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JWDD074-FM JWDD074-Hernandez December 5, 2007 4:41 Char Count= 0

Localized Waves

Edited by

HUGO E. HERNÁNDEZ-FIGUEROAMICHEL ZAMBONI-RACHEDERASMO RECAMI

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InnodataFile Attachment9780470168974.jpg


Localized Waves

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Localized Waves

Edited by

HUGO E. HERNÁNDEZ-FIGUEROAMICHEL ZAMBONI-RACHEDERASMO RECAMI

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Copyright C© 2008 by John Wiley & Sons, Inc. All rights reserved.

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Library of Congress Cataloging-in-Publication Data:

Localized waves / edited by Hugo E. Hernández-Figueroa, Michel Zamboni-Rached,Erasmo Recami.

p. cm.ISBN 978-0-470-10885-7 (cloth)1. Localized waves–Research. I. Hernández-Figueroa, Hugo E.

II. Zamboni-Rached, Michel. III. Recami, Erasmo.QC157.L63 2007532′.0593—dc22

2007002548

Printed in the United States of America

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Contents

CONTRIBUTORS xiii

PREFACE xv

1 Localized Waves: A Historical and Scientific Introduction 1Erasmo Recami, Michel Zamboni-Rached, andHugo E. Hernández-Figueroa

1.1 General Introduction 2

1.2 More Detailed Information 61.2.1 Localized Solutions 9

Appendix: Theoretical and Experimental History 17Historical Recollections: Theory 17X-Shaped Field Associated with a Superluminal Charge 20A Glance at the Experimental State of the Art 23

References 34

2 Structure of Nondiffracting Waves and Some InterestingApplications 43Michel Zamboni-Rached, Erasmo Recami, andHugo E. Hernández-Figueroa

2.1 Introduction 43

2.2 Spectral Structure of Localized Waves 442.2.1 Generalized Bidirectional Decomposition 46

2.3 Space–Time Focusing of X-Shaped Pulses 542.3.1 Focusing Effects Using Ordinary X-Waves 55

2.4 Chirped Optical X-Type Pulses in Material Media 572.4.1 Example: Chirped Optical X-Type Pulse in Bulk

Fused Silica 62

v


vi CONTENTS

2.5 Modeling the Shape of Stationary Wave Fields: Frozen Waves 632.5.1 Stationary Wave Fields with Arbitrary Longitudinal

Shape in Lossless Media Obtained by SuperposingEqual-Frequency Bessel Beams 63

2.5.2 Stationary Wave Fields with Arbitrary LongitudinalShape in Absorbing Media: Extending the Method 70

References 76

3 Two Hybrid Spectral Representations and Their Applicationsto the Derivations of Finite-Energy Localized Waves andPulsed Beams 79Ioannis M. Besieris and Amr M. Shaarawi

3.1 Introduction 79

3.2 Overview of Bidirectional and SuperluminalSpectral Representations 803.2.1 Bidirectional Spectral Representation 813.2.2 Superluminal Spectral Representation 83

3.3 Hybrid Spectral Representation and Its Applicationto the Derivation of Finite-Energy X-ShapedLocalized Waves 843.3.1 Hybrid Spectral Representation 843.3.2 (3 + 1)-Dimensional Focus X-Wave 853.3.3 (3 + 1)-Dimensional Finite-Energy X-Shaped

Localized Waves 86

3.4 Modified Hybrid Spectral Representation and Its Applicationto the Derivation of Finite-Energy Pulsed Beams 893.4.1 Modified Hybrid Spectral Representation 893.4.2 (3 + 1)-Dimensional Splash Modes and Focused

Pulsed Beams 89

3.5 Conclusions 93References 93

4 Ultrasonic Imaging with Limited-Diffraction Beams 97Jian-yu Lu

4.1 Introduction 97

4.2 Fundamentals of Limited-Diffraction Beams 994.2.1 Bessel Beams 994.2.2 Nonlinear Bessel Beams 1014.2.3 Frozen Waves 1014.2.4 X-Waves 1014.2.5 Obtaining Limited-Diffraction Beams with Variable

Transformation 102


CONTENTS vii

4.2.6 Limited-Diffraction Solutions to the Klein–GordonEquation 103

4.2.7 Limited-Diffraction Solutions to the SchrödingerEquation 106

4.2.8 Electromagnetic X-Waves 1084.2.9 Limited-Diffraction Beams in Confined Spaces 1094.2.10 X-Wave Transformation 1144.2.11 Bowtie Limited-Diffraction Beams 1154.2.12 Limited-Diffraction Array Beams 1154.2.13 Computation with Limited-Diffraction Beams 115

4.3 Applications of Limited-Diffraction Beams 1164.3.1 Medical Ultrasound Imaging 1164.3.2 Tissue Characterization (Identification) 1164.3.3 High-Frame-Rate Imaging 1164.3.4 Two-Way Dynamic Focusing 1164.3.5 Medical Blood-Flow Measurements 1174.3.6 Nondestructive Evaluation of Materials 1174.3.7 Optical Coherent Tomography 1174.3.8 Optical Communications 1174.3.9 Reduction of Sidelobes in Medical Imaging 117


5 Propagation-Invariant Fields: Rotationally Periodic andAnisotropic Nondiffracting Waves 129Janne Salo and Ari T. Friberg

5.1 Introduction 1295.1.1 Brief Overview of Propagation-Invariant Fields 1305.1.2 Scope of This Chapter 133

5.2 Rotationally Periodic Waves 1345.2.1 Fourier Representation of General RPWs 1355.2.2 Special Propagation Symmetries 1355.2.3 Monochromatic Waves 1365.2.4 Pulsed Single-Mode Waves 1385.2.5 Discussion 142

5.3 Nondiffracting Waves in Anisotropic Crystals 1425.3.1 Representation of Anisotropic Nondiffracting Waves 1435.3.2 Effects Due to Anisotropy 1465.3.3 Acoustic Generation of NDWs 1485.3.4 Discussion 149



viii CONTENTS

6 Bessel X-Wave Propagation 159Daniela Mugnai and Iacopo Mochi

6.1 Introduction 159

6.2 Optical Tunneling: Frustrated Total Reflection 1606.2.1 Bessel Beam Propagation into a Layer:

Normal Incidence 1606.2.2 Oblique Incidence 164

6.3 Free Propagation 1696.3.1 Phase, Group, and Signal Velocity: Scalar

Approximation 1696.3.2 Energy Localization and Energy Velocity:

A Vectorial Treatment 172

6.4 Space–Time and Superluminal Propagation 180References 181

7 Linear-Optical Generation of Localized Waves 185Kaido Reivelt and Peeter Saari


7.2 Definition of Localized Waves 186

7.3 The Principle of Optical Generation of LWs 191

7.4 Finite-Energy Approximations of LWs 193

7.5 Physical Nature of Propagation Invariance of PulsedWave Fields 195

7.6 Experiments 1987.6.1 LWs in Interferometric Experiments 1987.6.2 Experiment on Optical Bessel X-Pulses 2007.6.3 Experiment on Optical LWs 203


8 Optical Wave Modes: Localized and Propagation-InvariantWave Packets in Optically Transparent Dispersive Media 217Miguel A. Porras, Paolo Di Trapani, and Wei Hu


8.2 Localized and Stationarity Wave Modes Within the SVEA 2198.2.1 Dispersion Curves Within the SVEA 2218.2.2 Impulse-Response Wave Modes 222

8.3 Classification of Wave Modes of Finite Bandwidth 2248.3.1 Phase-Mismatch-Dominated Case: Pulsed Bessel

Beam Modes 2268.3.2 Group-Velocity-Mismatch-Dominated Case:

Envelope Focus Wave Modes 227


CONTENTS ix

8.3.3 Group-Velocity-Dispersion-Dominated Case:Envelope X- and Envelope O-Modes 229

8.4 Wave Modes with Ultrabroad Bandwidth 2318.4.1 Classification of SEWA Dispersion Curves 233

8.5 About the Effective Frequency, Wave Number, and PhaseVelocity of Wave Modes 236

8.6 Comparison Between Exact, SEWA, and SVEA Wave Modes 238


9 Nonlinear X-Waves 243Claudio Conti and Stefano Trillo


9.2 NLX Model 245

9.3 Envelope Linear X-Waves 2479.3.1 X-Wave Expansion and Finite-Energy Solutions 250

9.4 Conical Emission and X-Wave Instability 252

9.5 Nonlinear X-Wave Expansion 2559.5.1 Some Examples 2559.5.2 Proof 2569.5.3 Evidence 257

9.6 Numerical Solutions for Nonlinear X-Waves 2579.6.1 Bestiary of Solutions 259

9.7 Coupled X-Wave Theory 2629.7.1 Fundamental X-Wave and Fundamental Soliton 2649.7.2 Splitting and Replenishment in Kerr Media as a

Higher-Order Soliton 264

9.8 Brief Review of Experiments 2659.8.1 Angular Dispersion 2659.8.2 Nonlinear X-Waves in Quadratic Media 2659.8.3 X-Waves in Self-Focusing of Ultrashort Pulses in

Kerr Media 266


10 Diffraction-Free Subwavelength-Beam Optics on aNanometer Scale 273Sergei V. Kukhlevsky


10.2 Natural Spatial and Temporal Broadening of Light Waves 275

10.3 Diffraction-Free Optics in the Overwavelength Domain 281


x CONTENTS

10.4 Diffraction-Free Subwavelength-Beam Optics on aNanometer Scale 286

10.5 Conclusions 292Appendix 292References 293

11 Self-Reconstruction of Pulsed Optical X-Waves 299Ruediger Grunwald, Uwe Neumann, Uwe Griebner, Günter Steinmeyer,Gero Stibenz, Martin Bock, and Volker Kebbel


11.2 Small-Angle Bessel-Like Waves and X-Pulses 300

11.3 Self-Reconstruction of Pulsed Bessel-Like X-Waves 303

11.4 Nondiffracting Images 306

11.5 Self-Reconstruction of Truncated UltrabroadbandBessel–Gauss Beams 307


12 Localization and Wannier Wave Packets in Photonic CrystalsWithout Defects 315Stefano Longhi and Davide Janner


12.2 Diffraction and Localization of Monochromatic Waves inPhotonic Crystals 31712.2.1 Basic Equations 31712.2.2 Localized Waves 319

12.3 Spatiotemporal Wave Localization in Photonic Crystals 32412.3.1 Wannier Function Technique 32512.3.2 Undistorted Propagating Waves in Two- and

Three-Dimensional Photonic Crystals 329


13 Spatially Localized Vortex Structures 339Zdeněk Bouchal, Radek Čelechovský, and Grover A. Swartzlander, Jr.


13.2 Single and Composite Optical Vortices 342

13.3 Basic Concept of Nondiffracting Beams 346

13.4 Energetics of Nondiffracting Vortex Beams 350

13.5 Vortex Arrays and Mixed Vortex Fields 352


CONTENTS xi

13.6 Pseudo-nondiffracting Vortex Fields 354

13.7 Experiments 35713.7.1 Fourier Methods 35713.7.2 Spatial Light Modulation 358

13.8 Applications and Perspectives 361References 363

INDEX 367


xii


Contributors

Ioannis M. Besieris, The Bradley Department of Electrical and Computer Engi-neering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia

Martin Bock, Max-Born-Institute for Nonlinear Optics and Short-Pulse Spec-troscopy, Berlin, Germany

Zdeněk Bouchal, Department of Optics, Palacký University, Olomouc, CzechRepublic

Radek Čelechovský, Department of Optics, Palacký University, Olomouc, CzechRepublic

Claudio Conti, Research Center Enrico Fermi, Rome, Italy, and Research CenterSOFT INFM-CNR, University La Sapienza, Rome, Italy

Paolo Di Trapani, Dipartimento di Fisica e Matematica, Università degli Studidell’Insubria sede di Como, Como, Italy

Ari T. Friberg, Department of Microelectronics and Applied Physics, RoyalInstitute of Technology, Kista, Sweden

Uwe Griebner, Max-Born-Institute for Nonlinear Optics and Short-Pulse Spec-troscopy, Berlin, Germany

Ruediger Grunwald, Max-Born-Institute for Nonlinear Optics and Short-PulseSpectroscopy, Berlin, Germany

Hugo E. Hernández-Figueroa, Faculdade de Engenharia Elétrica e de Com-putação, Departamento de Microonda e Óptica, Universidade Estadual de Campinas,Campinas, SP, Brazil

Wei Hu, Laboratory of Photonic Information Technology, School for Informa-tion and Optoelectronic Science and Technology, South China Normal University,Guangzhou, P. R. China

Davide Janner, Dipartimento di Fisica, Politecnico di Milano, Milan, Italy

xiii


xiv CONTRIBUTORS

Volker Kebbel, Automation and Assembly Technologies GmbH, Bremen,Germany

Sergei V. Kukhlevsky, Department of Experimental Physics, Institute of Physics,University of Pécs, Pécs, Hungary

Stefano Longhi, Dipartimento di Fisica, Politecnico di Milano, Milan, Italy

Jian-yu Lu, Ultrasound Laboratory, Department of Bioengineering, The Universityof Toledo, Toledo, Ohio

Iacopo Mochi, Nello Carrara Institute of Applied Physics–CNR, FlorenceResearch Area, Sesto Fiorentino, Italy

Daniela Mugnai, Nello Carrara Institute of Applied Physics–CNR, FlorenceResearch Area, Sesto Fiorentino, Italy

Uwe Neumann, Max-Born-Institute for Nonlinear Optics and Short-Pulse Spec-troscopy, Berlin, Germany

Miguel A. Porras, Departamento de Fı́sica Aplicada, Escuela Técnica Superior deIngenieros de Minas, Universidad Politécnica de Madrid, Madrid, Spain

Erasmo Recami, Facoltà di Ingegneria, Università degli Studi di Bergamo,Bergamo, Italy and INFN–Sezione di Milano, Milan, Italy

Kaido Reivelt, Institute of Physics, University of Tartu, Tartu, Estonia

Peeter Saari, Institute of Physics, University of Tartu, Tartu, Estonia

Janne Salo, Laboratory of Physics, Helsinki University of Technology, Espoo,Finland

Amr M. Shaarawi, Department of Physics, The American University of Cairo,Cairo, Egypt

Günter Steinmeyer, Max-Born-Institute for Nonlinear Optics and Short-PulseSpectroscopy, Berlin, Germany

Gero Stibenz, Max-Born-Institute for Nonlinear Optics and Short-Pulse Spec-troscopy, Berlin, Germany

Grover A. Swartzlander, Jr., College of Optical Sciences, University of Arizona,Tucson, Arizona

Stefano Trillo, Department of Engineering, University of Ferrara, Ferrara, Italy,and Research Center SOFT INFM-CNR, University La Sapienza, Rome, Italy

Michel Zamboni-Rached, Centro de Ciências Naturais e Humanas, UniversidadeFederal do ABC, Santo André, SP, Brazil


Preface

Diffraction and dispersion effects have been well known for centuries and are rec-ognized to be limiting factors in many industrial and technology applications based,for example, on electromagnetic beams and pulses. Diffraction is an always-presentphenomenon, affecting two- or three-dimensional waves traveling in nonguiding me-dia. Arbitrary pulses and beams contain plane-wave components that propagate indifferent directions, causing a progressive increase in their spatial width along prop-agation. Dispersion is due to the dependence of the material media (refractive index)with frequency: therefore, each pulse’s spectral component propagates with a differ-ent phase velocity, so that an electromagnetic pulse will suffer a progressive increasein its temporal width along propagation. It is clear that these two effects may be aserious restriction for applications where it is highly desirable that the beam keepsits transverse localization or the pulse keeps its transverse localization and/or tem-poral width along propagation, which might be desirable, for example, in free-spacemicrowave, millimetric wave, terahertz and optical communications, microwave andoptical images, optical lithography, and optical tweezers. As a consequence, the de-velopment of techniques capable of alleviating signal degradation effects caused bythese two effects is of crucial importance.

Localized waves, also known as nondiffractive waves, arose initially as an at-tempt to obtain beams and pulses capable of resisting diffraction in free space forlong distances. Such waves were obtained initially theoretically as solutions to thewave equation in the early 1940s (J. A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941), and were demonstrated experimentally in 1987 (J. Durnin,J. J. Miceli, and J. H. Eberly, Diffraction-free beams, Phys. Rev. Lett., vol. 58, pp.1499–1501, 1987). Nowadays localized waves constitute a growing and dynamic re-search topic, not only in relation to nondispersive free space (or vacuum), but also fordispersive, nonlinear, and lossy nonguiding media. Taking into account the significantamount of exciting and impressive results published especially in the last five yearsor so, we decided to edit a book on this topic, the first of its kind in the literature. Thebook is composed of 13 chapters authored by the most productive researchers in thefield, with a well-balanced presentation of theory and experiment.

xv


xvi PREFACE

In Chapter 1, Recami et al. present a thorough review of localized waves, em-phasizing the theoretical foundations along with historical aspects and the intercon-nections of this subject with other technology and scientific areas.

In Chapter 2, Zamboni-Rached et al. discuss in detail the theoretical structure oflocalized waves, and some applications are presented, among which frozen waves areof particular interest.

In Chapter 3, Besieris and Shaarawi present a hybrid spectral representation methodwhich permits a smooth transition between two seemingly disparate classes of finite-energy spatiotemporally localized wave solutions to the three-dimensional scalar waveequation in free space: superluminal (X-shaped) and luminal (FWM-type) pulsedwaves. An additional advantage of the hybrid form is that it obviates the presenceof backward wave components, propagating at the luminal speed c, that have tobe minimized in practical applications. A modified hybrid spectral representationmethod has also been presented which permits a seamless transition from superluminallocalized waves to exact luminal splash modes. Within the framework of a certainparametrization, the latter are rendered indistinguishable from the paraxial luminalfinite-energy-focused pulsed beam solutions.

In Chapter 4, Jian-yu Lu describes X-waves in depth, providing generalized meth-ods for obtaining such waves through proper transformations, related primarily to theLorentz transformation. X-wave solutions to Schrödinger and Klein–Gordon equa-tions are also provided. In addition, the potential application of X-waves in medicalultrasound imaging is demonstrated experimentally.

In Chapter 5, Salo and Friberg show theoretically that diffraction-free wave prop-agation can also be achieved in anisotropic crystalline media. They explicitly analyzethe effect of arbitrary anisotropies on both continuous-wave and pulsed nondiffractingfields. Due to beam steering and other effects, generation of nondiffracting waves inanisotropic media poses new challenges, and the authors propose an efficient schemefor the generation and detection of a continuous-wave beam in a crystal wafer.

In Chapter 6, Mugnai and Mochi explore Bessel X-waves’ ability to provide local-ized energy and to exhibit superluminal propagation in both phase and group velocities(as verified experimentally). The authors also describe the ability of such waves totravel through a classically forbidden region (tunneling region) with no shift in thedirection of propagation, which makes them different and unique with respect toordinary waves.

In Chapter 7, Reivelt and Saari focus on the physical nature and experimentalimplementation or generation of localized waves. The authors demonstrate that theangular spectrum representation and the tilted pulse representation of localized wavesare suitable tools for achieving these purposes. They explain the concepts and resultsof their experiments, where the realizability of Bessel X-waves and focus wave modeswas verified for the first time.

In Chapter 8, Porras et al. present an interesting discussion of linear bullets, three-dimensional localized waves or particlelike waves propagating across a host medium,defeating diffraction spreading and dispersion broadening. Special attention is givento the generation of these bullets in practical settings by optical devices or by nonlinearmeans, showing the intimate relation between linear and nonlinear approaches to wave


PREFACE xvii

bullets, as in light filaments. The advantage of linear bullets with respect to standardwave packets (Gaussian-like) is also demonstrated for a variety of applications, suchas laser writing in thick media, ultraprecise microhole drilling for photonic-crystalfabrication, and laser micromachining.

In Chapter 9, the theory of X-waves in nonlinear materials is discussed thoroughlyby Conti and Trillo. Potential applications in light-matter interactions at high laserintensities in quantum optics and on the theoretical prediction of X-waves in Bose–Einstein condensates are pointed out.

In Chapter 10, by Kukhlevsky, the problem of spatial localization of light in freespace on a nanometer scale is presented in detail. The author shows that a sub-wavelength nanometer-sized beam propagating without diffractive broadening canbe produced by the interference of multiple beams of a Fresnel light source of therespective material waveguide. The results demonstrate theoretically the feasibility ofdiffraction-free subwavelength-beam optics on a nanometer scale for both continuouswaves and ultrashort (near-single-cycle) pulses. The approach extends the operationalprinciple of near-field subwavelength-beam optics, such as near-field scanning opticalmicroscopy, to the “not-too-distant” field regime (up to about 0.5 wavelength). Thechapter includes theoretical illustrations to facilitate an understanding of the naturalspatiotemporal broadening of light waves and the physical mechanisms that contributeto the diffraction-free propagation of subwavelength beams in free space.

In Chapter 11, Grunwald et al. show experimentally that ultraflat thin-film axi-cons enable the real physical approximation of nondiffracting beams and X-pulsesof extremely narrow angular spectra. By self-apodized truncation of Bessel–Gausspulses (coincidence of first field minimum with the rim of an aperture), needle-shaped propagation zones of large axial extension can be obtained without additionaldiffraction effects. The signature of undistorted progressive waves was indicated forsuch needle beams by the fringe-free propagation characteristics and ultrabroadbandspatio-spectral transfer functions.

In Chapter 12, Longhi and Janner provide a general overview of wave localiza-tion (in a weak sense) for an important and novel class of inhomogeneous periodicdielectric media (i.e., in photonic crystals), which have received increasing attentionin recent years. Compared to wave localization in homogeneous media, such as ina vacuum, the presence of a periodic dielectric permittivity strongly modifies thespace–time dispersion surfaces and hence the types of localized waves that may beobserved in photonic crystals.

In Chapter 13, Bouchal et al. focus on theoretical and experimental problems ofnondiffracting and singular optics. Particular attention is devoted to physical prop-erties, methods of experimental realization, and potential applications of single andcomposed vortex fields carried by a pseudo-nondiffracting background beam. Theunique propagation effects of vortex fields are pointed out, and consequences oftheir spiral phase singularities manifested by a transfer of the orbital angular mo-mentum are also discussed. The complex vortex structures whose parameters andproperties are controlled dynamically by a spatial light modulation provide advancedmethods of encoding and recording of information and can be utilized effectivelyin optical manipulations. Spatially localized vortex structures can be extended into


xviii PREFACE

nonstationary optical fields where novel spatiotemporal effects and applications can beexpected.

Acknowledgments

The editors are grateful to all the contributors to this volume for their efforts inproducing stimulating high-quality chapters in an area that is not yet well knownoutside the community of experts, always with the aim of making the area moreeasily accessible to interested physicists and engineers. For useful discussions theyare grateful to, among others, R. Bonifacio, M. Brambilla, R. Chiao, C. Cocca,C. Conti, A. Friberg, G. Degli Antoni, F. Fontana, G. Kurizki, M. Mattiuzzi,P. Milonni, P. Saari, A. Shaarawi, R. Ziolkowski, M. Tygel, and L. Ambrosio.

Preparation of the manuscript was facilitated greatly by George J. Telecki, RachelWitmer, and Angioline Loredo from Wiley; we thank them for their fine professionalwork. We are indebted to Kai Chang, Ioannis M. Besieris, and Richard W. Ziolkowskifor their crucial and inspirational encouragement. We would also like to thank ourwives: Marli de Freitas Gomes Hernández, Jane Marchi Madureira, and Marisa T.Vasconselos, for their continuous loving support.

The Editors

JWDD074-c01 JWDD074-Hernandez December 5, 2007 5:30 Char Count= 0

CHAPTER ONE

Localized Waves: A Historicaland Scientific Introduction

ERASMO RECAMIUniversità degli Studi di Bergamo, Bergamo, Italy, and INFN–Sezione diMilano, Milan, Italy

MICHEL ZAMBONI-RACHEDCentro de Ciências Naturais e Humanas, Universidade Federal do ABC,Santo André, SP, Brazil

HUGO E. HERNÁNDEZ-FIGUEROAUniversidade Estadual de Campinas, Campinas, SP, Brazil

In the first part of this introductory chapter, we present general and formal (simple)introductions to the ordinary Gaussian waves and to the Bessel waves, by explic-itly separating the cases of the beams from the cases of the pulses; and, finally, ananalogous introduction is presented for the localized waves (LW), pulses or beams.Always we stress the very different characteristics of the Gaussian with respect tothe Bessel waves and to the LWs, showing the numerous and important properties ofthe latter w.r.t. the former ones: Properties that may find application in all fields inwhich an essential role is played by a wave-equation (like electromagnetism, optics,acoustics, seismology, geophysics, gravitation, elementary particle physics, etc.). Inthe second part of this chapter (namely, in its Appendix), we recall at first how, in theseventies and eighties, the geometrical methods of special relativity (SR) predicted—in the sense below specified—the existence of the most interesting LWs, i.e., of theX-shaped pulses. At last, in connection with the circumstance that the X-shaped waves

Localized Waves, Edited by Hugo E. Hernández-Figueroa, Michel Zamboni-Rached, and Erasmo RecamiCopyright C© 2008 John Wiley & Sons, Inc.

1


2 LOCALIZED WAVES: A HISTORICAL AND SCIENTIFIC INTRODUCTION

are endowed with superluminal group-velocities (as carefully discussed in the firstpart of this chapter), we mention briefly the various experimental sectors of physicsin which superluminal motions seem to appear: In particular, a bird’s-eye view is pre-sented of the experiments till now performed with evanescent waves (and/or tunnelingphotons), and with the “localized superluminal solutions” to the wave equations.

1.1 GENERAL INTRODUCTION

Diffraction and dispersion have long been known as phenomena that limit the ap-plication of (e.g., optical) beams or pulses. Diffraction is always present, affectingany waves that propagate in two- or three-dimensional media, even when homoge-neous. Pulses and beams comprise waves traveling along different directions whichproduce gradual spatial broadening [6]. This effect is a limiting factor whenever apulse is needed that maintains its transverse localization (e.g., in free-space commu-nications [7], image forming [8], optical lithography [9,10], electromagnetic tweez-ers [11,12]).

Dispersion acts on pulses propagating in material media, causing mainly tempo-ral broadening: an effect known to be due to the variation in refraction index withfrequency, so that each spectral component of the pulse possesses a different phasevelocity. This entails gradual temporal widening, which constitutes a limiting fac-tor when a pulse is needed that maintains its time width (e.g., in communicationsystems [13]).

It is important, therefore, to develop any techniques able to reduce those phenom-ena. Localized waves, also known as nondiffracting waves, are indeed able to resistdiffraction for a long distance in free space. Such solutions to the wave equations(and, in particular, to Maxwell’s equations, under weak hypotheses) were predictedtheoretically long ago [14–17] (cf. also [18] and the Appendix to this chapter), con-structed mathematically in more recent times [19,20] and soon after produced exper-imentally [21–23]. Today, localized waves are well established both theoretically andexperimentally and are being used in innovative applications not only in vacuum butalso in material (linear or nonlinear) media, showing to be able to resist also disper-sion. As we mentioned, their potential applications are being explored intensively,always with surprising results, in such fields as acoustics, microwaves, and optics,and are also promising in mechanics, geophysics, and even in gravitational waves andelementary particle physics. Also worth noting are the possible applications of the“frozen waves,” discussed in Chapter 2, and the ones already obtained, for instance,in high-resolution ultrasound scanning of moving organs in the human body [24,25].

Restricting ourselves to electromagnetism, we cite present-day studies on electro-magnetic tweezers [26–29], optical (or acoustic) scalpels, optical guiding of atoms or(charged or neutral) corpuscles [30–32], optical litography [26,33], optical (or acous-tic) images [34], communications in free space [19,35–37], remote optical align-ment [38], and optical acceleration of charged corpuscles, among others.

Next we describe briefly the theory and applications of localized beams and pulses.

Localized (Nondiffracting) Beams The word beam refers to a monochromaticsolution to a wave equation, with transverse localization of its field. To fix our ideas,


1.1 GENERAL INTRODUCTION 3

we refer explicitly to the optical case, but our considerations hold for any waveequation (vectorial, spinorial, scalar—in particular, for the acoustic case).

The most common type of optical beam is the Gaussian beam, whose transversebehavior is described by a Gaussian function. But all the common beams suffer adiffraction, which spoils the transverse shape of their field, widening it graduallyduring propagation. As an example, the transverse width of a Gaussian beam doubleswhen it travels a distance zdif =

√3 π�ρ20/λ0, where �ρ0 is the beam initial width

and λ0 is its wavelength. One can verify that a Gaussian beam with an initial transverseaperture of the order of its wavelength will double its width after having traveled onlya few wavelengths.

It was generally believed that the only wave devoid of diffraction was the planewave, which does not undergo any transverse change, but some authors had shown thatit is not the only one. For instance, in 1941, Stratton [15] obtained a monochromatic so-lution to the wave equation whose transverse shape was concentrated in the vicinity ofits propagation axis and represented by a Bessel function. Such a solution, now calleda Bessel beam, was not subject to diffraction, since no change in its transverse shapetook place with time. Later, Courant and Hilbert [16] demonstrated how a large classof equations (including the wave equations) admit “nondistorted progressing waves”as solutions; and as early as 1915, Bateman [17] and others [39] showed the existenceof soliton-like, wavelet-type solutions to Maxwell’s equations. But all such literaturedid not attract the attention it deserved. In Stratton’s work [15] this can be partiallyjustified since the Bessel beam was endowed with infinite energy (as much as the planewaves). An interesting problem, therefore, was that of investigating what would hap-pen to the ideal Bessel beam solution when truncated by a finite transverse aperture.

Not until 1987 did a heuristical answer came from an actual experiment, whenDurnin et al. [40] showed that a realistic Bessel beam endowed with wavelengthλ0 = 0.6328 µm and central spot† �ρ0 = 59 µm, passing through an aperture withradius R = 3.5 mm, is able to travel about 85 cm keeping its transverse intensityshape approximately unchanged (in the region ρ



Annular aperture

f

a

Bessel beam

R

Lens

δa


1.1 GENERAL INTRODUCTION 5

a−NaN−1

aN

δa c), and have been studiedin a number of papers. Actually, when the phase velocity does not depend on the



P1 P2

R

R τc

ξ=τv

FIGURE 1.3 An X-shaped wave: that is, a localized superluminal pulse. It is an X-wave,possessing the velocity V > c, and the figure illustrates the fact that if its vertex or central spotis located at P1 at time t0, it will reach position P2 at time t + τ , where τ = |P2 − P1|/V <|P2 − P1|/c. This is 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 lightin the electromagnetic case, the sound speed in acoustics, etc.).

frequency, it is known that such a phase velocity becomes the group velocity! Re-membering how a superposition of Bessel beams is generated (e.g., by a discrete orcontinuous set of annular slits or transducers), it is clear that the energy forming theX-waves coming from those rings travels at the ordinary speed c of plane waves inthe medium considered [20,60–62]. (Here, c, representing the velocity of the planewaves in the medium, is the speed of sound in the acoustic case, the speed of light inthe electromagnetic case, and so on.) Nevertheless, the peak of the X-shaped wavesis faster than c.

It is possible to generate (in addition to the “classic” X-wave produced by Luet al. in 1992) infinite sets of new X-shaped waves, with their energy concentrated moreand more in a spot corresponding to the vertex region [42]. It may therefore appearrather intriguing that such a spot (even if no violation of special relativity is obviouslyimplied: all the results come from Maxwell’s equations or from the wave equations[73,74]) travels superluminally when the waves are electromagnetic. We shall call allthe X-shaped waves superluminal even when, for example, the waves are acoustic. InFig. 1.3, we illustrate the fact that if its vertex or central spot is located at P1 at timet1, it will reach position P2 at time t + τ , where τ = |P2 − P1|/V < |P2 − P1|/c.We discuss all these points below.

Soon after having constructed their “classic” acoustic X-wave mathematically andexperimentally, Lu et al. started applying them to ultrasonic scanning, obtaining, aswe already said, very high quality images. Subsequently, in a 1996 e-print and re-port, Recami et al. (see, e.g., [20] and references therein) published the analogousX-shaped solutions to Maxwell’s equations: by constructing scalar superluminal lo-calized solutions for each component of the Hertz potential. That showed, by theway, that localized solutions to scalar equation can also be used, under very weakconditions, for obtaining localized solutions to Maxwell’s equations (actually,


1.2 MORE DETAILED INFORMATION 7

Ziolkowski et al. [43] had found something similar, which they called slingshot pulses,for the simple scalar case, but their solution had gone practically unnoticed). In 1997,Saari and Reivelt [22] announced, in an important paper, the production in the lab ofan X-shaped wave in the optical realm, thus proving experimentally the existence ofsuperluminal electromagnetic pulses. Three years later, in 2000, Mugnai et al. [23]produced, experimentally, superluminal X-shaped waves in the microwave region(their paper aroused various criticisms, to which the authors responded).

1.2 MORE DETAILED INFORMATION

Let us consider [5] the differential equation known as homogeneous wave equa-tion: simple, but so important in acoustics, electromagnetism (microwaves, optics,etc.), geophysics, and even, as we said, gravitational waves and elementary particlephysics:

(∂2

∂x2+ ∂

2

∂y2+ ∂

2

∂z2− 1

c2∂2

∂t2

)ψ(x, y, z; t) = 0. (1.1)

Let us write it in the cylindrical coordinates (ρ, φ, z) and, for simplicity’s sake, confineourselves to axially symmetric solutions ψ(ρ, z; t). Then, Eq. (1.1) becomes

(∂2

∂ρ2+ 1

ρ

∂

∂ρ+ ∂

2

∂z2− 1

c2∂2

∂t2

)ψ(ρ, z; t) = 0. (1.2)

In free space, the solution ψ(ρ, z; t) can be written in terms of a Bessel–Fouriertransform with respect to the variable ρ, and two Fourier transforms with respect tovariables z and t :

ψ(ρ, z, t) =∫ ∞

0

∫ ∞−∞

∫ ∞−∞

kρ J0(kρ ρ) eikz z e−iωt ψ̄(kρ, kz, ω) dkρ dkz dω, (1.3)

where J0(·) is an ordinary zero-order Bessel function and ψ̄(kρ, kz, ω) is the transformof ψ(ρ, z, t).

Substituting Eq. (1.3) into Eq. (1.2), one finds that the relation

ω2

c2= k2ρ + k2z (1.4)

among ω, kρ , and kz has to be satisfied. In this way, by using condition (1.4) inEq. (1.3), any solutions to the wave equation (1.2) can be written

ψ(ρ, z, t) =∫ ω/c

0

∫ ∞−∞

kρ J0(kρ ρ) eiz√

ω2/c2 − k2ρ e−iωt S(kρ, ω) dkρ dω, (1.5)

where S(kρ, ω) is the spectral function chosen.



S(kρ,ωο) k

kz

kρ

êρ

êz

FIGURE 1.4 Visual interpretation of the integral solution (1.5) with the spectrum (1.6) interms of a superposition of plane waves.

The general integral solution (1.5) yields, for instance, (nonlocalized) Gaussianbeams and pulses, to which we shall refer to illustrate the differences between localizedwaves and them.

Gaussian Beam A very common (nonlocalized) beam is the Gaussian beam [76],corresponding to the spectrum

S(kρ, ω) = 2a2e−a2k2ρ δ(ω − ω0). (1.6)

In Eq. (1.6), a is a positive constant that will be shown to depend on the transverseaperture of the initial pulse.

Figure 1.4 illustrates the interpretation of integral solution (1.5) with spectralfunction (1.6) as a superposition of plane waves. From Fig. 1.4 one can easily realizethat this case corresponds to plane waves propagating in all directions (always withkz ≥ 0), the most intense being those directed along (positive) z. Notice that in theplane-wave case, �kz is the longitudinal component of the wave vector, �k = �kρ + �kz ,where �kρ = �kx + �ky .

On substituting Eq. (1.6) into Eq. (1.5) and adopting the paraxial approximation,one obtains the Gaussian beam:

ψGauss(ρ, z, t) = 2a2 exp[−ρ2/4(a2 + i z/2k0)]

2(a2 + i z/2k0)eik0(z−ct), (1.7)

where k0 = ω0/c. We can verify that such a beam, which suffers transverse diffrac-tion, doubles the initial width �ρ0 = 2a after having traveled the distance zdif =√

3 k0 �ρ20/2, called the diffraction length. The more concentrated a Gaussian beamhappens to be, the more rapidly it gets spoiled.


1.2 MORE DETAILED INFORMATION 9

Gaussian Pulse The most common (nonlocalized) pulse is the Gaussian pulse,which is obtained from Eq. (1.5) using the spectrum [75]

S(kρ, ω) = 2ba2

√π

e−a2k2ρ e−b

2(ω−ω0)2 , (1.8)

where a and b are positive constants. Indeed, such a pulse is a superposition ofGaussian beams of different frequencies.

Now, on substituting Eq. (1.8) into Eq. (1.5), and again adopting the paraxialapproximation, one gets the Gaussian pulse

ψ(ρ, z, t) = a2 exp[−ρ2/4(a2 + i z/2k0)]exp[−(z − ct)2/4c2b2]

a2 + i z/2k0 , (1.9)

endowed with speed c and temporal width �t = 2b, and suffering a progressiveenlargement of its transverse width so that its initial value is already doubled atposition zdif =

√3 k0 �ρ20/2, with �ρ0 = 2a.

1.2.1 Localized Solutions

Finally, let’s look at the construction of the two most renowned localized waves: theBessel beam and the ordinary X-shaped pulse [5].

It is interesting, first, to observe that, when superposing (axially symmetrical)solutions of the wave equation in vacuum, three spectral parameters (ω, kρ, kz) comeinto play which have however to satisfy constraint (1.4), deriving from the waveequation itself. Consequently, only two of them are independent, and here we choose†

ω and kρ . Such freedom in choosing ω and kρ was apparent in the spectral functionsgenerating Gaussian beams and pulses, which are the product of two functions, onedepending only on ω and the other on kρ .

We are going to see, moreover, that particular relations can be imposed between ωand kρ (or analogously, between ω and kz) in order to get interesting and unexpectedresults, such as localized waves.

Bessel Beam Let us start by imposing a linear coupling between ω and kρ (it couldactually be shown [41] that it is the unique coupling leading to localized solutions).Let us consider the spectral function

S(kρ, ω) = δ(kρ − (ω/c) sin θ )kρ

δ(ω − ω0), (1.10)

which implies that kρ = (ω sin θ )/c, with 0 ≤ θ ≤ π/2, a relation that can be regardedas a space–time coupling. Let us add that this linear constraint between ω and kρ ,

†Elsewhere we chose ω and kz .



k

kx

kzky

y

x

z

θ

FIGURE 1.5 An axially symmetric Bessel beam is created by the superposition of planewaves whose wave vectors lay on the surface of a cone having the propagation axis as itssymmetry axis and angle equal to θ (axicon angle).

together with relation (1.4), yields kz = (ω cos θ )/c. This is an important fact, since ithas been shown elsewhere [42] that an ideal localized wave must contain a couplingof the type ω = V kz + b, where V and b are arbitrary constants.

The interpretation of the integral function (1.5), this time with the spectrum (1.10),as a superposition of plane waves is visualized in Fig. 1.5, which shows that an axiallysymmetric Bessel beam is nothing but the result of the superposition of plane waveswhose wave vectors lay on the surface of a cone having the propagation axis as itssymmetry axis and an angle equal to θ , the axicon angle.

By inserting Eq. (1.10) into Eq. (1.5), one gets the mathematical expression of aBessel beam:

ψ(ρ, z, t) = J0(ω0

csin θ ρ

)exp

[iω0

ccos θ

(z − c

cos θt)]

. (1.11)

This beam possesses phase velocity vph = c/ cos θ and field transverse shape repre-sented by a Bessel function J0(·), so that its field is concentrated in the area surroundingthe propagation axis z. Moreover, Eq. (1.11) tells us that the Bessel beam keeps itstransverse shape (which is therefore invariant) while propagating, with central spot�ρ = 2.405c/(ω sin θ ).

The ideal Bessel beam is, however, not a square-integrable function and thuspossesses infinite energy (i.e., it cannot be produced experimentally).

But we can have recourse to truncated Bessel beams, generated by finite apertures.In this case the (truncated) Bessel beams are still able to travel a long distance whilemaintaining their transfer shape, as well as their speed, approximately unchanged[40,69,70]; that is, they still possess a large field depth. For instance, the depth offield of a Bessel beam generated by a circular finite aperture with radius R is givenby

Zmax = Rtan θ

, (1.12)

hugo e. hernandez-figueroa´ michel …...2.4 chirped optical x-type pulses in material media 57...

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