbitrary spectra S(ω) in a Fourier series in the interval ω ω ω + : S(ω) = n= A n exp [+in 2π ] ω, (128) where (we went back, now, from the s to the ω variable), A n = 1 ω+ ω dω S(ω) exp [ in 2π ] ω (129) quantity being defined as above.

90 324 Erasmo Recami and Michel Zamboni-Rached On recounting the special solution in Eq. (127), we can infer from expansion (126) that, for any arbitrary spectral function S(ω), a rather general axially symmetric analytic solution for the subluminal case can be worked out as [ (ρ, ζ, η) = 2bβγ 2 exp i b ] c βγ2 η n= [ A n exp in π ] b sinc 2 ( ) b 2 β c 2 γ2 ρ 2 + c γ2 ζ + nπ, (13) in which the coefficients A n are still given by Eq. (129). We repeat that our solution is expressed in terms of the particular Eq. (127), which is a Mackinnon-type solution. The present approach presents many advantages. We can easily choose spectra localized within the prefixed frequency interval (optical waves, microwaves, and so on) and endowed with the desired bandwidth. Moreover, as we have seen, spectra can now be chosen such that they have zero value in the region ω ω b, which is responsible for the backward-traveling components of the subluminal pulse. We stress that, even when the adopted spectrum S(ω) does not possess a known Fourier series [so that the coefficients A n cannot be exactly evaluated via Eq. (129)], one can calculate approximately such coefficients without any problem, since our general solutions in Eq. (13) will still be exact solutions. Some examples are presented below Examples In general, optical pulses generated in the laboratory possess a spectrum centered at some frequency value, ω, called the carrier frequency. For instance, the pulses can be ultrashort, when ω/ω 1; or quasimonochromatic, when ω/ω << 1, where ω is the spectrum bandwidth. These types of spectra can be mathematically represented by a Gaussian function or functions with similar behavior. Examples 1 and 2. Let us first consider a Gaussian spectrum S(ω) = a π exp [ a 2 (ω ω ) 2 ], (131) whose values are negligible outside the frequency interval ω <ω<ω + over which the Bessel beams superposition in Eq. (117) is made, it being ω = b/(1 + β) and ω + = b/(1 β). Of course, the relation in Eq. (116) must

91 Localized Waves: A Review 325 still be satisfied, with b >, to obtain an ideal subluminal localized solution. Notice that, with the spectrum in Eq. (131), the bandwidth (actually, the full-width half-maximum (FWHM)) results in ω = 2/a. We emphasize that, once v and b have been fixed, the values of a and ω can afterward be selected in order to eliminate the backward-traveling components that correspond to ω<b. The Fourier expansion in Eq. (128), which yields, with the above spectral function (131), the coefficients A n 1 [ in W exp 2π ] [ ] ω n 2 π 2 exp a 2 W 2, (132) constitutes an excellent representation of the Gaussian spectrum [Eq. (131)] in the interval ω <ω<ω + (provided that, as we requested, the Gaussian spectrum does get negligible values outside the frequency interval ω < ω<ω + ). In other words, on choosing a pulse velocity v < c and a value for the parameter b, a subluminal pulse with the above frequency spectrum [Eq. (131)] can be written as Eq. (129), with the coefficients A n given by Eq. (132) the evaluation of such coefficients A n being rather simple. Even if the values of A n are obtained via a (rather good, by the way) approximation, our basis is the exact solution Eq. (13). For instance, exact solutions representing subluminal pulses for optical frequencies can be obtained. Let us determine the subluminal pulse with velocity v =.99 c, carrier angular frequency ω = Hz (that is, λ =.785 μm), and bandwidth (FWHM) ω = ω /2 = Hz, which is an optical pulse of 24 fs (which is the FWHM of the pulse intensity). For a complete pulse characterization, the value of the frequency b must be chosen; let it be b = Hz; as a consequence ω = Hz and ω + = Hz. (This is exactly a case in which the considered pulse is not plagued by the presence of backward-traveling components, since the chosen spectrum possesses totally negligible values for ω<b.) The construction of the resulting pulse is already satisfactory when considering about 51 terms ( 25 n 25) in the series entering Eq. (13). Figures 36 show the pulse evaluated by summing the 51 terms. Figure 36a depicts the orthogonal projection of the pulse intensity, and Figure 36b shows the 3D intensity pattern of the real part of the pulse, which reveals the carrier wave oscillations. The ball-like shape 7 for the field intensity typically should be associated with all the subluminal LWs, whereas the typical superluminal 7 It can be noted that each term of the series in Eq. (13) corresponds to an ellipsoid or, more specifically, to a spheroid, for each velocity v.

92 326 Erasmo Recami and Michel Zamboni-Rached FIGURE 36 (a) The intensity orthogonal projection for the pulse corresponding to Eqs. (131) and (132) in the case of an optical frequency (see text). (b) The 3D intensity pattern of the real part of the same pulse, which reveals the carrier wave oscillations. ones are known to be X-shaped (Lu and Greenleaf (1992a); Recami (1998); Recami, Zamboni-Rached, and Dartora (24)), as predicted by SR in its non-restricted version (see Barut, Maccarrone, and Recami (1982); Recami (1986, 1998); Recami et al. (23) and references therein). For instance, a second spectrum S(ω) would be the inverted parabola, centered at the frequency ω ; that is, S(ω) = 4 [ω (ω ω/2)][ω (ω + ω/2)] ω 2 otherwise, for ω ω/2 ω ω + ω/2 (133) where ω, the distance between the two zeros of the parabola, can be regarded as the spectrum bandwidth. One can expand S(ω), given by Eq. (133), in the Fourier series (128), for ω ω ω +, with coefficients A n that even if straightforwardly calculable are complicated, so we omit reporting them here. Here we mention only that spectrum (133) may be easily used to determine, for instance, an ultrashort (femtosecond) optical nondiffracting pulse, with satisfactory results even when considering very few terms in expansion (128). Example 3. As a third interesting example, we consider the simple case when (within the integration limits ω, ω + ) the complex exponential spectrum in Eq. (124) is replaced by the real function (still linear in ω) S(ω) = a 1 exp [ a(ω + ω )] exp [a(ω ω +)], (134)

93 Localized Waves: A Review 327 with a a positive number (for a =, one reverts to the Mackinnon case). The spectrum in Eq. (134) is exponentially concentrated in the proximity of ω +, where it reaches its maximum value and becomes increasingly concentrated (on the left of ω + ) as the arbitrarily chosen value of a increases, its frequency bandwidth being ω = 1/a. Recall that, on their turn, quantities ω + and ω depend on the pulse velocity v and on the arbitrary parameter b. By performing the integration as in the case of Eq. (126), instead of the solution in Eq. (127), in the present case one eventually reaches the solution (ρ, ζ, η) = 2abβγ2 exp [abγ 2 ] exp [ aω + ] 1 exp [ a(ω + ω )] [ exp i b ] [ b c βγ2 η sinc c γ2 γ 2 ρ 2 (av + iζ) ]. 2 (135) After Mckinnon s, Eq. (135) appears to be the simplest closed-form solution, since both do not require any recourse to series expansions. In a sense, our solution in Eq. (135) may be regarded as the subluminal analog of the (superluminal) X-wave solution; the difference being that the standard X-shaped solution has a spectrum starting with, where it assumes its maximum value, while in the present case the spectrum starts at ω and increases afterward, until ω +. More important is that the Gaussian spectrum has a priori two advantages with respect to Eq. (134): It may be more easily centered around any value ω of ω, and, when increasing its concentration in the surrounding of ω, the spot transverse width does not increase indefinitely, but tends to the spot width of a Bessel beam with ω = ω and k z = (ω b)/v, at variance with what occurs with Eq. (134). Regardless, Eq. (135) is noteworthy since it is the simplest solution. Figure 37 shows the intensity of the real part of the subluminal pulse corresponding to this spectrum, with v =.99 c, with b = Hz (which results in ω = Hz and ω + = Hz), and with ω/ω + = 1/1 (i.e., a = 1). This is an optical pulse of.2 ps A Second Method for Constructing Subluminal Localized Pulses The previous method appears to be very efficient for determining analytic subluminal LWs, but it loses its validity in the limiting case v, since for v = itisω ω + and the integral in Eq. (117) degenerates, furnishing a null value. By contrast, we also are interested in the v = case, since it corresponds to some of the most interesting, and potentially useful, LWs namely, the stationary solutions to the wave equations endowed with a static envelope, which we have called frozen waves.

94 328 Erasmo Recami and Michel Zamboni-Rached (Re C) (mm) (mm).5 FIGURE 37 The intensity of the real part of the subluminal pulse corresponding to spectrum (134), with v =.99 c, with b = Hz (which results in ω = Hz and ω = Hz), and with ω/ω + = 1/1 (i.e., a = 1)..1 Before proceeding, recall that the theory of FWs was initially developed in Zamboni-Rached, Recami, and Hernández-Figueroa (25) by having recourse to discrete superpositions to bypass the need for numerical simulations. In the case of continuous superpositions, some numerical simulations were performed by Dartora et al. (26). However, the method presented in this section does allow determining analytic exact solutions (with no further need for numerical simulations) even for FWs consisting of continuous superpositions. We will show that the present method works regardless of the chosen field intensity shape and in regions with size of the order of the wavelength. It is possible to reach such results by starting from Eq. (115), with the constraint in Eq. (114), but going on (this time) to integrals over k z, instead of over ω. It is enough to write Eq. (116) in the form k z = 1 (ω b) (116 ) v for expressing the exact solutions in Eq. (115) as (ρ, z, t) = exp [ ibt] kz max k z min dk z S(k z ) J (ρk ρ ) exp [iζk z ], (136)

95 Localized Waves: A Review 329 with and with k z min = b 1 c 1 + β k z max = b 1 c 1 β (135) k 2 ρ = k2 z γ 2 + 2b c βk z + b2 c 2, (138) where quantity ζ is still defined according to Eq. (119), always with v < c. The unique exact solution previously known, Mackinnon (1978), may be rewritten in the form of Eq. (136) with S(k z ) = constant. Then, on following the same procedure in our first method (previous Section 5.3), we can obtain the new exact solutions corresponding to where [ ] i2nπkz S(k z ) = exp, (139) K K k z max k z min, by performing the change of variable [analogous, in its finality, to the one in Eq. (121)] k z b c γ2 (s + β). (14) At the end, the exact subluminal solution corresponding to spectrum in Eq. (139) results as (ρ, ζ, η) = 2 b [ c γ2 exp i b ] c βγ2 η exp [inπβ] sinc b 2 c 2 γ2 ρ 2 + ( ) b 2 c γ2 ζ + nπ, (141) We again observe that any spectra S(k z ) can be expanded, in the interval k z min < k z < k z max, in the Fourier series: S(k z ) = n= [ A n exp +in 2π ] K k z, (142)

96 33 Erasmo Recami and Michel Zamboni-Rached with coefficients given now by A n = 1 K kz max k z min [ dk z S(k z ) exp in 2π ] K k z, (143) quantity K having been defined previously. At the end of the process, the general exact solution representing a subluminal LW, for any spectra S(k z ), can be eventually written as (ρ, ζ, η) = 2 b [ c γ2 exp [i b ] c βγ2 η n= A n exp [inπβ] sinc b 2 c 2 γ2 ρ 2 + ( ) b 2 c γ2 ζ + nπ, (144) whose coefficients are expressed in Eq. (143), and where quantity η is defined as above, in Eq. (125). Interesting examples could easily be worked out, as at the end of the previous Section Stationary Solutions with Zero-Speed Envelopes Here we refer to the (second) method, expounded in the previous Section 5.4. Our solution in Eq. (144), for the case of envelopes at rest, that is, in the case v = (which implies ζ = z), becomes (ρ, z, t) = 2 b c exp [ ibt] n= A n sinc b 2 c 2 ρ2 + ( ) b 2 c z + nπ, (145) with coefficients A n given by Eq. (143) with v =, so that its integration limits simplify into b/c and b/c, respectively. Thus, A n = c 2b b/c b/c [ dk z S(k z ) exp [ in cπ ] b k z. (143 ) Equation (145) is a new exact solution, corresponding to stationary beams with a static intensity envelope. Observe, however, that even in this case there is energy propagation, as it can be easily verified from the power flux S s = R R / t (scalar case) or from the Poynting

97 Localized Waves: A Review 331 vector S v = (E H) (vectorial case: the condition being that R be a single component, A z, of the vector potential A) (Recami, 1998). We have indicated by R the real part of. For v =, Eq. (116) becomes ω = b ω, so that the particular subluminal waves endowed with null velocity are actually monochromatic beams. It may be stressed that this method does yield exact solutions, without any need for the paraxial approximation, which is so often used when looking for expressions representing beams like the Gaussian ones. We recall that, when having recourse to the paraxial approximation, the obtained beam expressions are valid only when the envelope sizes (e.g., the beam spot) vary in space much more slowly than the beam wavelength. For instance, the usual expression for a Gaussian beam, Newell and Molone (1992), holds only when the beam spot ρ is much larger than λ ω /(2πc) = b/(2πc): so that those beams cannot be very much localized. By contrast, our method overcomes such problems, since we have seen that it yields exact expressions for (well-localized) beams with sizes of the order of their wavelength. Notice, moreover, that the already known exact solutions for instance, the Bessel beams are nothing but particular cases of our solution in Eq. (145). Example. On choosing (with q < q + 1) the spectral double-step function c for q ω /c k z q + ω /c ω S(k z ) = (q + q ) elsewhere, (146) the coefficients of Eq. (145) become A n = ic 2πnω (q + q ) [ e iq +πn e iq πn ]. (147) The double-step spectrum in Eq. (146) corresponds, with regard to the longitudinal wave number, to the mean value k z = ω (q + + q )/2c and to the width k z = ω (q + q )/c. From such relations, it follows that k z /k z = 2(q + q )/(q + + q ). For values of q and q + that do not satisfy the inequality k z /k z << 1, the resulting solution will be a nonparaxial beam.

98 332 Erasmo Recami and Michel Zamboni-Rached FIGURE 38 (a) Orthogonal projection of the 3D intensity pattern of the beam (a null-speed subluminal wave) corresponding to spectrum (146); (b) 3D plot of the field intensity. The beam considered in this example is highly nonparaxial. Figures 38 shows the exact solution corresponding to ω = Hz (i.e., λ = 1 μm), and to q =.3 and to q + =.9. It results as a beam with a spot-size diameter of.6 μm, and, moreover, with a rather good longitudinal localization. In the case of Eqs. (144) and (145), about 21 terms ( 1 n 1) in the sum entering Eq. (143) are already satisfactory for a good evaluation of the series. The beam considered in this example is highly nonparaxial (with k z /k z = 1 ), and therefore could not have been obtained by ordinary Gaussian beam solutions (which are valid only in the paraxial regime) A New Approach to Frozen Waves A noticeable property of our present method is that it allows a spatial modeling of even monochromatic fields (that correspond to envelopes at rest; so that, in electromagnetic cases, one can speak, for example, of the modeling of light fields at rest ). Recall that such a modeling (rather interesting, especially for applications, Zamboni-Rached et al. (25)) was already performed in Zamboni- Rached (24, 26); Zamboni-Rached, Recami, and Hernández-Figueroa (25) and has been exploited at the beginning of the previous section [compare with Eqs. (99) and (1)], in terms of discrete superpositions of Bessel beams. However, the method presented in the previous subsections allows use of continuous superpositions to obtain a predetermined longitudinal (onaxis) intensity pattern, inside a desired space interval < z < L. In fact, the 8 Here we consider only scalar wave fields. In the case of nonparaxial optical beams, the vector character of the field has to be taken into account.

99 Localized Waves: A Review 333 continuous superposition, analogous to Eq. (1), now can be written as (ρ, z, t) = e iω t ω /c ω /c dk z S(k z ) J (ρk ρ ) e izk z, (148) which is nothing but the previous Eq. (136) with v = (and therefore ζ = z). In other words, Eq. (149) does represent a null-speed subluminal wave. To clarify, recall that the FWs were expressed in the past as discrete superposition because it was not then known how to analytically treat a continuous superposition like that in Eq. (148). However, the previous approach to FWs can now be extended to the case of integrals, without numerical simulations, but in terms of analytic solutions. Indeed, the exact solution of Eq. (148) is given by Eq. (145), with coefficients in Eq. (143 ); and one can choose the spectral function S(k z ) in such a way that assumes the on-axis prechosen static intensity pattern F(z) 2. Namely, the equation to be satisfied by S(k z ), to such an aim, is derived by associating Eq. (148) with the requirement (ρ =, z, t) 2 = F(z) 2, which entails the integral relation ω /c ω /c dk z S(k z ) e izk z = F(z). (149) Equation (149) would be trivially solvable in the case of an integration between and +, since it would merely be a Fourier transformation, but obviously this is not the case, because its integration limits are finite. Actually, there are functions F(z) for which Eq. (149) is not solvable at all, in the sense that no spectra S(k z ) exist obeying the last equation. For instance, if we consider the Fourier expansion F(z) = dk z S(kz ) e izk z, when S(k z ) does assume nonnegligible values outside the interval ω /c < k z <ω /c, then in Eq. (149) no S(k z ) can forward that particular F(z) as a result. However, some procedures can be devised, such that one can nevertheless find a function S(k z ) that approximately (but satisfactorily) complies with Eq. (149). The first procedure consists of writing S(k z ) in the form S(k z ) = 1 K n= ( ) 2nπ F e i2nπkz/k, (15) K

100 334 Erasmo Recami and Michel Zamboni-Rached where, as before, K = 2ω /c. Then, Eq. (15) can easily be verified as guaranteeing that the integral in Eq. (149) yields the values of the desired F(z) at the discrete points z = 2nπ/K. Indeed, the Fourier expansion (15) is already of the same type as Eq. (142), so that in this case the coefficients A n of our solution in Eq. (145), appearing in Eq. (143 ), do simply become A n = 1 ( K F 2nπ ). (151) K This is a powerful way to obtain a desired longitudinal (on-axis) intensity pattern, especially for tiny spatial regions, because it is not necessary to solve any integral to determine the coefficients A n, which by contrast are given directly by Eq. (151). Figures 39 depicts some interesting applications of this method. A few desired longitudinal intensity patterns F(z) 2 have been chosen, and the corresponding FWs are calculated by using Eq. (145) with the coefficients A n given in Eq. (151). The desired patterns are enforced to exist within very small spatial intervals only, to show the capability of the method to model (Zamboni-Rached and Recami, 28a) the field intensity shape under such strict requirements. In the four examples below, we considered a wavelength λ =.6 μm, which corresponds to ω = b = Hz. The first longitudinal (on-axis) pattern considered by us is that given by { e a(z Z) for z Z F(z) = elsewhere, that is, a pattern with an exponential increase, starting from z = until z = Z. The chosen values of a and Z are Z = 1 μm and a = 3/Z. The intensity of the corresponding FW is shown in Figure 39a. The second longitudinal pattern (on-axis) taken into consideration is the Gaussian one, given by F(z) = { e q( z Z )2 for Z z Z elsewhere, with q = 2 and Z = 1.6 μm. The intensity of the corresponding FW is shown in Figure 39b.

101 Localized Waves: A Review 335 FIGURE 39 Frozen Waves with the on-axis longitudinal field pattern chosen as (a) exponential, (b) Gaussian, (c) super-gaussian, and (d) zero-order Bessel function. In the third example, the desired longitudinal pattern is supposed to be a super-gaussian: { e q( Z z )2m for Z z Z F(z) = elsewhere, where m controls the edge sharpness. Here we have chosen q = 2, m = 4, and Z = 2 μm. The intensity of the FW obtained in this case is shown in Figure 39c. Finally, in the fourth example, we chose the longitudinal pattern as being the zero-order Bessel function { J (qz) for Z z Z F(z) = elsewhere,

102 336 Erasmo Recami and Michel Zamboni-Rached with q = m 1 and Z = 15 μm. The intensity of the corresponding FW is shown in Figure 39d. Any static envelopes of this type can be easily transformed into propagating pulses by the mere application of LTs. Another procedures exists for evaluating S(k z ), based on the assumption that S(k z ) S(k z ), (152) which constitutes a good approximation whenever S(k z ) assumes negligible values outside the interval [ ω /c, ω /c]. In such a case, there is recourse to the method associated with Eq. (142), and S(k z ) itself can be expanded in a Fourier series, eventually yielding the relevant coefficients A n by Eq. (143). Let us recall that it is still K k z max k z min = 2ω /c. It is worthwhile to call attention to the circumstance that, when constructing FWs in terms of a sum of discrete superpositions of Bessel beams (as done in our Section 5.1 and in the references Zamboni-Rached (24, 26); Zamboni-Rached, Recami, and Hernández-Figueroa (25); Zamboni-Rached et al. (25)), it was easy to obtain extended envelopes like, for example, cigars where easy means by using only a few terms of the sum. By contrast, when we construct FWs as continuous superpositions, then it is easy to get highly localized (concentrated) envelopes. Moreover, the method presented in this subsection furnishes FWs that are no longer periodic along the z-axis (a situation that, with our old method Zamboni-Rached (24, 26); Zamboni-Rached, Recami, and Hernández- Figueroa (25), was obtainable only when the periodicity interval tended to infinity) The Role of Special Relativity and Lorentz Transformations On one hand, strict connections exist between the principles and structure of SR and, on the other hand, the entire subject of subluminal, luminal, and superluminal localized waves, in the sense that it was expected for years that a priori they are transformable one into the other via suitable LTs (compare references Barut, Maccarrone, and Recami (1982); Mignani and Recami (1973); Recami (1986), in addition to ours in progress). 9 We first confine ourselves to the cases faced in this text section. Our subluminal localized pulses, which may be called wave bullets, behave 9 We call attention to a paper by Saari and Reivelt (24) only recently noticed by us, wherein the relativistic connections among the LWs were also investigated in terms of suitable LTs. We are happy to quote such a reference here because it appears inspired by a philosophy, which going back in part to papers like Barut, Maccarrone, and Recami (1982); Recami (1986) has been constantly shared by our old, and recent, papers. Let us mention also a further interesting article by Besieris et al. (1998), wherein ordinary LTs were already used, correctly, in the context of subluminal LWs (although the superluminal LTs used in that paper seem, however, to be partially defective).

103 Localized Waves: A Review 337 as particles. Indeed, our subluminal pulses (as well as the luminal and superluminal [X-shaped] ones, that have been amply investigated in the past literature) do exist as solutions of any wave equations, ranging from electromagnetism and acoustics or geophysics, to elementary particle physics (and even, as we discovered recently, to gravitation physics). From the kinematical point of view, the velocity composition relativistic law holds also for them. The same is true, more generally, for any LWs (pulses or beams). For simplicity we start by considering, in an initial reference frame O, just a (ν-order) Bessel beam: (ρ, φ, z, t) = J ν (ρk ρ ) e iνφ e izk z e iωt. (153) In a second reference frame O, moving with respect to O with speed u along the positive z-axis and in the positive direction, for simplicity it will be observed (Saari and Reivelt, 24) the new Bessel beam (ρ, φ, z, t ) = J ν (ρ k ρ ) eiνφ e iz k z e iω t, (154) obtained by applying the appropriate LT (a Lorentz boost ) with γ = [ 1 u 2 /c 2 ] 1 : k ρ = k ρ; k z = γ(k z uω/c 2 ); ω = γ(ω uk z ); (155) for example, this can be easily seen by putting ρ = ρ ; z = γ(z + ut ); t = γ(t + uz /c 2 ) (156) directly into Eq. (154). We now pass to subluminal pulses. We can investigate the action of a LT, by expressing them either via the first method (Section 4.3) or via the second one (Section 4.4). Let us consider for instance, in the frame O, a v-speed (subluminal) pulse, given by Eq. (155). We proceed to a second observer O moving with the same speed v with respect to frame O, and, still for the sake of simplicity, passing through the origin O of the initial frame at time t =. The new observer O will see the pulse (Saari and Reivelt, 24) (ρ, z, t ) = e it ω ω+ ω dω S(ω) J (ρ k ρ ) eiz k z, (157)

104 338 Erasmo Recami and Michel Zamboni-Rached with k z = γ 1 ω/v γb/v; ω = γb; k ρ = ω /c 2 k z 2, (158) as determined from the LTs in Eq. (155) or in Eq. (156), with u = v [and γ given by Eq. (123)]. Notice that k z is a function of ω, as expressed by the first of the three relations in Eq. (158), and that here ω is a constant. If we explicitly insert into Eq. (157) the relation ω = γ(vk z + γb), which is nothing but rewriting the first relation of Eq. (158), then Eq. (157) becomes ω (ρ, z, t /c ) = γv e it ω dk z S(k z ) J (ρ k ρ ) eiz k z, (159) ω /c where S is expressed in terms of the previous function S(ω), entering Eq. (157), as follows: S(k z ) = S(γvk z + γ2 b). (16) Equation (159) describes monochromatic beams with axial symmetry (and does coincide with what derived in our second method, in Section 13, when posing v = ). The remarkable conclusion is that a subluminal pulse, given by our Eq. (117), which appears as a v-speed pulse in a frame O, will appear (Saari and Reivelt, 24) in another frame O (traveling with respect to observer O with the same speed v in the same direction z) just as the monochromatic beam in Eq. (159) endowed with angular frequency ω = γb, whatever be the pulse spectral function in the initial frame O: even if the kind of monochromatic beam to which we arrive does of course depend 1 on the chosen S(ω). The opposite is also true, in general. Let us set forth explicitly an observation that until now has been noticed only by Zamboni-Rached and Recami (28a). Namely, we mention that, when starting not from Eq. (117) but from the most general solutions which, as already seen, are sums of solutions (117) over the various values b m of b, then a LT will lead to a sum of monochromatic beams actually, of harmonics (rather than to a single monochromatic beam). In particular, if the goal is to obtain a sum of harmonic beams, a LT must be applied to more general subluminal pulses. 1 One gets in particular a Bessel-type beam when S is a Dirac delta function: S(ω) = δ(ω ω ). Moreover, let us notice that, on applying a LT to a Bessel beam, one obtains another Bessel beam, with a different axicon angle.

105 Localized Waves: A Review 339 In additions, the various superluminal localized pulses are transformed (Saari and Reivelt, 24) one into the other by the mere application of ordinary LTs, while it may be expected that the subluminal and the superluminal LWs are to be linked (apart from some known technical difficulties that require particular caution) by the superluminal Lorentz transformations expounded long ago (see Barut, Maccarrone, and Recami, 1982; Mignani and Recami, 1973; Recami, 1986; Recami and Rodrigues, 1985 and references therein.) 11 Recall that in the years , the theory of SR, in its nonrestricted version, predicted that, while the simplest subluminal object is obviously a sphere (or, in the limit, a space point), the simplest superluminal object is, on the contrary, an X-shaped pulse (or, in the limit, a double cone) (see Figure 11, 12 from Barut, Maccarrone, and Recami, 1982; Recami, 1986). The circumstance that the localized solutions to the wave equations follow the same behavior is rather interesting, and is expected to be useful for example, in the case of elementary particles and quantum physics, for a deeper comprehension of de Broglie s and Schrödinger s wave mechanics. With regard to the fact that the simplest subluminal LWs, solutions to the wave equation, are ball-like, let us present in Figure 4, in ordinary 3D space, the general shape of Mackinnon s solutions, as expressed by Eq. (124) for v << c: In such figures we graphically depict the field isointensity surfaces, which results (as expected) to be just spherical in the considered case. We have also seen, among the others, that, even if our first method (Section 5.3) cannot directly yield zero-speed envelopes, such envelopes at rest, Eq. (145), however can be obtained by applying a v-speed LT to Eq. (13). In this way, one starts from many frequencies [Eq. (13)] and ends with one frequency only [Eq. (145)], since b gets transformed into the frequency of the monochromatic beam. 11 The topic of superluminal LTs is a delicate one, to the extent that the majority of the recent attempts to readdress this question and its applications seem to be defective (sometimes they do not even maintain the necessary covariance of the wave equation itself). 12 Recall, more specifically, that Figure 11 depicts the following. Let us start from an object that is intrinsically spherical (i.e., that is a sphere in its rest frame [Panel (a)]. Then, after a generic subluminal LT along x (i.e., under a subluminal x-boost), it is predicted by SR to appear as ellipsoidal due to Lorentz contraction [panel (b)]. After a superluminal x-boost (Mignani and Recami, 1973; Recami, 1986; Recami and Rodrigues, 1985) (namely, when this object moves (Recami, Zamboni-Rached, and Dartora, 24) with superluminal speed V), it is predicted by SR in its nonrestricted version, extended relatively (ER) to appear (Barut, Maccarrone, and Recami, 1982) as in panel [d], i.e., as occupying the cylindrically symmetric region bounded by a two-sheeted rotation hyperboloid and an indefinite double cone. The whole structure, according to ER, is expected to move rigidly and, of course, with the speed V, the cotangent square of the cone semi-angle being (V/c) 2 1. Panel [c] refers to the limiting case when the boost speed tends to c, either from the left or from the right [for simplicity, a space axis is skipped]). It is remarkable that the shape of the localized (subluminal and superluminal) pulses, solutions to the wave equations, appears to follow the same behavior; this can also play a role for better comprehension of de Broglie and Schrödinger wave mechanics. See also Figure 4.

106 34 Erasmo Recami and Michel Zamboni-Rached FIGURE 4 In Figure 39 we saw how SR, in its nonrestricted version (ER), predicted (Barut, Maccarrone, and Recami, 1982; Recami, 1986) that, while the simplest subluminal object is obviously a sphere (or, in the limit, a space point), the simplest superluminal object is, on the contrary, an X-shaped pulse (or, in the limit, a double cone). The circumstance that the localized solutions to the wave equations do follow the same pattern is rather interesting and is expected to be useful for example, in the case of elementary particles and quantum physics for a deeper comprehension of de Broglie and Schrödinger wave mechanics. With regard to the fact that the simplest subluminal LWs, solutions to the wave equations, are ball-like, let us depict in these figures, in ordinary 3D space, the general shape of Mackinnon solutions as expressed by Eq. (124), numerically evaluated for v << c. Panels (a) and (b) graphically represent the field iso-intensity surfaces, which in the considered case are (as expected) just spherical Nonaxially Symmetric Solutions: Higher-Order Bessel Beams To this point, we have directed attention to exact solutions representing axially symmetric (subluminal) pulses only: that is, to pulses obtained by suitable superpositions of zero-order Bessel beams. However, it is interesting to look for analytic solutions representing nonaxially symmetric subluminal pulses, which can be constructed in terms of superpositions of ν-order Bessel beams, with ν a positive integer (ν > ). This can be attempted both in the case of Section 4.3 (first method) and in the case of Section 4.4 (second method). For brevity s sake, let us consider only the first method (Section 4.3). One is immediately confronted with the difficulty that no exact solution is known for the integral in Eq. (122) when J (.) is replaced with J ν (.). This difficulty can be overcome by following a simple method that allows obtaining higher-order subluminal waves in terms of the axially symmetric ones. Indeed, it is well known that, if (x, y, z, t) is an exact solution to the ordinary wave equation, then / x and / y are also

107 Localized Waves: A Review 341 exact solutions. 13 By contrast, when working with cylindrical coordinates, if (ρ, φ, z, t) is a solution to the wave equation, quantities / ρ and / φ generally are not solutions. Nevertheless, it is not difficult to reach the noticeable conclusion that, once (ρ, φ, z, t) is a solution, then ( (ρ, φ, z, t) = e iφ ρ + i ) (161) ρ φ also is an exact solution. For instance, for an axially symmetric solution of the type = J (k ρ ρ) exp[ik z ] exp[ iωt], Eq. (161) yields = k ρ J 1 (k ρ ρ) exp[iφ] exp[ik z ] exp[ iωt], which is actually one more analytic solution. In other words, it is enough to start for simplicity from a zero-order Bessel beam, and to apply Eq. (161), successively, ν times, to get as a new solution = ( k ρ ) ν J ν (k ρ ρ) exp[iνφ] exp[ik z ] exp[ iωt], which is a ν-order Bessel beam. In this mannar, when applying ν times Eq. (161) to the (axially symmetric) subluminal solution (ρ, z, t) in Eqs. (128 13) [obtained from Eq. (117) with spectral function S(ω)], we get the subluminal nonaxially symmetric pulses ν (ρ, φ, z, t) as new analytic solutions, consisting, as expected, of superpositions of ν-order Bessel beams: n (ρ, φ, z, t) = ω+ ω dω S (ω) J ν (k ρ ρ) e iνφ e ik zz e iωt, (162) where k ρ (ω) is given by Eq. (118), and quantities S (ω) = ( k ρ (ω)) ν S(ω) are the spectra of the new pulses. If S(ω) is centered at a certain carrier frequency (it is a Gaussian spectrum, for instance), then S (ω) also will approximately be the same type. If we wish the new solution ν (ρ, φ, z, t) to possess a predefined spectrum S (ω) = F(ω), we can first take Eq. (117) and put S(ω) = F(ω)/( k ρ (ω)) ν in its solution [Eq. (13)] and afterward apply to it, ν times, the operator U exp[iφ] [ / ρ + (i/ρ) / φ)]. As a result, we obtain the desired pulse, ν (ρ, φ, z, t), endowed with S (ω) = F(ω). Example. On starting from the subluminal axially symmetric pulse (ρ, z, t), given by Eq. (13) with the Gaussian spectrum in Eq. (131), we can get the subluminal, nonaxially symmetric, exact solution 1 (ρ, φ, z, t) by simply calculating 1 (ρ, φ, z, t) = ρ eiφ, (163) 13 Let us mention that even n / z n and n / t n will be exact solutions.

108 342 Erasmo Recami and Michel Zamboni-Rached which actually yields the first-order pulse 1 (ρ, φ, z, t), which can be more compactly written in the form: 1 (ρ, φ, η, ζ) = 2 b [ c v γ2 exp i b ] c βγ2 η with n= [ A n exp in π ] β ψ 1n (164) where ψ 1n (ρ, φ, η, ζ) b2 c 2 γ2 ρ Z 3 [Z cos Z sin Z] e iφ, (165) Z b 2 c 2 γ2 ρ 2 + ( ) b 2 c γ2 ζ + nπ. (166) This exact solution corresponds to the superposition in Eq. (162), with S (ω) = k ρ (ω)s(ω), quantity S(ω) being given by Eq. (131). It is represented in Figure 41. The pulse intensity has a donut-like shape Concluding Remarks for Section 5 In this section we started by developing, by suitable superpositions of equal-frequency Bessel beams, a first 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 z L of the propagation axis z. Their intensity envelope remains static (i.e., with velocity v = ), so we named these new solutions to the wave equations (and, in particular, to the Maxwell equations) Frozen Waves (FW). Inside the envelope of a FW only the carrier wave propagates. In addition, the longitudinal shape, within the interval z L, can be chosen in such a way that no nonnegligible field exists outside the predetermined region (consisting, e.g., of one or more high-intensity peaks). Such 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 apparatus (mainly for destroying cancerous cells), and so on. We later addressed the more general subject of the subluminal LWs and showed that, as in the well-known superluminal case (Hernández- Figueroa, Zamboni-Rached, and Recami, 28), the subluminal solutions can be obtained by superposing Bessel beams (Zamboni-Rached and Recami, 28a). Such solutions were barely considered in the past because the superposition integral must run in this case over a finite interval (which

109 Localized Waves: A Review 343 (L.m) c(,,, ) (L.m) FIGURE 41 Orthogonal projection of the field intensity corresponding to the higher-order subluminal pulse represented by the exact solution Eq. (163), quantity being given by Eq. (128) with the Gaussian spectrum in Eq. (131). The pulse intensity here happens to have a donut-like shape. makes it mathematically difficult to determine analytic expressions for them). However, we have shown how it is possible to obtain, in a simple way, nondiffracting subluminal pulses as exact analytic solutions to the wave equations: for arbitrarily chosen frequencies and bandwidths, and avoiding any recourse to the backward-traveling components. Until recently, only one closed-form subluminal LW solution, ψ cf,tothe wave equations was known, Mackinnon (1978). It was obtained by choosing in the relevant integration a constant weight function S(ω), whereas all other solutions had previously got only by numerical simulations. On the contrary, a subluminal LW can be obtained in closed form by adopting, for instance, any spectra S(ω) that are expansions in terms of ψ cf.in fact, the initial disadvantage dealing with a limited bandwidth may be turned into an advantage because, in the case of truncated integrals, the spectrum S(ω) can be expanded in a Fourier series. More generally, it has been shown how exact solutions can be determined both by integration over the Bessel beams angular frequency ω and by integration over their longitudinal wave number k z. Both methods have been expounded above.