HUGO E. HERNÁNDEZ-FIGUEROA, PHD, is a Full Professor in the School of Electrical and Computer Engineering of the State University of Campinas (UNICAMP), Brazil. He is a Senior Member of the IEEE, an Associate Editor of the IEEE/OSA Journal of Lightwave Technology, and a Member of the Editorial Board of the IEEE Transactions on Microwave Theory and Techniques. His research interests concentrate on a wide variety of wave electromagnetics phenomena and applications mainly in photonics and microwaves. MICHEL ZAMBONI-RACHED, PHD, is a Professor in the Centro de Ciências Naturais e Humanas, Universidade Federal do ABC, Brazil. His research interests are electromagnetic field theory, theory and applications of localized waves (in electromagnetism, acoustics, and wave mechanics), optics, optical communications, and some topics in theoretical physics. ERASMO RECAMI, PHD, has been a Professor of Physics (currently at Bergamo State University, Italy) for the past forty years. His current research includes the structure of leptons, tunneling times, the application of the GR methods to strong interactions, extended SR, and, in particular, the superluminal group velocities associated with evanescent waves and with the localized solutions to Maxwell's equations.

• Contributors xiiiPreface xv1 Localized Waves: A Historical and Scientific Introduction 1Erasmo Recami, Michel Zamboni-Rached, and Hugo E. Hernández-Figueroa1.1 General Introduction 21.2 More Detailed Information 61.2.1 Localized Solutions 9Appendix: Theoretical and Experimental History 17Historical Recollections: Theory 17X-Shaped Field Associated with a Superluminal Charge 20A Glance at the Experimental State of the Art 23References 342 Structure of Nondiffracting Waves and Some Interesting Applications 43Michel Zamboni-Rached, Erasmo Recami, and Hugo E. Hernández-Figueroa2.1 Introduction 432.2 Spectral Structure of Localized Waves 442.2.1 Generalized Bidirectional Decomposition 462.3 Space–Time Focusing of X-Shaped Pulses 542.3.1 Focusing Effects Using Ordinary X-Waves 552.4 Chirped Optical X-Type Pulses in Material Media 572.4.1 Example: Chirped Optical X-Type Pulse in Bulk Fused Silica 622.5 Modeling the Shape of Stationary Wave Fields: Frozen Waves 632.5.1 Stationary Wave Fields with Arbitrary Longitudinal Shape in Lossless Media Obtained by Superposing Equal-Frequency Bessel Beams 632.5.2 Stationary Wave Fields with Arbitrary Longitudinal Shape in Absorbing Media: Extending the Method 70References 763 Two Hybrid Spectral Representations and Their Applications to the Derivations of Finite-Energy Localized Waves and Pulsed Beams 79Ioannis M. Besieris and Amr M. Shaarawi3.1 Introduction 793.2 Overview of Bidirectional and Superluminal Spectral Representations 803.2.1 Bidirectional Spectral Representation 813.2.2 Superluminal Spectral Representation 833.3 Hybrid Spectral Representation and Its Application to the Derivation of Finite-Energy X-Shaped Localized 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 863.4 Modified Hybrid Spectral Representation and Its Application to 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 893.5 Conclusions 93References 934 Ultrasonic Imaging with Limited-Diffraction Beams 97Jian-yu Lu4.1 Introduction 974.2 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 1024.2.6 Limited-Diffraction Solutions to the Klein–Gordon Equation 1034.2.7 Limited-Diffraction Solutions to the Schrödinger Equation 1064.2.8 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 1154.3 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 1174.4 Conclusions 117References 1185 Propagation-Invariant Fields: Rotationally Periodic and Anisotropic Nondiffracting Waves 129Janne Salo and Ari T. Friberg5.1 Introduction 1295.1.1 Brief Overview of Propagation-Invariant Fields 1305.1.2 Scope of This Chapter 1335.2 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 1425.3 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 1495.4 Conclusions 150References 1516 Bessel X-Wave Propagation 159Daniela Mugnai and Iacopo Mochi6.1 Introduction 1596.2 Optical Tunneling: Frustrated Total Reflection 1606.2.1 Bessel Beam Propagation into a Layer: Normal Incidence 1606.2.2 Oblique Incidence 1646.3 Free Propagation 1696.3.1 Phase, Group, and Signal Velocity: Scalar Approximation 1696.3.2 Energy Localization and Energy Velocity: A Vectorial Treatment 1726.4 Space–Time and Superluminal Propagation 180References 1817 Linear-Optical Generation of Localized Waves 185Kaido Reivelt and Peeter Saari7.1 Introduction 1857.2 Definition of Localized Waves 1867.3 The Principle of Optical Generation of LWs 1917.4 Finite-Energy Approximations of LWs 1937.5 Physical Nature of Propagation Invariance of Pulsed Wave Fields 1957.6 Experiments 1987.6.1 LWs in Interferometric Experiments 1987.6.2 Experiment on Optical Bessel X-Pulses 2007.6.3 Experiment on Optical LWs 2037.7 Conclusions 211References 2138 Optical Wave Modes: Localized and Propagation-Invariant Wave Packets in Optically Transparent Dispersive Media 217Miguel A. Porras, Paolo Di Trapani, and Wei Hu8.1 Introduction 2178.2 Localized and Stationarity Wave Modes Within the SVEA 2198.2.1 Dispersion Curves Within the SVEA 2218.2.2 Impulse-Response Wave Modes 2228.3 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 2278.3.3 Group-Velocity-Dispersion-Dominated Case: Envelope X- and Envelope O-Modes 2298.4 Wave Modes with Ultrabroad Bandwidth 2318.4.1 Classification of SEWA Dispersion Curves 2338.5 About the Effective Frequency, Wave Number, and Phase Velocity of Wave Modes 2368.6 Comparison Between Exact, SEWA, and SVEA Wave Modes 2388.7 Conclusions 240References 2409 Nonlinear X-Waves 243Claudio Conti and Stefano Trillo9.1 Introduction 2439.2 NLX Model 2459.3 Envelope Linear X-Waves 2479.3.1 X-Wave Expansion and Finite-Energy Solutions 2509.4 Conical Emission and X-Wave Instability 2529.5 Nonlinear X-Wave Expansion 2559.5.1 Some Examples 2559.5.2 Proof 2569.5.3 Evidence 2579.6 Numerical Solutions for Nonlinear X-Waves 2579.6.1 Bestiary of Solutions 2599.7 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 2649.8 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 2669.9 Conclusions 266References 26710 Diffraction-Free Subwavelength-Beam Optics on a Nanometer Scale 273Sergei V. Kukhlevsky10.1 Introduction 27310.2 Natural Spatial and Temporal Broadening of Light Waves 27510.3 Diffraction-Free Optics in the Overwavelength Domain 28110.4 Diffraction-Free Subwavelength-Beam Optics on a Nanometer Scale 28610.5 Conclusions 292Appendix 292References 29311 Self-Reconstruction of Pulsed Optical X-Waves 299Ruediger Grunwald, Uwe Neumann, Uwe Griebner, Günter Steinmeyer, Gero Stibenz, Martin Bock, and Volker Kebbel11.1 Introduction 29911.2 Small-Angle Bessel-Like Waves and X-Pulses 30011.3 Self-Reconstruction of Pulsed Bessel-Like X-Waves 30311.4 Nondiffracting Images 30611.5 Self-Reconstruction of Truncated Ultrabroadband Bessel–Gauss Beams 30711.6 Conclusions 310References 31112 Localization and Wannier Wave Packets in Photonic Crystals Without Defects 315Stefano Longhi and Davide Janner12.1 Introduction 31512.2 Diffraction and Localization of Monochromatic Waves in Photonic Crystals 31712.2.1 Basic Equations 31712.2.2 Localized Waves 31912.3 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 32912.4 Conclusions 334References 33513 Spatially Localized Vortex Structures 339Zdeněk Bouchal, Radek Č elechovsk, and Grover A. Swartzlander, Jr.13.1 Introduction 33913.2 Single and Composite Optical Vortices 34213.3 Basic Concept of Nondiffracting Beams 34613.4 Energetics of Nondiffracting Vortex Beams 35013.5 Vortex Arrays and Mixed Vortex Fields 35213.6 Pseudo-nondiffracting Vortex Fields 35413.7 Experiments 35713.7.1 Fourier Methods 35713.7.2 Spatial Light Modulation 35813.8 Applications and Perspectives 361References 363Index 367

HUGO E. HERNÁNDEZ-FIGUEROA, PHD, is a Full Professor in the School of Electrical and Computer Engineering of the State University of Campinas (UNICAMP), Brazil. He is a Senior Member of the IEEE, an Associate Editor of the IEEE/OSA Journal of Lightwave Technology, and a Member of the Editorial Board of the IEEE Transactions on Microwave Theory and Techniques. His research interests concentrate on a wide variety of wave electromagnetics phenomena and applications mainly in photonics and microwaves. MICHEL ZAMBONI-RACHED, PHD, is a Professor in the Centro de Ciências Naturais e Humanas, Universidade Federal do ABC, Brazil. His research interests are electromagnetic field theory, theory and applications of localized waves (in electromagnetism, acoustics, and wave mechanics), optics, optical communications, and some topics in theoretical physics. ERASMO RECAMI, PHD, has been a Professor of Physics (currently at Bergamo State University, Italy) for the past forty years. His current research includes the structure of leptons, tunneling times, the application of the GR methods to strong interactions, extended SR, and, in particular, the superluminal group velocities associated with evanescent waves and with the localized solutions to Maxwell's equations.

• Contributors xiiiPreface xv1 Localized Waves: A Historical and Scientific Introduction 1Erasmo Recami, Michel Zamboni-Rached, and Hugo E. Hernández-Figueroa1.1 General Introduction 21.2 More Detailed Information 61.2.1 Localized Solutions 9Appendix: Theoretical and Experimental History 17Historical Recollections: Theory 17X-Shaped Field Associated with a Superluminal Charge 20A Glance at the Experimental State of the Art 23References 342 Structure of Nondiffracting Waves and Some Interesting Applications 43Michel Zamboni-Rached, Erasmo Recami, and Hugo E. Hernández-Figueroa2.1 Introduction 432.2 Spectral Structure of Localized Waves 442.2.1 Generalized Bidirectional Decomposition 462.3 Space–Time Focusing of X-Shaped Pulses 542.3.1 Focusing Effects Using Ordinary X-Waves 552.4 Chirped Optical X-Type Pulses in Material Media 572.4.1 Example: Chirped Optical X-Type Pulse in Bulk Fused Silica 622.5 Modeling the Shape of Stationary Wave Fields: Frozen Waves 632.5.1 Stationary Wave Fields with Arbitrary Longitudinal Shape in Lossless Media Obtained by Superposing Equal-Frequency Bessel Beams 632.5.2 Stationary Wave Fields with Arbitrary Longitudinal Shape in Absorbing Media: Extending the Method 70References 763 Two Hybrid Spectral Representations and Their Applications to the Derivations of Finite-Energy Localized Waves and Pulsed Beams 79Ioannis M. Besieris and Amr M. Shaarawi3.1 Introduction 793.2 Overview of Bidirectional and Superluminal Spectral Representations 803.2.1 Bidirectional Spectral Representation 813.2.2 Superluminal Spectral Representation 833.3 Hybrid Spectral Representation and Its Application to the Derivation of Finite-Energy X-Shaped Localized 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 863.4 Modified Hybrid Spectral Representation and Its Application to 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 893.5 Conclusions 93References 934 Ultrasonic Imaging with Limited-Diffraction Beams 97Jian-yu Lu4.1 Introduction 974.2 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 1024.2.6 Limited-Diffraction Solutions to the Klein–Gordon Equation 1034.2.7 Limited-Diffraction Solutions to the Schrödinger Equation 1064.2.8 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 1154.3 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 1174.4 Conclusions 117References 1185 Propagation-Invariant Fields: Rotationally Periodic and Anisotropic Nondiffracting Waves 129Janne Salo and Ari T. Friberg5.1 Introduction 1295.1.1 Brief Overview of Propagation-Invariant Fields 1305.1.2 Scope of This Chapter 1335.2 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 1425.3 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 1495.4 Conclusions 150References 1516 Bessel X-Wave Propagation 159Daniela Mugnai and Iacopo Mochi6.1 Introduction 1596.2 Optical Tunneling: Frustrated Total Reflection 1606.2.1 Bessel Beam Propagation into a Layer: Normal Incidence 1606.2.2 Oblique Incidence 1646.3 Free Propagation 1696.3.1 Phase, Group, and Signal Velocity: Scalar Approximation 1696.3.2 Energy Localization and Energy Velocity: A Vectorial Treatment 1726.4 Space–Time and Superluminal Propagation 180References 1817 Linear-Optical Generation of Localized Waves 185Kaido Reivelt and Peeter Saari7.1 Introduction 1857.2 Definition of Localized Waves 1867.3 The Principle of Optical Generation of LWs 1917.4 Finite-Energy Approximations of LWs 1937.5 Physical Nature of Propagation Invariance of Pulsed Wave Fields 1957.6 Experiments 1987.6.1 LWs in Interferometric Experiments 1987.6.2 Experiment on Optical Bessel X-Pulses 2007.6.3 Experiment on Optical LWs 2037.7 Conclusions 211References 2138 Optical Wave Modes: Localized and Propagation-Invariant Wave Packets in Optically Transparent Dispersive Media 217Miguel A. Porras, Paolo Di Trapani, and Wei Hu8.1 Introduction 2178.2 Localized and Stationarity Wave Modes Within the SVEA 2198.2.1 Dispersion Curves Within the SVEA 2218.2.2 Impulse-Response Wave Modes 2228.3 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 2278.3.3 Group-Velocity-Dispersion-Dominated Case: Envelope X- and Envelope O-Modes 2298.4 Wave Modes with Ultrabroad Bandwidth 2318.4.1 Classification of SEWA Dispersion Curves 2338.5 About the Effective Frequency, Wave Number, and Phase Velocity of Wave Modes 2368.6 Comparison Between Exact, SEWA, and SVEA Wave Modes 2388.7 Conclusions 240References 2409 Nonlinear X-Waves 243Claudio Conti and Stefano Trillo9.1 Introduction 2439.2 NLX Model 2459.3 Envelope Linear X-Waves 2479.3.1 X-Wave Expansion and Finite-Energy Solutions 2509.4 Conical Emission and X-Wave Instability 2529.5 Nonlinear X-Wave Expansion 2559.5.1 Some Examples 2559.5.2 Proof 2569.5.3 Evidence 2579.6 Numerical Solutions for Nonlinear X-Waves 2579.6.1 Bestiary of Solutions 2599.7 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 2649.8 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 2669.9 Conclusions 266References 26710 Diffraction-Free Subwavelength-Beam Optics on a Nanometer Scale 273Sergei V. Kukhlevsky10.1 Introduction 27310.2 Natural Spatial and Temporal Broadening of Light Waves 27510.3 Diffraction-Free Optics in the Overwavelength Domain 28110.4 Diffraction-Free Subwavelength-Beam Optics on a Nanometer Scale 28610.5 Conclusions 292Appendix 292References 29311 Self-Reconstruction of Pulsed Optical X-Waves 299Ruediger Grunwald, Uwe Neumann, Uwe Griebner, Günter Steinmeyer, Gero Stibenz, Martin Bock, and Volker Kebbel11.1 Introduction 29911.2 Small-Angle Bessel-Like Waves and X-Pulses 30011.3 Self-Reconstruction of Pulsed Bessel-Like X-Waves 30311.4 Nondiffracting Images 30611.5 Self-Reconstruction of Truncated Ultrabroadband Bessel–Gauss Beams 30711.6 Conclusions 310References 31112 Localization and Wannier Wave Packets in Photonic Crystals Without Defects 315Stefano Longhi and Davide Janner12.1 Introduction 31512.2 Diffraction and Localization of Monochromatic Waves in Photonic Crystals 31712.2.1 Basic Equations 31712.2.2 Localized Waves 31912.3 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 32912.4 Conclusions 334References 33513 Spatially Localized Vortex Structures 339Zdeněk Bouchal, Radek Č elechovsk, and Grover A. Swartzlander, Jr.13.1 Introduction 33913.2 Single and Composite Optical Vortices 34213.3 Basic Concept of Nondiffracting Beams 34613.4 Energetics of Nondiffracting Vortex Beams 35013.5 Vortex Arrays and Mixed Vortex Fields 35213.6 Pseudo-nondiffracting Vortex Fields 35413.7 Experiments 35713.7.1 Fourier Methods 35713.7.2 Spatial Light Modulation 35813.8 Applications and Perspectives 361References 363Index 367