Localized States in Physics: Solitons and Patterns

Orazio Descalzi • Marcel Clerc • Stefania Residori Gaetano Assanto Editors

Editors Orazio Descalzi Universidad de los Andes Facultad de Ingeniería y Cs. Aplicadas Av. San Carlos de Apoquindo 2200 Las Condes Santiago Chile [emailprotected] Stefania Residori Institut Non-linéaire de Nice route de Lucioles 1361 06560 Valbonne France [emailprotected]

Marcel Clerc Universidad de Chile Fac. de Cs. Físicas y Mat. Depto. de Física Casilla 487 - 3 Santiago Chile [emailprotected] Gaetano Assanto University Roma Tre Nonlinear Optics and OptoElectronics Lab Via della Vasca Navale 84 00146 Roma Italy [emailprotected]

ISBN 978-3-642-16548-1 e-ISBN 978-3-642-16549-8 DOI 10.1007/978-3-642-16549-8 Springer Heidelberg Dordrecht London New York © Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: eStudio Calamar S.L. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Physical systems driven far from thermodynamic equilibrium can give rise to a variety of dissipative spatial structures through spontaneous breaking of symmetries. A fascinating feature of these pattern-forming systems is their tendency to originate spatially confined states. Such localized states can exist as wave packets or propagating entities through space and/or time. Observed in many different branches of science, localized states appear to be ubiquitous in nature and characterized by common macroscopic properties, independently of the specific physical laws governing the underlying field and/or matter interactions. Even though Localized States in Physics can be found in such different domains as hydrodynamics, optics, granular matter, reaction-diffusion systems, neural networks, plasmas, Bose-Einstein condensates etc., books on the topic are still very rare and often devoted to a particular type. This Book is based on a series of lectures given at a workshop on the subject: it reflects the spirit and the breadth of the meeting, held in 2008 at the University of los Andes, Santiago, Chile. Its main motivations stem from the need to bring together - coolate and compare - various approaches to the description of localized states in physics, offering a comprehensive panorama of confined states, from localized patterns to solitons, convectons, oscillons, pulses, etc., aimed at establishing a common - or at least shared - comprehension of these physical states. In fluids, for instance, convecting regions can coexist stably with non-convecting regions in uniformly heated cells. Localized hexagonal patterns have also been observed in a parametrically excited layer of fluid. In chemical systems, autocatalytic reactions on metallic surfaces can lead to solitary waves with partial and full annihilation after collision of pulses traveling in opposite directions. In granular matter, vertically driven layers of particles (sand, rice, stones, metal balls, etc.) reveal that, for peak acceleration exceeding a critical value, standing wave patterns spontaneously form and oscillate at half the excitation frequency. Square, stripe, hexagonal and spiral patterns can emerge, depending on the oscillation frequency and amplitude of the forcing, including coherent states such as localized standing waves or oscillons. Localized states are also relevant in neural systems, where action potentials propagate along axons or networks of thalamic neurons exhibit activity waves, just to mention two examples. In optics, the interplay between dispersion/diffraction v

Preface

and the medium nonlinearity leads to light propagation in space/time self-confined beams, the so-called optical solitons. In the presence of feedback, optical localized structures such as cavity solitons have been identified as transverse solutions encompassing bistability; they have been observed in several media and controlled by suitable addressing protocols. Finally, coherence and interference properties of atomic clouds of Bose-Einstein condensates, as well as localized structures in population models, have been investigated. The book covers quite a few of the most active and interesting contemporary aspects of Localized States in Physics, providing both review elements and current information on the latest research in the field. It consists of thirteen chapters discussing localized objects in optics, fluids and neural networks. The first four chapters are mostly dedicated to fundamental research in light localization. Reports on the state-of-the-art in optical spatial solitons, self-confined light and optical turbulence are presented with particular emphasis on experimental observations. The related theoretical work is treated in a general way and recent nonlinear optical experiments are reported to support the various predictions. The next three chapters deal with localized structures as localized solutions of pattern-forming systems. Analogies are drawn between fluids and optics, with a chapter dedicated to confined convective states in fluids and another one to optical transverse structures in liquid crystal light-valves. The recent theoretical developments in pattern localization are treated in a dedicated chapter, where crystal-like hexagonal structures are shown to localize according to the symmetry of the underlying grid. In the second part of the book special attention is paid to the potentials of localized states towards applications. Four chapters are devoted to optical systems and their use for controllable light pixels. Finally, excitability and localized states are treated in the last two chapters, where pulse localization is illustrated with examples in a nonlinear optical cavity and in neural networks. The Book as a whole is intended for an audience of senior and junior researchers and graduate students working in the field of pattern formation, instabilities and spatio-temporal dynamics of macroscopic systems far from equilibrium. It provides an overview of the state-of-the-art in localized states to a readership of physicists, mathematicians, electrical/electronic engineers. We trust that a number of scientists from neighbouring areas, such as e.g. biology, sociology, environment science and meteorology, will find its contents stimulating and informative. Santiago de Chile, August 2010

Orazio Descalzi Marcel G. Clerc Stefania Residori Gaetano Assanto

Acknowledgements

We wish to thank the following national and international institutions for the their financial support, that made possible Localized states in Physics: a focused workshop 2008: • Facultad de Ingenier´ıa y Cs. Aplicadas, Universidad de los Andes, Chile. • Fondo de Ayuda a la Investigaci´on, Universidad de los Andes, Chile. • Departamento de F´ısica, Facultad de Cs. F´ısicas y Matem´aticas, Universidad de Chile, Chile. • Nonlinear Optics and OptoElectronics Lab, CNISM, University of Rome ”Roma Tre”, Italy. • Institut Non Lin´eaire de Nice, France. • The Center for Advanced Interdisciplinary Research in Materials – CIMAT (Chile). • The Consortium of the Americas for Interdisciplinary Science, University of New Mexico (USA). • Programa Bicentenario de Ciencia y Tecnolog´ıa, Anillo ACT 15, ”Dynamics, Singularities and Geometry of Matter out of Equilibrium”.

vii

Contents

Part I Solitons, self-confined light and optical turbulence 1

Light Self-trapping in Nematic Liquid Crystals . . . . . . . . . . . . . . . . . . 3 Miroslaw A. Karpierz and Gaetano Assanto 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Reorientational Self-focusing in Nematic Liquid Crystals . . . . . . . . 4 1.3 Spatial Optical Solitons in Purely Nematic Liquid Crystals . . . . . . . 7 1.4 Spatial Optical Solitons in Chiral Nematic Liquid Crystals . . . . . . . 11 1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Photonic Plasma Instabilities and Soliton Turbulence in Spatially Incoherent Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dmitry V. Dylov and Jason W. Fleischer 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Basic Theory and Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Wigner Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Initial Stages of Instability. Linear Perturbation Theory . . 2.2.3 Growth Rate and Conditions for Weak/Strong Turbulence 2.2.4 Debye Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Quasi-Linear Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 General Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Bump-on-Tail Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Numerical Results for BOT Instability . . . . . . . . . . . . . . . . 2.4.2 Numerical Results for Multiple BOT Instability . . . . . . . . 2.5 Experimental Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Single Bump-on-Tail Instability . . . . . . . . . . . . . . . . . . . . . 2.5.3 Holographic Readout of Dynamics . . . . . . . . . . . . . . . . . . .

17 18 19 19 21 22 24 25 26 27 28 28 29 30 30 31 34

Contents

2.5.4

Multiple Bump-on-Tail Instability and Long-Range Turbulence Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.6 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3

Gap-Acoustic Solitons: Slowing and Stopping of Light . . . . . . . . . . . . Richard S. Tasgal, Roman Shnaiderman, and Yehuda B. Band 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Derivation of the Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Electromagnetic Field Equations with Phonon Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Acoustic Wave Equations with Electrostrictive Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 The Bragg-Brillouin-Kerr System . . . . . . . . . . . . . . . . . . . . 3.3 Lagrangian, Hamiltonian, and Conserved Quantities . . . . . . . . . . . . 3.3.1 Dimensionless Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Gap-Acoustic Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Soliton Stability and Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optical Wave Turbulence and Wave Condensation in a Nonlinear Optical Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jason Laurie, Umberto Bortolozzo, Sergey Nazarenko and Stefania Residori 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Evolution Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Long-Wave Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 The Fjørtoft Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Hamiltonian Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Canonical Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.6 The Kinetic Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.7 Modulational Instability and the Creation of Solitons . . . . 4.4 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Experimental and Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Direct cascade of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 42 45 45 48 51 51 53 54 57 64 66 67

68 69 70 70 71 72 74 75 76 79 80 80 84 85 86 86

Contents

Part II Localized structures in pattern forming systems 5

Localized Structures in the Liquid Crystal Light Valve Experiment . 91 Umberto Bortolozzo, Marcel G. Clerc, Ren´e G. Rojas, Florence Haudin and Stefania Residori 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.2 The Liquid Crystal Light Valve Experiment . . . . . . . . . . . . . . . . . . . 93 5.2.1 Description of the setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.2.2 The optical feedback: model equations . . . . . . . . . . . . . . . . 95 5.3 Experimental Observations of Optical Localized Structures . . . . . . 97 5.3.1 Round localized structures: interaction and dynamics . . . . 97 5.3.2 Triangular localized structures: bistability and phase singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.3.3 Bipatterns and localized peaks . . . . . . . . . . . . . . . . . . . . . . . 100 5.3.4 1D spatially forced model . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.4 Control of Optical Localized Structures . . . . . . . . . . . . . . . . . . . . . . . 102 5.4.1 Pinning range and localized structures . . . . . . . . . . . . . . . . 102 5.4.2 Controlled storage of localized structures matrices . . . . . . 103 5.5 Propagation Properties of Optical Localized Structures . . . . . . . . . . 105 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Convectons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Arantxa Alonso, Oriol Batiste, Edgar Knobloch and Isabel Mercader 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.2 Convectons with periodic boundary conditions . . . . . . . . . . . . . . . . . 112 6.3 Convectons with ICCBC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.4 Multiconvectons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.5 Localized traveling waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.6 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Morphological Characterization of Localized Hexagonal Patterns . . . 127 Daniel Escaff Dixon 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.2 Prototypical Model for Hexagon Formation . . . . . . . . . . . . . . . . . . . 129 7.3 Localized Hexagonal States: Geometrical Considerations and Morphological Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.4 Heuristic Description of the Localization Process . . . . . . . . . . . . . . . 133 7.5 The Case of a Localized Line of Cells . . . . . . . . . . . . . . . . . . . . . . . . 135 7.6 Conclusions and Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

xii

Contents

Part III Localized structures for optical applications 8

Cavity Solitons in Vertical Cavity Surface Emitting Lasers and their Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Massimo Giudici, Francesco Pedaci, Emilie Caboche, Patrice Genevet, Stephane Barland, Jorge Tredicce, Giovanna Tissoni and Luigi Lugiato 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 8.2 CS motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 8.2.1 Numerical Analysis of CS motion in a constant phase gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 8.2.2 Experimental Evidence of CS motion in a constant phase gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 8.3 Applications of CS movement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 8.3.1 CS drift in a constant gradient . . . . . . . . . . . . . . . . . . . . . . . 150 8.3.2 Experimental realization of reconfigurable CS arrays . . . . 151 8.4 CS motion and device defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 8.4.1 CS force microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 8.4.2 Modeling of an inhom*ogeneous device . . . . . . . . . . . . . . . 157 8.4.3 Interaction between phase gradient and defects: the CS tap158 8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Cavity Soliton Laser based on coupled micro-resonators . . . . . . . . . . . 169 Patrice Genevet, St´ephane Barland, Massimo Giudici, and Jorge R. Tredicce 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 9.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 9.3 Bistability regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 9.3.1 Multistable Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 9.3.2 Local bifurcation diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 176 9.3.3 Towards the whole bifurcation diagram . . . . . . . . . . . . . . . 178 9.4 Coherence properties of laser solitons . . . . . . . . . . . . . . . . . . . . . . . . 181 9.4.1 Modal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 9.5 Conclusions and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

10 Cavity soliton laser based on a VCSEL with saturable absorber . . . . 187 Giovanna Tissoni, Keivan M. Aghdami, Franco Prati, Massimo Brambilla and Luigi A. Lugiato 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 10.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 10.2.1 Bistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 10.2.2 Plane wave instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 10.2.3 Pattern forming instabilities . . . . . . . . . . . . . . . . . . . . . . . . . 193

Contents

xiii

10.3

CS switching techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 10.3.1 Switching dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 10.3.2 Switching energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 10.4 Stability of the CS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 10.5 Motion of the CS in a finite device . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 10.5.1 Circular pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 10.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 11 Dynamic Control of Localized Structures in a Nonlinear Feedback Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Mousa Ayoub, Bj¨orn G¨utlich, Cornelia Denz, Francesco Papoff, Gian-Luca Oppo, and William J. Firth 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 11.2 Self-organized localized structures in feedback systems . . . . . . . . . 215 11.3 Localized structures in a single-feedback system using a liquid crystal light valve as a nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 11.3.1 Formation of localized structures . . . . . . . . . . . . . . . . . . . . 221 11.4 Boundary-induced localized structures in LCLV . . . . . . . . . . . . . . . . 223 11.5 Dynamic and static position control of feedback localized states . . 227 11.6 Gradient induced motion control of feedback localized structures . 231 11.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Part IV Excitability and localized states 12 Interaction of oscillatory and excitable localized states in a nonlinear optical cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Dami`a Gomila, Adri´an Jacobo, Manuel A. Mat´ıas, and Pere Colet 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 12.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 12.3 Overview of the behavior of localized states . . . . . . . . . . . . . . . . . . . 243 12.3.1 Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 12.3.2 Saddle-loop bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 12.3.3 Excitability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 12.4 Interaction of two oscillating localized states . . . . . . . . . . . . . . . . . . 246 12.4.1 Full system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 12.4.2 Simple model: two coupled Landau-Stuart oscillators . . . 251 12.5 Interaction of excitable localized states: logical gates . . . . . . . . . . . 259 12.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 13 Lurching waves in thalamic neuronal networks . . . . . . . . . . . . . . . . . . 265 Jaime E. Cisternas, Thomas M. Wasylenko, and Ioannis G. Kevrekidis 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

xiv

Contents

13.2

The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 13.2.1 Smooth and Lurching waves . . . . . . . . . . . . . . . . . . . . . . . . 270 13.3 Exploration of parameter space and continuation . . . . . . . . . . . . . . . 272 13.3.1 Direct time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 13.3.2 Continuation using Newton method . . . . . . . . . . . . . . . . . . 274 13.3.3 Pseudo-arclength continuation . . . . . . . . . . . . . . . . . . . . . . . 278 13.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

List of Contributors

Miroslaw A. Karpierz Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00662 Warsaw-Poland, e-mail: [emailprotected] Gaetano Assanto NooEL, Nonlinear Optics and OptoElectronics Lab, CNISM, University of Rome ”Roma Tre”, Via della Vasca Navale 84, 00146 Rome - Italy, e-mail: [emailprotected] Mousa Ayoub Institut f¨ur Angewandte Physik and Center for Nonlinear Science, Westf¨alische Wilhelms-Universit¨at M¨unster, Corrensstr. 2/4, 48149 M¨unster, Germany, e-mail: [emailprotected] Bj¨orn G¨utlich Institut f¨ur Angewandte Physik and Center for Nonlinear Science, Westf¨alische Wilhelms-Universit¨at M¨unster, Corrensstr. 2/4, 48149 M¨unster, Germany Cornelia Denz Institut f¨ur Angewandte Physik and Center for Nonlinear Science, Westf¨alische Wilhelms-Universit¨at M¨unster, Corrensstr. 2/4, 48149 M¨unster, Germany, e-mail: [emailprotected] Francesco Papoff SUPA, Department of Physics, University of Strathclyde, 107 Rottenrow, Glasgow, G4 0NG, Scotland, U.K. e-mail: [emailprotected] Gian-Luca Oppo SUPA, Department of Physics, University of Strathclyde, 107 Rottenrow, Glasgow, G4 0NG, Scotland, U.K. e-mail: [emailprotected] William Firth SUPA, Department of Physics, University of Strathclyde, 107 Rottenrow, Glasgow, G4 0NG, Scotland, U.K. e-mail: [emailprotected] xv

xvi

List of Contributors

Umberto Bortolozzo INLN, Universit´e de Nice Sophia-Antipolis, CNRS, 1361 route des Lucioles 06560 Valbonne, France e-mail: [emailprotected] Marcel G. Clerc Departamento de F´ısica, Facultad de Ciencias F´ısicas y Matem´aticas, Universidad de Chile, Casilla 487-3, Santiago, Chile, e-mail: [emailprotected] Ren´e G. Rojas Instituto de F´ısica, Pontificia Universidad Cat´olica de Valpara´ıso, Casilla 4059, Valparaıso, ´ Chile e-mail: [emailprotected] Florence Haudin INLN, Universit´e de Nice Sophia-Antipolis, CNRS, 1361 route des Lucioles 06560 Valbonne, France e-mail: [emailprotected] Stefania Residori INLN, Universit´e de Nice Sophia-Antipolis, CNRS, 1361 route des Lucioles 06560 Valbonne, France, e-mail: [emailprotected] Daniel Escaff Dixon Complex Systems Group, Facultad de Ingenier´ıa y Cs. Aplicadas, Universidad de los Andes, Av. San Carlos de Apoquindo 2200, Santiago, Chile e-mail: [emailprotected] Giovanna Tissoni CNISM and INFM–CNR, Dipartimento di Fisica e Matematica, Universit`a dell’Insubria, Como, Italy e-mail: [emailprotected] Franco Prati CNISM and INFM–CNR, Dipartimento di Fisica e Matematica, Universit`a dell’Insubria, Como, Italy e-mail: [emailprotected] Luigi A. Lugiato CNISM and INFM–CNR, Dipartimento di Fisica e Matematica, Universit`a dell’Insubria, Como, Italy e-mail: [emailprotected] Keivan M. Aghdami Physics Department, Payame Noor University, Mini City, 19569 Tehran, Iran Massino Brambilla INFM–CNR, Dipartimento di Fisica Interateneo, Politecnico di Bari, Italy e-mail: [emailprotected] Arantxa Alonso Departament de F´ısica Aplicada, Universitat Polit`ecnica de Catalunya, Barcelona, Spain e-mail: [emailprotected] Oriol Batiste Departament de F´ısica Aplicada, Universitat Polit`ecnica de Catalunya, Barcelona, Spain e-mail: [emailprotected]

List of Contributors

xvii

Isabel Mercader Departament de F´ısica Aplicada, Universitat Polit`ecnica de Catalunya, Barcelona, Spain e-mail: [emailprotected] Edgar Knobloch Department of Physics, University of California, Berkeley CA 94720, USA e-mail: [emailprotected] Dami`a Gomila IFISC, Instituto de F´ısica Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus Universitat Illes Balears, 07122 Palma de Mallorca, Spain, e-mail: [emailprotected] Adri´an Jacobo IFISC, Instituto de F´ısica Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus Universitat Illes Balears, 07122 Palma de Mallorca, Spain, e-mail: [emailprotected] Manuel A. Mat´ıas IFISC, Instituto de F´ısica Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus Universitat Illes Balears, 07122 Palma de Mallorca, Spain, e-mail: [emailprotected] Pere Colet IFISC, Instituto de F´ısica Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus Universitat Illes Balears, 07122 Palma de Mallorca, Spain, e-mail: [emailprotected] Dmitry V. Dylov Department of Electrical Engineering, Princeton University, Olden Street, Princeton, New Jersey 08544, USA e-mail: [emailprotected] Jason W. Fleischer Department of Electrical Engineering, Princeton University, Olden Street, Princeton, New Jersey 08544, USA e-mail: [emailprotected] Richard S. Tasgal Departments of Chemistry and Electro-Optics, and the Ilse Katz Center for Nano-Science, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel e-mail: [emailprotected] Roman Shnaiderman Departments of Chemistry and Electro-Optics, and the Ilse Katz Center for Nano-Science, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel e-mail: [emailprotected] Yehuda B. Band Departments of Chemistry and Electro-Optics, and the Ilse Katz Center for Nano-Science, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel e-mail: [emailprotected]

xviii

List of Contributors

Jason Laurie Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom e-mail: [emailprotected] Sergey Nazarenko Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom e-mail: [emailprotected] Massimo Giudici Universit´e de Nice Sophia Antipolis, Centre National de la Recherche Scientifique, Institut Non Lin´eaire de Nice, 1361 route des lucioles, Valbonne, France e-mail: [emailprotected] Francesco Pedaci Universit´e de Nice Sophia Antipolis, Centre National de la Recherche Scientifique, Institut Non Lin´eaire de Nice, 1361 route des lucioles, Valbonne, France Emilie Caboche Universit´e de Nice Sophia Antipolis, Centre National de la Recherche Scientifique, Institut Non Lin´eaire de Nice, 1361 route des lucioles, Valbonne, France Patrice Genevet Universit´e de Nice Sophia Antipolis, Centre National de la Recherche Scientifique, Institut Non Lin´eaire de Nice, 1361 route des lucioles, Valbonne, France e-mail: [emailprotected] Stephane Barland Universit´e de Nice Sophia Antipolis, Centre National de la Recherche Scientifique, Institut Non Lin´eaire de Nice, 1361 route des lucioles, Valbonne, France Jorge Tredicce Universit´e de Nice Sophia Antipolis, Centre National de la Recherche Scientifique, Institut Non Lin´eaire de Nice, 1361 route des lucioles, Valbonne, France e-mail: [emailprotected] Jaime E. Cisternas Complex Systems Group, Facultad de Ingenier´ıa y Ciencias Aplicadas, Universidad de los Andes, Santiago, Chile e-mail: [emailprotected] Thomas M. Wasylenko Department of Chemical Engineering, Princeton University, Princeton, U.S.A. Current address: Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, U.S.A. e-mail: [emailprotected] Ioannis G. Kevrekidis Department of Chemical Engineering and Program of Applied and Computational Mathematics, Princeton University, Princeton, U.S.A. e-mail: [emailprotected]

Part I

Solitons, self-confined light and optical turbulence

Chapter 1

Light Self-trapping in Nematic Liquid Crystals Miroslaw A. Karpierz and Gaetano Assanto

Abstract We review the most important achievements and recent progress in the area of light-beam self-localization into optical spatial solitons in reorientational molecular media, with specific focus on nematic liquid crystals in planarly aligned, twisted and chiral arrangements.

1.1 Introduction Liquid crystals are widely used in displays for a large and ever growing number of applications, including high resolution television sets. It is less known to the general public that these molecular materials are employed and studied with a much larger set of scientific objectives, including electro-optic modulators and nonlinear photonics, particularly in devices for light switching and all-optical circuits towards novel generations of optical telecom systems. These molecular dielectrics are fluid, transparent, damage resistant, temperature and voltage tunable, etc. [1, 2, 3]. When the organic molecules, typically large and rod-shaped, are ordered in the so-called nematic phase, liquid crystals tend to exhibit a large (optical and radio-frequency) birefringence and their optical properties can also be modified by light through a nonlinear reorientational response, i.e. their constituent organic molecules can rotate and reorientate in space based on the optical excitation [1, 2, 3]. The latter nonlinear response is known as optical reorientation and has been exploited to investigate light-beam self-localization in non-diffracting filaments or spatial solitons, i.e. beams which do not spread upon propagation, maintain an invariant transverse Miroslaw A. Karpierz Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00662 Warsaw-Poland e-mail: [emailprotected] Gaetano Assanto NooEL, Nonlinear Optics and OptoElectronics Lab, CNISM, University of Rome ”Roma Tre” Via della Vasca Navale 84, 00146 Rome - Italy, e-mail: [emailprotected]

O. Descalzi et al. (eds.), Localized States in Physics: Solitons and Patterns, DOI 10.1007/978-3-642-16549-8_1, © Springer-Verlag Berlin Heidelberg 2011

Miroslaw A. Karpierz and Gaetano Assanto

profile via a power-driven increase in refractive index and are able to guide weaker signals [4, 5, 6, 7]. The molecular nonlinearity of nematic liquid crystals (NLC) is large (i.e. it requires low powers), depends on field polarization but is substantially independent on wavelength in the whole transparency range, typically from visible to mid-infrared; being associated to molecular motion in a fluid, it is rather slow in time, this latter drawback requiring a careful choice/selection of potential applications in reconfigurable circuits. The light-driven NLC reorientational response supports various phenomena [2, 3] and, as anticipated, the generation and propagation of stable and robust self-trapped spatial solitary waves, also named Nematicons [4]. The first solitons in NLC were observed in hollow capillaries filled with dye-doped materials [8, 9, 10]; they exploited the thermal response through absorption and, in some cases, a phase transition from nematic to isotropic states. Recently, the most studied geometries for optical solitons have been planar cells with non-absorbing NLC, various boundary conditions and applied voltage biases. In the next section we give a brief account of the basics of the reorientation response and the excitation of nematicons. In Section 11.3 we overview the main recent results on nematicons in undoped nematic liquid crystals in planar geometries. In Section 7.4 we discuss spatial optical solitons in twisted and chiral NLC.

1.2 Reorientational Self-focusing in Nematic Liquid Crystals Optical reorientation in nematic liquid crystals relies on the structure of the medium, which consist of anisotropic elongated non-polar molecules in a fluid state [1]. In the isotropic phase these molecules are disordered in position and orientation, the latter usually defined by the angle of their major axis or director n, a unity vector corresponding to the optic axis. In the nematic phase, the NLC director exhibits an average angular orientation, typically mediated by molecular anchoring at the boundaries of a cell. The director distribution in an NLC cell can be modified by anchoring at the surfaces, applied electromagnetic fields, medium temperature, light beams. NLC, in fact, maintain their fluid state despite the anchoring, with changes in director orientation related to the free energy of density: 1 1 1 fF = K11 (∇ · n)2 + K22 (n · ∇ × n + G)2 + K33 (n × ∇ × n)2 , 2 2 2

(1.1)

with Kii the elastic (Frank) constants for the three basic spatial distortions of the molecules: splay (K11 ), twist (K22 ) and bend (K33 ). In most common NLC K33 > K11 > K22 and are of the order of a few pN units [1, 2, 3]. Expression (1) is often simplified by taking K11 = K22 = K33 = K. The parameter G in the second term on the RHS of (1) describes chirality with pitch p: G = 0 for standard NLC and G = 2π /p for chiral nematic liquid crystals (ChNLC). Owing to the elongated shape of molecules in NLC, valence electrons have a larger mobility along the major axis; hence, the dielectric permittivity is higher for field vectors parallel to n. In the nematic phase, therefore, typical NLC are birefringent dielectrics with uni-

1 Light Self-trapping in Nematic Liquid Crystals

axial properties, i. e. ∆ε = εk − ε⊥ , εk and ε⊥ being the components of the electric permittivity (at low and/or optical frequencies) for extraordinary and ordinary polarizations, respectively. An electric field E, either externally applied (e.g. a voltage) or associated to a propagating light beam, can rotate the main molecular axis by a Coulombian torque, the latter trying to align n along the field vector despite anchoring at the boundaries and the free energy (1). This latter interaction energy has density E ε 0 ∆ε D (n · E)2 (1.2) fel = − 2 Since energy (2) is minimized when n is parallel to the E field vector, the reorientational response is a saturable one: the maximum nonlinearity corresponds to a change from ε⊥ to εk or from ordinary index n⊥ to extraordinary index nk . An intense enough extraordinarily polarized beam, such that E, propagation wavevector k and NLC optic axis n are coplanar, can alter the molecular orientation, increase the electric permittivity and the extraordinary index of refraction, give rise to self-focusing. Such effect can be modeled by deriving the Euler-Lagrange-Rayleigh equations from the minimization of the total free energy, which includes the deformation energy, the interaction energy with external fields, the interaction energy with the boundaries and the dissipation energy [7]. The latter has density fR 1 fR = γ 2

∂n ∂t

¶2 (1.3)

with γ the orientational viscosity. The density fR is required in time-dependent analyses. When the director at the boundaries and the electric field E lie in the same plane, the angle θ defining the orientation of the director n with respect to the propagation coordinate z can describe the reorientation in two-dimensional problems, as sketched in Fig. 1.1. The refractive index for an extraordinarily polarized field varies with θ according to the usual ne (θ ) = q

n⊥ nk n2⊥ cos2 θ + n2k sin2 θ

(1.4)

For the geometry sketched in Fig. 1.1 the director n = (sin θ , 0, cos θ ). Assuming an initial orientation θ = θ0 , nonlinear beam propagation at wavelength λ (frequency ω ) is ruled by the coupled system 4K∇2⊥ θ + ε0 ∆ε sin (2θ ) |A|2 = 0 2ik

ω2

∂ A + ∇2⊥ A + 2 n2e (θ ) A − k2 A = 0 ∂z c

(1.5) (1.6)

with A the slowly-varying beamq envelope,∇2⊥ = ∂ 2 /∂ x2 + ∂ 2 /∂ y2 the Laplacian in the transverse plane, k ≈ (ω /c)

n2⊥ + ∆ ε sin2 θ0 , c the light speed in vacuum [11].

Miroslaw A. Karpierz and Gaetano Assanto

Eq. (6) is a saturable (θ cannot exceed π /2) nonlinear Schr¨odinger-like equation with an index increase limited by ∆ n = nk − n⊥ . In Eq. (5) and Eq. (6), the initial value θ0 can represent the effect of a fixed pre-tilt or a tilt induced by a voltage V applied to the NLC thickness across x. A small initial tilt, such that n and E are not perpendicular to one another, prevents the threshold effect known as Fr´eedericksz transition. Eq. (6) holds valid with θ0 = constant for narrow beams in thick cells, i. e., in NLC regions far from the anchoring boundaries. System (5)-(6) models a nonlocal nonlinear response, whereby the reorientational index change extends beyond the transverse size of the beam field envelope [12]. Nonlocality, as well as saturation, allow nematicons to be stable and robust in 2+1 dimensions [13, 14]. Fig. 1.1(b) displays the calculated distribution of θ (x) as compared to a bell-shaped electric field excitation for various intensities I0 . It is apparent that the boundary conditions affect the nonlocal response and its strength depending on the waist of the beam [5, 15, 16]. (c)

x E n

q E // x k z

Fig. 1.1 (a) Geometry of a bounded NLC layer in a planar cell with voltage bias across the thickness. (b) The input E field belongs to the principal plane xz and is coplanar with n and k k z. (c) Distribution of θ (x) for various electric field excitations. The field profile is the solid curve. In this 1D calculations we used an NLC layer thickness of 50 µm and boundary conditions θ (−d/2) = θ (d/2) = 0.01

The reorientational response of NLC, stemming from the shape of the constituent non polar molecules determines the properties of spatial optical solitons. The nonlinearity is polarization sensitive, self-focusing, saturable, non instantaneous and nonlocal; hence, it supports stable two-dimensional propagating solitons. Moreover, owing to the refractive index increase counterbalancing diffraction, co-polarized signals of different wavelengths can be guided within the soliton channels [11, 17]. By using the correct input beam polarization, applying suitable boundary conditions at the interfaces containing the layer of NLC, e.g. by mechanical rubbing, appropriate pre-tilt θ0 can maximize the nonlinear response and allow mW power excitations to generate nematicons with propagation over several Rayleigh distances, i. e. to define reconfigurable signal interconnects. In the remaining of this chapter we will illustrate the main properties of low-power reorientational nematicons in threshold-less configurations. Nematicons have been investigated in various cell geometries, from hollow capillaries to thick cells with fiber in/out connections, to thin planar waveguides. The main planar cells for 2D nematicons are sketched in Fig. 1.2 and consist of glass

1 Light Self-trapping in Nematic Liquid Crystals

Fig. 1.2 Most common NLC planar cells for the study of optical spatial solitons: (a) planar anchoring with external voltage bias V applied by means of Indium Tin Oxide (ITO) thin film electrodes, (b) twisted or chiral NLC arrangement. The wide arrows along z indicate the excitation field wavevector of amplitude E, the thinner arrows refer to the input linear polarization

plates with proper rubbing at the internal interfaces [11]. The plates are held parallel and separated by spacers. Thin film transparent electrodes (e.g. Indium Tin Oxide) can be used to apply the desired low-frequency bias and tune the initial orientation θ0 . Input and output glass slabs can also be used to seal the cells and avoid meniscus formation and beam depolarization. When the NLC thickness is much larger than the waist of the input beam, the NLC layer can be treated as a bulk and the observation of (2+1) dimensional spatial solitons is possible [4, 7, 11, 13]. Conversely, if the thickness is comparable with the beam waist, then the structure is better modelled as a planar waveguide and can support (1+1) dimensional nematicons [5, 6].

1.3 Spatial Optical Solitons in Purely Nematic Liquid Crystals The basic geometry adopted for demonstrating Nematicons in a planar glass cell containing undoped NLC (specifically, the Merck mixture known as E7) is sketched in Fig. 1.1(a) and Fig. 1.2(a) [7, 11]. The excitation was a linearly polarized Gaussian beam with the electric field parallel to x, i.e. extraordinarily-polarized. Surface anchoring and applied voltage across the d = 75 µm thickness guaranteed a threshold-less all-optical response in the uniaxial dielectric, as described by Eq. (4) and Eq. (5). In the presence of the external bias V = dErf , neglecting walk-off in the plane xz and non-paraxial effects, the evolution of the slowly-varying beam amplitude A propagating along z in the mid-plane of the cell is modelled by [7, 11]: ´¡ ¢ ∂ ω2 ³ A + ∇2⊥ A + 2 n2k − n2⊥ sin2 θ − sin2 θ0 A = 0 (1.7) ∂z c ´ ∂ 2θ 1 1 ³ K 2 + K∇2⊥ θ + ∆ εRF Erf2 sin (2θ ) + ε0 n2k − n2⊥ |A|2 sin (2θ ) = 0 (1.8) ∂z 2 4 2ik

with θ0 the pre-tilt and θ the director orientation due to both light and voltage, ∆εRF the dielectric anisotropy in the low frequency region.

Miroslaw A. Karpierz and Gaetano Assanto

System (7), for narrow nematicons with respect to the cell and small waist compared to the extent of the nonlocal response, reduces to a saturable nonlinear Schr¨odinger equation with nonlocal and stable 2D+1 self-localized solutions [12]. In this limit, nematicons exhibit the features attributed to ”accessible” solitons by Snyder and Mitchell [14], with a breathing character resulting in the (quasi) periodic oscillation of their waist and peak intensity [12, 13, 14]. This breathing is excitationdependent and can be reduced by exciting the solitons with power and waist close to their existence curve. In several experimental scenarios, nematicons often appear as transversely invariant beams with a slowly decaying intensity due to Rayleigh scattering in the medium. Self-localized solutions in the ”local” regime can also be found for beams of waist comparable to the nonlocal range [18, 19]. Fig. 1.3(a) displays actual (colour-coded) images of individual 2mW Gaussian green (514.5nm) beams in the ordinary (top panel) and extraordinary (bottom) polarizations, resulting in linear (diffraction) and nonlinear (self-localized) propagation, respectively, as observed by collecting the out-of-plane scattered photons with a camera. The linear behaviour in the ordinary polarization corresponds to an E-field orthogonal to n; hence, to lack of reorientation below the Freedericks threshold. The nematicon (Fig. 1.3(a, lower panel) remains invariant over distances exceeding 20 diffraction lengths. Fig. 1.3(c) shows the corresponding evolution of a red (632.8 nm ) probe (signal) co-polarized and co-launched with the pump: as a nematicon is generated, the weak signal is confined in the soliton waveguide despite its longer wavelength: another demonstration of the nonlocal nature of nematicons, inasmuch as the numerical aperture of the solitary channel exceeds that associated to the spatial extent of the self-localized solution. Another effect of nonlocality is low-pass filtering. In the case of spatial incoherent excitations, e.g. a speckled beam produced by a diffuser, nonlocality can eliminate the high frequency wave-vector components and allow a spatial soliton to be formed, even if at the price of a larger power [20, 21, 23]. Fig. 1.3(b) and Fig. 1.3(d) display diffraction and self-localization of pump (Fig. 1.3(b)) and probe (Fig. 1.3(d)) in ordinary (top panels) and extraordinary (bottom) polarizations, respectively. It is apparent that an excitation of 2.7 mW (versus 2.0 mW in the coherent case) suffices to compensate the larger diffraction associated to the speckled input. The incoherent character of nonlocal solitons also allows the formation of vector nematicons with two (or more) co-polarized wavelength components [24, 25], as well as the mutual attraction between nematicons, propagating either in plane [26, 27, 28] or out of plane [29, 30, 31, 32]. Fig. 1.4 illustrates a couple of simple in plane interactions between two solitons excited by equi-power beams propagating at a small angle. Since the initial separation does not exceed the nonlocal range, the nematicons tend to attract as the refractive perturbation links the two self-induced waveguides, until the in initially diverging beams become parallel (Fig. 1.4(b)). At higher input powers (Fig. 1.4(c)) the mutual attraction becomes strong enough to make the two solitons cross and interleave, exchanging their position along y. This power-dependent interaction can be exploited for all-optical switching and logic gates.

Linear (o-)

y [mm]

100

z [mm]

Incoh. Pump (514nm, 2.7mW)

z [mm]

Incoh. Probe (632.8nm)

0 100

y [mm]

(b)

-100

Soliton (e-)

(c)

Probe (632.8nm)

-100 0

y [m m]

Soliton (e-)

(a)

Pump (514nm, 2mW)

100

y [m m]

Linear (o -)

1 Light Self-trapping in Nematic Liquid Crystals

z[mm] z[mm]

z[mm]

(d)

-100 0

z [mm] z[mm]

z [mm]

Fig. 1.3 Colour coded images of beam propagation from an Ar+ laser (left) and a collinear copolarized He-Ne laser (right) in a planar NLC cell with E7. (a) Top row: linear diffraction when injecting an ordinary polarization. Bottom: soliton propagation in the extraordinary polarization (k x); (b) corresponding linear (top) and nonlinear (bottom) evolution of the probe (100 µW). (c-d) Spatially incoherent beam propagation as in (a) and (b), respectively, but for a (c) 2.7 mW pump and (d) an equally incoherent probe

(a) (b)

z[mm]

Fig. 1.4 Color coded images of (a) a single nematicon in a planar cell; (b) two identical nematicons launched by ≈2 mW beams forming a mutual angle of 1.7o ; (c) same as in (b) but with launch powers ≈4 mW

Miroslaw A. Karpierz and Gaetano Assanto

Figure 1.5 summarizes a few other cases of interactions between nematicons. If the initial separation and/or angle are large enough, the two spatial solitons cross each other (Fig. 1.5(a)) while maintaining straight trajectories, as for onedimensional Kerr solitons [21]. Fig. 1.5(b) illustrates the formation of a few nematicons using a wide beam focused well inside the cell [33]: transverse modulational instability mediates the formation of a number of solitons depending on the size and power of the optical excitation [22, 34, 35]. Several nematicons can also be the byproduct of a dispersive shock wave or undular bore [36]. Fig. 1.5(c) illustrates the interaction of two equi-power solitons lauched skew in the cell: mutual attraction gives rise to a cluster of nematicons with spiralling trajectories and angular momentum. Since the latter is proportional to the photon content of each soliton, as the excitation increases the cluster rotates faster, as displayed in Fig. 1.5(d) showing the output images of the two light spots versus input power [30, 31]. A similar behaviour has been also predicted with clusters of nematicons of different wavelengths and with more than two components.

y [mm]

(a)

0 -50 0

200

(b)

0 -200

z [mm]

2.5

(c)

(d)

Fig. 1.5 (a) In plane crossing of two identical nematicons. (b) Multiple soliton generation by a focused light beam. (c) Out-of-plane attraction between two skew nematicons. (d) The photographs taken at the output of the cell show that, as the excitation increases, the cluster rotates faster in propagation, with a power-dependent angular change (π , in this set)

1 Light Self-trapping in Nematic Liquid Crystals

Nematicons are extraordinarily polarized wave-packets in uniaxials; hence, they undergo walk-off, i. e. their photon flux forms an angle ³ ´   n2k − n2⊥ sin (2θ )  ³ ´ δ (θ ) = tan−1  (1.9) n2k + n2⊥ + n2k − n2⊥ cos (2θ ) with the wave-vector. Such angle can be as large as 7 − 9o in typical NLC, and suitable launch conditions need be adopted to prevent or reduce it not to make the soliton hit the cell boundaries [37]. Boundaries contribute to define the potential landscape for soliton propagation and can play an important role in the actual nematicon path within finite cells [38, 39, 40]. Since walk-off depends on the angle θ which, in turn, can be controlled by the external bias, the applied voltage can also be used to change nematicon trajectory by varying δ , either in the whole cell or in specific regions of it [41, 42, 43, 44]. In the latter case, graded interfaces can be formed and support soliton refraction or total internal reflection [43, 44]. Examples of refraction and total internal reflection in a cell with two electrodes defining regions of higher and lower optical densities are shown in Fig. 1.6 (a-b). Analogous effects can be obtained by illuminating NLC regions and inducing reorientation along the path of the nematicon. This has been demonstrated with lens-like perturbations in undoped NLC and Azo-NLC [45, 46], through dye-mediated absorption and surface anchoring [47], in liquid crystal light valves by means of a photoconductive layer altering the voltage drop across the NLC [48]. Finally, owing to the large walkoff, double refraction in uniaxial NLC can originate negative refraction at the input glass-NLC interface, with the soliton propagating in the same half-plane of the incident wave vector, as visible in Fig. 1.6(c) comparing ordinary and extraordinary (self-confined) beam propagation [49].

1.4 Spatial Optical Solitons in Chiral Nematic Liquid Crystals In twisted and chiral nematic cells the molecular director is parallel to the glass plates (interfaces) and twisted within the film thickness (Fig. 1.2(b)) [50, 51]. Such an orientation is typically induced by the boundary conditions (in twisted nematics, TNLC) and by the chiral properties (in cholesteric liquid crystals, ChNLC). For light polarized along y the refractive index varies across the sample thickness from the ordinary value n⊥ in planes where θ = 0 to the extraordinary nk in planes where θ = π /2. A self-trapped light beam propagates in the thin layer where the refractive index is the largest (close to nk ). In ChNLC several such layers occur throughout the liquid crystal and their number depends on the chirality pitch and the thickness of the film. In the configurations investigated a light beam propagates in the z-direction parallel to the glass plates and is initially linearly polarized with the electric field vector E = yEy also parallel to the interfaces. Because the birefringence axis rotates across

Miroslaw A. Karpierz and Gaetano Assanto

z [mm]

Fig. 1.6 (a-b) A planar cell with suitable director orientation in the plane yz and two sets of electrodes can be used to define two dielectric regions separated by a graded interface (dashed line). If the nematicon, injected from the left, reaches the interface from an optically rarer region, it can undergo refraction as in (a). If the input region is denser, the soliton can undergo total internal reflection, as in (b). The overall change in angle from refraction to reflection in this experiment is 18 + 22 = 40o . (c) Double refraction in NLC: the ordinary beam component (upper) undergoes positive refraction while diffracting; the extraordinary beam nematicon propagates with negative refraction (towards y < 0) at the walk-off angle with respect to k (along the ordinary beam)

the layer, during propagation all components of electric and magnetic fields appear. However, only Ey and Ez are important for reorientation. Using the Euler-Lagrange equation for energy minimization, the following partial differential equation can be obtained: K

h³ i ¯ ¯2 ´ ¡ ¢ ∂ 2θ 1 + ε0 ∆ ε |Ez |2 − ¯Ey ¯ sin (2θ ) + Ey∗ Ez + Ez∗ Ey cos (2θ ) = 0 (1.10) 2 ∂x 4

where θ (x) = θ0 + 2π x/p is the initial orientation (without electric field) and p is the chirality pitch. The description of light propagation in a twisted or a chiral NLC layer can be simplified by assuming that the beam profile along x is roughly constant, a hypothesis which is correct at some distance from the input, where the self-guided mode is formed. In this limit, taking Ez 50 times the Rayleigh length (a few millimeters). The relatively large power required to form nematicons in TNLC can be reduced by decreasing the film thickness d. This can be obtained in ChNLC with a smaller pitch, where nematicons are substantially similar to those in TNLC [55, 56]. However, the former offer some new opportunities because the width of a guiding layer (in x) is not only determined by the sample thickness (as in TNLC) but also by the chirality pitch. As a result, in ChNLC it is easier to change the thickness of a layer and -as a consequence- the nonlinear strength. It is also possible to utilize multilayers for the propagation of independent or interacting nematicons, as schematically shown in Fig. 1.7(a).

Fig. 1.7 (a) ChNLC cell geometry and (b) experimental results showing the formation of nematicons in different layers across the film, as obtained by launching the input beam in distinct vertical positions ∆x.

Typical results in ChNLC are presented in Fig. 1.7(b) for a Ti:Sapphire (λ = 790 nm) laser beam with input waist of about 2 µm . Spatial solitons were excited in a cell with pitch p = 25 µm at powers P ≈ 30 mW (66-67). Nematicons were size invariant for about 2 mm of propagation (> 80 Rayleigh lengths). Due to the finite thickness of each layer in ChNLC, self focusing could balance diffraction and give rise to self-trapped solitons only in a limited waist range. Moreover, by changing the vertical input position along x, it was possible to launch as many solitons as layers in the ChNLC structure, as reported in Fig. 1.7(b) corresponding to the four layers

Miroslaw A. Karpierz and Gaetano Assanto

Fig. 1.8 Experimental results on spatial solitons in ChNLC: (a) light beam propagation for various external electric fields (marked on photos) and (b) corresponding intensity profiles at a distance z = 0.6 mm.

of a cell about 50 µm thick and with a 25 µm pitch. The four nematicons could be injected independently, separated by 10 − 12 µm. Using the smaller pitch p = 10 µm the power required to form soliton was reduced to P < 10 mW. Additionally, similar to the nematicons discussed in Sect. 11.3, even in TNLC and ChNLC we verified that the solitary waveguide was able to confine different wavelength signals, specifically a co-polarized low-power probe from a Helium-Neon laser (λ = 633 nm). When two nematicons were launched close to one another in the same layer, they attracted and eventually merges into a single self-trapped beam. Small changes in input beam polarization caused nematicons to change direction of propagation. However, if the polarization is sufficiently rotated, then diffraction in a non-soliton polarization prevailed over self-focusing. Similar results were observed when the external electric field was applied perpendicularly to the layer. The direction of nematicon propagation could be controlled by a voltage V, as presented in Fig. 1.8. For larger values of the external electric field (in our example for V> 6 V) the induced reorientation prevented an effective self-trapping of light.

1.5 Conclusions Nematic liquid crystals, organic self-assembling molecular fluid dielectrics with anisotropic optical properties, are ideal material systems for studying self-localization of light into spatial solitons at milliwatt powers and over millimeter distances. Their reorientational response, saturable nonlocal and polarization sensitive, supports several types of self confinement and soliton all-optical effects, including signal trapping and routing, switching, processing. We predict that various other light localization phenomena, from dispersive shock waves to undular bores and soliton propagation in random potentials, will find experimental validation in these media.

1 Light Self-trapping in Nematic Liquid Crystals

Acknowledgements We are indebted with various collaborators, including U. Bortolozzo, M. Kaczmarek, I. C. Khoo, A. A. Minzoni, E. Nowinowski, S. Residori, M. Sierakowski, F. Simoni, N. F. Smyth, C. Umeton. We are grateful to our students and associates for the extensive contributions to this work: A. Alberucci, C. Conti, A. Fratalocchi, K. Jaworowicz, U. A. Laudyn, M. Kwasny, M. Peccianti, A. Piccardi and K. A. Rutkowska.

References 1. P. G. de Gennes, The physics of liquid crystals, (Clarendon Press, Oxford, 1974) 2. I. C. Khoo, N. T. Wu, Optics and nonlinear optics of liquid crystals, (World Scientific Publ., Singapore, 1993) 3. F. Simoni, Nonlinear Optical Properties of Liquid Crystals, (World Scientific Publ., London, 1997) 4. G. Assanto, M. Peccianti, C. Conti, Opt. Photon. News 14, 44–48 (2003) 5. M. A. Karpierz, M. Sierakowski, M. Swillo, T. R. Wolinski, Mol. Cryst. Liq. Cryst. 320, 157–164 (1998) 6. M. A. Karpierz, Phys Rev. E 66, 036603 (2002) 7. G. Assanto, M. Peccianti, IEEE J. Quantum Electron. 39, 13–21 (2003) 8. E. Braun, L. P. Faucheux, A. Libchaber, Phys. Rev. A 48, 611–622 (1993) 9. M. Warenghem, J. F. Henninot, G. Abbate, Opt. Express 2, 483–490 (1998) 10. F. Derrien, J. F. Henninot, M. Warenghem, G. Abbate, J. Opt. A - Pure Appl. Opt. 2, 332-337 (2000) 11. M. Peccianti, G. Assanto, A. De Luca, C. Umeton, I. C. Khoo, Appl. Phys. Lett. 77, 7–9 (2000) 12. C. Conti, M. Peccianti, G. Assanto, Phys. Rev. Lett. 91, 073901 (2003) 13. C. Conti, M. Peccianti, G. Assanto, Phys. Rev. Lett. 92, 113902 (2004) 14. A. W. Snyder, D. J. Mitchell, Science. 276, 1538–1541 (1997) 15. M. Peccianti, C, Conti, G. Assanto, Phys. Rev. E 68, 025602 (2003) 16. A. A. Minzoni, N. F. Smyth, A. L. Worthy, J. Opt. Soc. Am. B 24, 1549–1556 (2007) 17. U. A. Laudyn, M. Kwany, K. Jaworowicz, K. A. Rutkowska, M. A. Karpierz, G. Assanto, Photon. Lett. Pol. 1, 7–9 (2009) 18. C. Garcia-Reimbert, A. A. Minzoni, N. F. Smyth, A. L. Worthy, J. Opt. Soc. Am. B 23, 2551– 2558 (2006) 19. C. Garcia-Reimbert, A. A. Minzoni, N. F. Smyth, J. Opt. Soc. Am. B 23, 294–301 (2006) 20. M. Peccianti, G. Assanto, Opt. Lett. 26, 1791–1793 (2001) 21. M. Peccianti, G. Assanto, Phys. Rev. E 26, 035603–035606R (2002) 22. M. Peccianti, C. Conti, E. Alberici, G. Assanto, Laser Phys. Lett. 2, 25–29 (2005) 23. K. G. Makris, H. Sarkissian, D. N. Christodoulides, G. Assanto, J. Opt. Soc. Am. B 22, 1371– 1377 (2005) 24. A. Alberucci, M. Peccianti, G. Assanto, A. Dyadyusha, M. Kaczmarek, Phys. Rev. Lett. 97, 153903 (2006) 25. G. Assanto, N. F. Smyth, A. L. Worthy, Phys. Rev. A 78, 013832 (2008) 26. M. Peccianti, K. A. Brzdakiewicz, G. Assanto, Opt. Lett. 27, 1460–1462 (2002) 27. M. Peccianti, C. Conti, G. Assanto, A. De Luca, C, Umeton, Appl. Phys. Lett. 81, 3335–3337 (2002) 28. G. Assanto, M. Peccianti, K. A. Brzdakiewicz, A. De Luca, C. Umeton, J. Nonl. Opt. Phys. Mat. 12, 123–134 (2003) 29. A. Fratalocchi, M. Peccianti, C. Conti, G. Assanto, Mol. Cryst. Liq. Cryst. 421, 197–207 (2004) 30. A. Fratalocchi, A. Piccardi, M. Peccianti, G. Assanto, Opt. Lett. 32, 1447–1449 (2007) 31. A. Fratalocchi, A. Piccardi, M. Peccianti, G. Assanto, Phys. Rev. A 75, 063835 (2007)

16 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56.

Miroslaw A. Karpierz and Gaetano Assanto G. Assanto, N. F. Smyth, A. L. Worthy, Phys. Rev. A 78, 013832 (2008) M. Peccianti, C. Conti, G. Assanto, Opt. Lett. 28, 2231–2233 (2003) G. Assanto, M. Peccianti, C. Conti, IEEE J. Sel. Top. Quantum Electron. 10, 862–869 (2004) M. Peccianti, G. Assanto, Opt. Lett. 30, 2290–2292 (2005) G. Assanto, T. Marchant, N. Smyth, Phys. Rev. A 78, 063808 (2008) M. Peccianti, A. Fratalocchi, G. Assanto, Opt. Express 12, 6524–6529 (2004) A. Alberucci, G. Assanto, J. Opt. Soc. Am. B 24, 2314–2320 (2007) A. Alberucci, M. Peccianti, G. Assanto, Opt. Lett. 32, 2795–2797 (2007) A. Alberucci, G. Assanto, D. Buccoliero, A. S. Desyatnikov, T. R. Marchant, N. F. Smyth, Phys. Rev. A 79, 043816 (2009) M. Peccianti, C. Conti, G. Assanto, A. De Luca, C. Umeton, Nature A 432, 733–737 (2004) G. Assanto, C. Umeton, M. Peccianti, A. Alberucci, J. Nonl. Opt. Phys. Mat. 15, 33–42 (2006) M. Peccianti, A. Dyadyusha, M. Kaczmarek, G. Assanto, Nat. Phys. 2, 737–742 (2006) M. Peccianti, G. Assanto, A. Dyadyusha, M. Kaczmarek, Opt. Lett. 32, 271–273 (2007) A. Pasquazi, A. Alberucci, M. Peccianti, G. Assanto, Appl. Phys. Lett. 87, 261104 (2005) S. V. Serak, N. V. Tabiryan, M. Peccianti, G. Assanto, IEEE Photon. Techn. Lett. 18, 1287– 1289 (2006) A. Piccardi, G. Assanto, L. Lucchetti, F. Simoni Appl. Phys. Lett. 93, 171104 (2008) A. Piccardi, U. Bortolozzo, S. Residori, G. Assanto, Opt. Lett. 34, 737–739 (2009) M. Peccianti, G. Assanto, Opt. Express 15, 8021-8028 (2007) M.A. Karpierz, M. Sierakowski, T.R. Wolinski, Mol. Cryst. Liq. Cryst. 375, 313-320 (2002) M.A. Karpierz, K.A. Brzdakiewicz, Q.V. Nguyen, Acta Phys. Pol. A 103, 169-175 (2003) J. Baran, Z. Raszewski, R. Dabrowski, J. Kedzierski, J. Rutkowska, Mol. Cryst. Liq. Cryst. 123, 237-242 (1985) R. Dabrowski, J. Dziaduszek, T. Szczucinski, Mol. Cryst. Liq. Cryst. 124, 241-246 (1985) K. Jaworowicz, K.A. Brzdakiewicz, M.A. Karpierz, M. Sierakowski, Mol. Cryst. Liq. Cryst. 453, 301-307 (2006) U.A. Laudyn, K. Jaworowicz, M.A. Karpierz, Mol. Cryst. Liq. Cryst. 489, 214-221 (2008) U.A. Laudyn, M. Kwasny, M.A. Karpierz, Appl. Phys. Lett. 94, 091110 (2009)

Chapter 2

Photonic Plasma Instabilities and Soliton Turbulence in Spatially Incoherent Light Dmitry V. Dylov and Jason W. Fleischer

Abstract We develop a plasma theory of nonlinear statistical optics. In this model, partially spatially incoherent light is treated as an ensemble of speckles which can interact through the nonlinearity. A photonic plasma frequency is defined, as is a photonic Debye length. This approach unifies previous observations using partially coherent light and predicts a new class of optical phenomena. Examples include the two-scale energy transfer common to modulation instability and the continuous excitation of modes from the gradient-driven bump-on-tail instability. The latter example, well-known from plasma physics, represents a new regime for optical experiments. We observe it here by considering the nonlinear coupling of two partially coherent beams in a self-focusing photorefractive crystal. For weak wave coupling, determined by small modal density within a Debye sphere, we observe momentum exchange with no variation in intensity. For strong wave coupling, modulations in intensity appear, as does evidence for wave (Langmuir) collapse at large scales. To achieve a broader range of wave coupling, we consider a double bump-on-tail geometry. This system can be modeled as a pair of coupled single-hump instabilities whose interaction involves general issues of nonlinear competition, synchronization, etc. For the case of strong wave coupling, the multiple humps merge into a single-peaked profile with an algebraic k−2 inertial range. This self-similar spectrum, representing an ensemble of dynamically-interacting solitons atop a sea of radiation modes, is a definitive observation of soliton (Langmuir) turbulence.

Dmitry V. Dylov Department of Electrical Engineering, Princeton University, Olden Street, Princeton, New Jersey 08544, USA, e-mail: [emailprotected] Jason W. Fleischer Department of Electrical Engineering, Princeton University, Olden Street, Princeton, New Jersey 08544, USA, e-mail: [emailprotected]

O. Descalzi et al. (eds.), Localized States in Physics: Solitons and Patterns, DOI 10.1007/978-3-642-16549-8_2, © Springer-Verlag Berlin Heidelberg 2011

Dmitry V. Dylov and Jason W. Fleischer

2.1 Introduction Dynamical instabilities occur in every nonlinear wave system. Perhaps the simplest is modulation instability (MI), in which perturbations grow at the expense of a uniform background. For example, a plane wave propagating in a self-focusing medium will break up into stripes, with a characteristic period determined by a balance between diffraction/dispersion and nonlinearity. If the background is statistical, e.g. thermal, then attempts at growth are de-phased by the background, and there is a nonlinear threshold for instability [1, 2, 3, 4]. Put another way, mode coupling must be sufficiently strong to generate enough correlation for unstable growth. Once instability begins, the evolution is again characterized by a direct transfer between the background and a preferred scale (this time determined by the correlation length). The resulting array of solitons is then free to interact over longer evolution times/propagation distances. A contrasting process of energy transfer is one that occurs over a continuum of scales. This type of dynamics involves a local coupling between adjacent modes in wavenumber space, resulting in a self-similar cascade. This process gives an algebraic power spectrum and is typical of hom*ogeneous turbulence, such as that described by Kolmogorov theory [5, 6]. It is simpler, in some senses, as dimensional analysis and scaling arguments can be used to characterize the dynamics. The two methods of energy transfer represent complementary limits. Two-scale coupling can cascade, with higher-order effects appearing. This includes the generation of higher-order modes [7], condensation processes [8, 9, 10], and soliton clustering [11]. Likewise, local coupling can generate localized structures which can evolve dynamically [5, 12, 13]. This convergence of dynamics should not be surprising, as they are nothing but different pathways to the same asymptotic state. Until recently [14, 15, 16, 7, 17, 18], only two-scale dynamics had been demonstrated experimentally in optics, viz. the snake instability in self-defocusing media [19] and modulation instability in self-focusing media [3, 4]. Here, we outline our work on instabilities which cascade modes over a range of scales. As optical turbulence is our ultimate goal, we use light that is partially spatially incoherent. Such beams can be treated as an ensemble of speckles which, in a nonlinear medium, can be considered as quasi-particles that interact through large-scale modulation waves [14]. This description gives rise to a photonic plasma interpretation. It unifies all previous observation in nonlinear statistical optics and predicts a wide range of new dynamics. As a particular example, we consider an all-optical bump-on-tail instability. This instability, well-known in plasma physics [8, 9], is a gradient-driven effect which couples modes across a range of scales. We show that instability occurs whenever higher-momentum modes are more populated than lower ones, regardless of nonlinear coupling strength, and derive analytic dispersion relations for the growth rates. Experimentally, we observe the dynamics by considering the nonlinear coupling of two partially-coherent beams in a self-focusing photorefractive medium. For weak nonlinear interactions, the result is momentum (k) transfer without any observable

2 Photonic Plasma Instabilities and Soliton Turbulence in Spatially Incoherent Light

variation in intensity (x). For strong interactions, both x-space modulations and kspace dynamics appear. As the dynamics evolve, the growing perturbations start to back-react on their underlying source distribution. The source intensity becomes depleted and its spectral profile is modified. For this stage, linear theory is no longer adequate. To address this, we develop a quasi-linear theory and apply it to the bump-on-tail example, showing explicitly how modes grow until there is no more driving gradient. For even stronger wave growth, wave-wave interactions become dominant. That is, the perturbed modes interact with each other, independent of the background distribution. This is a highly nonlinear state, and it is difficult to achieve experimentally with limited nonlinearity and propagation distance. To facilitate the process, we consider a double bump-on-tail geometry, so that a broader range of unstable wavenumbers can grow and interact. We show that the dynamics can be treated as a coupled pair of individual bump-on-tail instabilities. It is thus a model system which can address a variety of general issues in nonlinear dynamics, including synchronization, competition, parametric pumping, and cascades of energy and momentum transfer [20]. In our case, we show analytically and experimentally that the momentum cascade leads to an algebraic k−2 power spectrum. The results highlight the difficulty of synchronized wave mixing inherent in noisy nonlinear systems and demonstrate a pathway towards all-optical studies of turbulence.

2.2 Basic Theory and Formalism Our starting equation is the nonlinear Shr¨odinger equation (NLS) for the slowly varying, partially coherent field packet ψ (rr , z), which reads i

∂ψ β 2 + ∇r ψ + κ G(hψ ∗ ψ i)ψ = 0. ∂z 2

(2.1)

Here, r is a diffraction\dispersion variable, the propagation is along z and coefficient β = λ /2π n0 is the diffraction (or second-order dispersion) coefficient for light of wavelength λ in a medium with base index of refraction n0 , κ is the nonlinear coefficient, and G(hψ ∗ ψ i) is the nonlinear response function of the medium. The bracket h...i denotes statistical ensemble average; it is valid when the medium’s response time is much longer than the characteristic time of the intensity fluctuations of the statistical wave packet.

2.2.1 Wigner Formalism There are many equivalent ways to treat such partially coherent light [21]; here, we use a full wave-kinetic approach via the Wigner formalism [22, 23]. In this

Dmitry V. Dylov and Jason W. Fleischer

method Eq. (2.1) is transformed by the Wigner function (including the Klimontovich statistical average), defined as ¿ µ ¶ µ ¶À Z +∞ ξ ξ f (rr , k , z) = (2π )−3 d 3 ξ · eikk ·ξ ψ ∗ r + ψ r− . (2.2) 2 2 −∞ R

+∞ 3 Equation (2.2) satisfies the intensity relation hψ ∗ (rr , z)ψ (rr , z)i = −∞ d k f (rr , k , z). Eq. (2.1) transformed by (2.2) takes the following form [22, 23, 24, 25, 14] "  ← → # ¡ ¢ ∂f 1 ∂ ∂  + β k · ∇r f + 2κ G h| ψ |2 i sin  · f = 0, (2.3) ∂z 2 ∂r ∂k

where the arrows in the sine operator indicate that the spatial derivative acts on the function G (to the left) and the momentum derivative acts on the distribution f (to the right) [24, 25]. In the geometrical optics approximation (the long-wavelength limit): ∆ k · ∆ r À 2π ), so that Eq. (2.3) reduces to

∂f + β k · ∇r f − κ E (rr , z) · ∇k f = 0, ∂z where the self-consistent driving field E (rr , z) is introduced as ¡ ¢ E (rr , z) = −∇r G h| ψ |2 i .

(2.4)

(2.5)

Equation (2.3) is known as a Wigner-Moyal equation for the evolution of the Wigner distribution function f (rr , k , z). Its simplification (2.4) has a form of the Vlasov transfer equation [8, 24] or, essentially, a radiation transfer equation [26, 27, 21]. It is valid for slow (long-wavelength) variations in the refractive index when the average speckle size (correlation length lc ) of the light is smaller than the beam envelope. Note that the usual form of this short-wave–long-wave dynamics is coupled but has been¡ reduced ¢ to a single equation by implicitly absorbing intensity fluctuations in G h| ψ |2 i . Equation (2.4) implies the conservation of the number of optical quasi-particles in {rr , k }-space. In this chapter we will treat these quasiparticles collectively and borrow language from plasma physics. Eqs. (2.4), (2.5) then become a starting point to account for spectral dynamics of localization, oscillations and instabilities in statistical, nonlinear optics. In plasma physics the self-consistent driving field defined in (2.5) is responsible for ponderomotive self-focusing, corresponding to the drift of electrons down a gradient in the plasma density [8, 6, 28, 29]. The corresponding nonlinear index change can be viewed as a divergence (variation) of E (rr , z) resulting from the local intensity inhom*ogeneity. Mathematically we can express it in terms of the generic condition for G(hψ ∗ ψ i): µ ¶ Z +∞ ∇r · E (rr , z) = κ hI0 i − d 3 k f (rr , k , z) , (2.6) −∞

2 Photonic Plasma Instabilities and Soliton Turbulence in Spatially Incoherent Light

where hI0 i is a uniform background intensity without any variations. Equation (2.6) is a Poisson equation [8] that implies that nonlinearity acts as a uniformly distributed volumetric ”charge”. It is valid if the following physically sensible conditions are met: the right-hand side has to be finite, the functions G and E have to be spatially continuous, and the medium has to have only smooth optical inhom*ogeneities (if any).

2.2.2 Initial Stages of Instability. Linear Perturbation Theory Initial stages of instabilities in nonlinear media can be studied by standard perturbation analysis. To illustrate the main points, we consider perturbations around a spatially uniform distribution f0 (kk ): f (rr , k , z) = f0 (kk ) + ∑ ρα (kk , z)eiα ·rr ,

(2.7)

with | ρα |¿| f0 | for all wavenumbers α 6= 0. In the unperturbed state, the nonlinear driving field (2.5) is assumed to be zero, so that E (rr , z) can be regarded as a small quantity (weak nonlinearity). In terms of the Fourier modes, E (rr , z) = ∑ E α (z)eiα ·rr .

(2.8)

Substituting (10.1) and (2.8) in Eqs. (2.4) and (2.6), noting that and linearizing in the perturbations yields

∂ ρα + iβ α · k ρα − κ E α · ∇k f0 = 0, ∂z Z +∞ κ E α = 2 iα d 3 k ρα . α −∞

R +∞ 3 k ) = hI0 i, −∞ d k f 0 (k

(2.9) (2.10)

One can solve these equations using the Laplace transformation along the propagation coordinate (∼ egz ). The resulting dispersion function is Dα (g) = 1 +

κ2 α 2β

Z +∞ −∞

d3k

α · ∇k f0 . ig − α · k

(2.11)

The stability of the partially coherent beam in a nonlinear medium is then determined by the zeros in g of the dispersion function (2.11).

Dmitry V. Dylov and Jason W. Fleischer

2.2.3 Growth Rate and Conditions for Weak/Strong Turbulence For simplicity, we reduce the problem to one transverse dimension. The dispersion relation (2.11) becomes [3, 24, 30, 25, 14] ± Z κ 2 +∞ ∂ f0 ∂ k Dα (g) = 1 + dk . (2.12) αβ −∞ ig − α k Initially, the function Dα (g) is defined for Re g > 0 and then is analytically continued into the rest of the plane. If there is a complex root g(α ) = gR (α ) − igI (α ) of Dα (g) = 0, and gR (α ) > 0, then any intensity perturbation will grow exponentially (instability). ¡ ¢ A Lorentzian distribution f0 (k) = (I0 ∆ k/π ) / k2 + ∆ k2 plugged into Eq. (2.12) allows an exact solution to the growth rate: s g α 4κ I0 = −∆ k + − 1, (2.13) β α2 2 β α2 where ∆ k = 2π /lc represents the spectral spread for a beam with correlation length lc . This gain coefficient, similar to that originally derived in Ref. [3], separates the effects of nonlinearity and statistics and shows a clear threshold value for the development of perturbations. As in plasma physics [1, 2], modulations will not appear unless the nonlinearity ∆ n is greater than the angular spread (effective temperature) (∆ k/k0 )2 . Below threshold, modulations are suppressed, a de-phasing which Fedele and Anderson et al. interpreted as a type of Landau damping due to the monotonically-decreasing distribution f0 [30, 24, 25]. However, they did not identify plasma-like parameters or consider the potential for inverse Landau damping (wave growth) when the distribution is non-monotonic. To treat the gradient-driven dynamics at initial stages of instability, we consider first weak growth (|gR | ¿ |gI |) in the long-wavelength limit (|ig| À α k). Expanding the denominator in Eq. (2.12) then gives " µ ¶ # Z +∞ g2 αβ 2 2 ≈κ dk f0 (k) 1 + 3 k . (2.14) β α2 g −∞ To be consistent with the quasi-thermal light used in the experiment [31], we consider a Gaussian beam profile (the detailed difference between this distribution and a Lorentzian will be addressed in Section 2.4 below): µ ¶ I0 k2 f0 (k) = √ exp − . (2.15) 2∆ k 2 2π ∆ k Note that a Lorentzian distribution will give the same results below, though more care is needed to handle the divergence of hk2 i in Eq. (2.14). The form (2.15) of the intensity is more true to the plasma mapping [18], in which the underlying

2 Photonic Plasma Instabilities and Soliton Turbulence in Spatially Incoherent Light

(b)

z→∞

F ( k, z )

(a)

∆k

| k | 2

f0 ( k )

Fig. 2.1 Quasi-thermal unstable light. (a) Typical doublehump distribution in k-space for BOT instability. (b) Asymptotic quasi-linear plateau and corresponding spectral energy density in the unstable region k1 ≤ k ≤ k2 .

distribution is Maxwell-Boltzmann. We note, however, that there is no true equilibrium distribution in the optical system, as there are no collisions available for relaxation [32]. Put another way, the dynamical system (2.4) conserves entropy, so that there are many possible steady-state profiles with which to start. Explicitly accounting for the principal value and pole in Eq. (2.14) gives s µ ¶ µ ¶¯ 3 2 2 π κ I0 ∂ f0 ¯¯ g(α ) = ig p 1 + α λD + κα , (2.16) 2 2 β ∂k ¯ k=g p /αβ

where g p is an effective plasma frequency and λD is an effective Debye length, with αλD ¿ 1. These parameters are s κ I0 β∆k gp = , λD = . (2.17) β gp The first term in Eq. (2.17) is a Bohm-Gross dispersion relation [8] for nonlinear statistical light, showing that optical speckles can interact via Langmuir-type waves [14, 18]. Growth or damping of these waves is a resonant process that depends on the relative (spatial) phase velocity of the underlying quasi-particles (speckles). From the second term in Eq. (2.16), it is clear that there are no growing modes if ∂ f0 /∂ k < 0, e.g., for a quasi-thermal Gaussian distribution, since on average more quasi-particles travel slower than the interaction wave than faster. However, the weak limit used to derive (2.16) breaks down when αλD ≈ 1, or κ I0 ≈ β h∆ k2 i; in this case case, the growth rate exceeds the rate of statistical dephasing (spectral bandwidth) of the background, causing intensity modulations to appear [1, 9]. Interestingly, this strong-coupling condition becomes the instability threshold reported earlier in [3]. Not considered before, however, was the possibility for optical instability by inverse Landau damping when ∂ f0 /∂ k > 0 (see Fig. 2.1(a) for a typical distribution). A prime example is the ”bump-on-tail” (BOT) instability, well-known from plasma physics [8], in which a non-equilibrium hump is added to one side of an equilibrium distribution. To our knowledge, the BOT instability has never before been demonstrated outside of a plasma context. However, it should be clear from the above

Dmitry V. Dylov and Jason W. Fleischer

derivation that BOT dynamics should occur in any wave-kinetic system, including hydrodynamics [33], optics, and (potentially) Bose-Einstein condensates. The dynamics within this photonic plasma depend on the spectral density of perturbation modes within a Debye sphere. For the weak-coupling regime considered above, defined by αλD ¿ 1 [1, 9, 3], the BOT instability is mostly a momentumspace effect [14]. Above this threshold, intensity modulations appear [1, 34, 3, 4], wave-wave coupling (vs. wave-speckle coupling) becomes dominant [14, 10, 9], and the perturbation method ceases to apply. Adopting plasma language, we define these two limits as regimes of weak and strong optical Langmuir turbulence [15].

2.2.4 Debye Scaling In all previous work, the statistics of the input beam and the nonlinearity of the medium have been considered separate parameters, as they are controlled separately in the experiments. However, Eq. (2.16) shows that they are joined in the composite parameter of the photonic Debye length λD . As in material plasma, λD signifies the amount of interaction wave inhibition (screening) due to the random de-phasing of waves [14]. The photonic Debye length provides a natural length scale for highly incoherent beams, i.e. beam for which the correlation length is significantly less than the beam width (lc ¿ w0 ). For example, narrow (Gaussian) beams that linearly expand as (w(z)/w0 )2 = 1 + z2 , where z = z/LD is the propagation distance measured in terms of the linear diffraction length LD , evolve nonlinearly as µ

¶2

³ ´ 2 = 1 + z2 1 − λ D ,

(b)

(a)

40 0

-0.025

0.000

0.025

(w(z)/w0)2

8 4 0

(2.18)

120

FWHM, µm

w(z) w0

30 20 10 0

n Fig. 2.2 Numerical and eperimental results for the nonlinear diffraction of a spatially incoherent beam. (a) Plot of full-width half-maximum versus δ n at a fixed propagation distance z = 1cm. (b) Plot of simulation results in (a) using scaling from Eq. (2.18); (c) Experimental results measured after propagation in a photorefractive crystal. The dashed line represents the highly-incoherent limit of Eq. (2.18).

2 Photonic Plasma Instabilities and Soliton Turbulence in Spatially Incoherent Light (b)

(a)

Stripe width, µm

Fig. 2.3 Period of MI pattern (”stripe width”) as a function of (a) nonlinearity and (b) Debye length. Data for two correlation lengths of 80 µ m (triangles) and 92 µ m (rhombus) are shown. Notice the collapse of data after Debye scaling.

48 44 40 36

0.55

0.65

0.75

0.85

where λ D is the Debye length normalized to the nonlinear length that characterizes a coherent soliton [35]. Numerical and experimental verification of this formula, for nonlinearities below the soliton limit, are shown in Fig. 2.2. Technically, the plasma formula (2.16) is only valid for weak perturbations, for which |gR | ¿ |gI |. On the other hand, it is reasonable to use the Langmuir modes from this theory as a basis for further interactions when the nonlinearity is increased [12, 13]. This suggests that the Debye length remains a valid scaling parameter. As shown in Fig. 2.3 for the case of incoherent modulation instability, this is indeed the case. Finally, these results can be generalized to more complex cases. For example, multiple-stream geometries in k-space can be represented in terms of Gaussian multi-hump distribution, with the ”humps” positioned at different spatial frequencies (angular separations) δ k01 , δ k12 , δ k23 ,...,δ kM−1 M : " # (k−δ k j−1 j )2 M 2 1 − k 2 − 2 2∆ k f0 (k) = √ I0 e 2∆ k + ∑ I j e , (2.19) 2π ∆ k j=1 then the system’s dynamics would still be described simply by Eq. (2.16), but with the Debye length scaled as [15] v u µ ¶ M δ k2 ∼ u β 3 ∆ k2 j−1 j λ D= t +∑ . (2.20) κ Itot Ij j=1 Note that due to the redistribution and reshaping of the total intensity Itot = ∑ j I j , the threshold for the appearance of intensity modulations will shift as well.

2.3 Quasi-Linear Approximation It was mentioned in the Section 2.2.4 that after the initial steps of instability, linear perturbation theory ceases to be valid [14, 36, 37]. The reason is that the shape of distribution function changes with time, due to energy depletion and back-reaction by the perturbations. In this section we will treat such dynamics as time-dependent

Dmitry V. Dylov and Jason W. Fleischer

(or, more rigorously, z-dependent), which means that it is necessary to consider evolution of perturbations as well.

2.3.1 General Derivation We proceed by returning to the one-dimensional case of Eq. (2.4). Using the Fourier decompositions (10.1) and (2.8), we rewrite (2.10) as

∂ ρα (k, z) ∂ ( f0 (k) + ρ0 (k, z)) + iβ α kρα (k, z) − κ Eα (z) = 0, ∂z ∂k

(2.21)

where we neglected the wave-wave interaction term ∑α 0 6=0 Eα −α 0 ∂k ρα 0 and took only first term ρ0 to account for z-dependence of Wigner distribution function. We assume that ρ0 is a slowly varying function of z, while f0 is a distribution giving rise to a weak instability (like in the double-hump case of Fig. 2.1(a)). The rate of change of ρ0 is given by ∂ ρ0 ∂ ρα = ∑ E−α . (2.22) ∂z ∂k α Now assume that ρα and Eα have the following form: µZ z ¶ ρα (k, z) = ρˆ α (k) exp d ζ [gR (ζ ) − igI (ζ )] , µZ z ¶ ˆ Eα (z) = Eα exp d ζ [gR (ζ ) − igI (ζ )] .

(2.23)

The solution to (2.21) is then

ρα =

iκ ∂ /∂ k( f0 + ρ0 ) Eα , β igR + gI − α k

(2.24)

This result can be plugged into Poisson’s equation, and analogous calculations of the dispersion relation and growth rate yield: Z

κ 2 +∞ ∂ /∂ k( f0 + ρ0 ) Dα (g) = 1 + dk , αβ −∞ igR + gI − α k s µ ¶¯ π κ I0 ∂ ( f0 + ρ0 ) ¯¯ gR = κα . ¯ 2 β ∂k k=g p /αβ

(2.25) (2.26)

Notice that gR became a slowly varying function of z through ρ0 . The long-term processes can be studied now by examining the long-term behavior of ρ0 . For this we substitute Eq. (2.24) into Eq. (2.22), which gives

2 Photonic Plasma Instabilities and Soliton Turbulence in Spatially Incoherent Light

∂ ρ0 κ2 ∂ gR ∂F = 2 ∑ |Eα |2 , ∂z β α ∂ k (gI − α k)2 + g2R ∂ k

(2.27)

where we used Eα = E−α , and F(k, z) ≡ f0 (k) + ρ0 (k, z). Note that Eq. (2.27) has the form of a diffusion equation. Turning from sums to continuous integrals and using (2.23) we get µ ¶ ∂F ∂ ∂F = Dk , (2.28) ∂z ∂k ∂k

∂ |Eα |2 = 2gR |Eα |2 , ∂z

(2.29)

where the k-space diffusion function is defined as

κ2 Dk = 2 β

Z +∞ −∞

d α |Eα |2

gR . (gI − α k)2 + g2R

(2.30)

Eqs. (2.28) and (2.29) are the basic equations of quasi-linear theory for statistical light. They govern the rate of change of the distribution F and spectral energy density |Eα |2 as the light propagates in a moderately nonlinear medium.

2.3.2 Bump-on-Tail Dynamics We now apply the quasi-linear formalism to the bump-on-tail instability shown in Fig. 2.1. Previously thought to exist only in plasma, the bump-on-tail (BOT) instability is a gradient-driven effect in which a non-equilibrium bump on the tail of a thermal distribution acts as a source of free energy [8]. As such, it requires an inverted population of statistical modes and is often considered a type of classical lasing [38]. In plasma, the effect occurs when a gas of charged particles interact through electrostatic, or Langmuir, waves. Recently, we showed that the same phenomenon could occur in the nonlinear propagation of statistical light, in which an ensemble of speckles interact through large-scale modulation waves [14]. The initial distribution may be written as µ ¶ µ ¶ I0 k2 I1 (k − δ k)2 exp − +√ exp − , f0 (k) = √ 2∆ k 2 2∆ k 2 2π ∆ k 2π ∆ k

(2.31)

with I1 < I0 and δ k ≥ ∆ k. Theoretically, initial stages of BOT instability can be fully described by (2.16), with s s µ ¶ κ (I0 + I1 ) β 3 ∆ k2 δ k2 gp = , λD = + . (2.32) β κ I0 + I1 I1

Dmitry V. Dylov and Jason W. Fleischer

This linearized theory, however, says nothing about the progression of the distribution after some distance of propagation, such as saturation (stabilization) of continued instability; it merely provides wavenumbers of unstable modes between the beams. The deficiency of standard linearized theory is that it considers f0 (k) zindependent, which is no longer valid as the dynamics evolve. Recalling that |gR | ¿ |gI |, the fraction in the diffusion function (2.30) can be approximated as ∼ πδ (gI − α k), yielding · ¸ ∂F κ2 ∂ 1 ∂F = 2 |Ek |2 . (2.33) ∂z β ∂ k |k| ∂k The asymptotic state of the instability can be found by considering the change of R the corresponding power spectrum W (z) = 1/2 dkF 2 (k, z). Using (2.33), we have

∂W = ∂z

Z +∞ −∞

∂F κ2 dkF =− 2 ∂z β

Z +∞ −∞

µ ¶2 1 2 ∂F dk |Ek | . |k| ∂k

(2.34)

Each term in the last integrand is positive, which means that W will decrease until either |Ek |2 = 0, or ∂ F/∂ k = 0 for each value of k. The initial growth of modes is described by (2.16) and (2.32), and since |Ek |2 grows in the region between the two beams, the distribution function should flatten out so that there is no driving gradient ∂ F/∂ k (Fig. 2.1(b)). This ”quasi-linear plateau” has been observed in recent experiments [14, 15] (to be discussed in Section 2.5). Lastly, using (2.16), (2.32), and (2.33) and neglecting ∂ |Ek |2 /∂ k at z = 0, we can calculate the asymptotic values of |Ek |2 and F as ¯ Z ¯ πβ 3 k 2¯ |Ek | ¯ = k dk[F(k, ∞) − f 0 (k)], (2.35) gp k1 z=∞ F(k, ∞) =

1 k2 − k1

Z k2 k1

dk f0 (k),

(2.36)

where k1 and k2 are the k-vectors of the unstable region, k1 ≤ k ≤ k2 , in which the plateau is established (see Fig. 2.1(b)). Expressions (2.35) and (2.36) provide the the effective overall gain of the flattening and the hight of the resulting plateau.

2.4 Numerical Analysis 2.4.1 Numerical Results for BOT Instability To check the validity of the Quasi-Linear Approximation and Eqs. (2.35), (2.36), we have carried out numerical simulations of Eq. (2.4) and (2.5) for the case of a Kerr medium (Fig. 2.4). The bump-on-tail configuration was created by launching two partially incoherent beams of fixed spectral width ∆ k/k0 = 1.7 × 10−3 at a

2 Photonic Plasma Instabilities and Soliton Turbulence in Spatially Incoherent Light

Fig. 2.4 Simulation of bump-on-tail propagation in a nonlinear Kerr-like medium. Shown are double-hump spectra for (a) Lorentzian and (b) Gaussian distributions. The total intensity and nonlinearity are kept constant (∆ n/n0 = 1.74 × 10−4 ) and only the shape of the statistics differs. Dashed circles highlight the initial driving gradient and formation of the quasilinear plateau, which occurs quicker in the Gaussian case.

relative angles δ k/k0 = 2.0 × 10−3 . Comparisons between Lorentzian and Gaussian profiles for the distribution f0 (k), at fixed total intensity hI0 i, show that the Gaussian distribution triggers the unstable dynamics faster. More details of this momentum transfer will be discussed in Section 2.5.

2.4.2 Numerical Results for Multiple BOT Instability Figure 2.5 shows numerical simulation of the dynamics and the corresponding gain curves calculated from Eqs. (2.16) and (2.20) when three beams are launched into the medium (multiple bump-on-tail instability). As expected, modes grow in the regions of positive spectral slope until there is no more driving gradient. We find that the system is described effectively as a pair of coupled BOT instabilities: one

1.0

(a)

(b)

(c)

(d)

Fig. 2.5 Numerical simulation of multiple bump-on-tail instability. (a,b) Input profiles, with corresponding gain curves (shaded graphs at baseline), with the middle hump shifted (a) to the left of the equal gain value and (b) to the right. (c,d) Output pictures after 1cm of propagation (∆ n/n0 = 1.74 × 10−4 ).

Intensity (a.U.)

0.5

0.0 1.0

0.5

0.0 -2

k / k0,10-3

Dmitry V. Dylov and Jason W. Fleischer

on the left and one on the right, with negligible coupling between the leftmost and rightmost Gaussians due to their separation distance [15]. In this case, competition between the gain curves implies that plateau formation happens sequentially, even though the initial slopes and nonlinearity in the two regions are identical. If the middle hump is closer to the left (main) distribution, then the left region goes unstable first, and vice versa with a right bias (Figs. 2.5(c,d)). The balanced situation, with the central hump equidistant from either side, has gain curves of the same peak value and is unstable. In simulations, this initial condition always degenerated into one of the two asymmetric scenarios, a result supported by analytic perturbations of δ k01 in Eq. (2.16).

2.5 Experimental Observation 2.5.1 Experimental Setup Experimentally, we explore the bump-on-tail dynamics by considering the nonlinear interaction of two partially-coherent spatial beams. The setup is shown in Fig. 2.6. A statistical light input is created by focusing light from a 532nm CW laser onto a ground-glass diffuser and then imaging into a photorefractive SBN:60 (Sr0.6 Ba0.4 Nb2 O6 ) crystal [31]. The correlation length, and correspondingly the spectral bandwidth, can be changed by varying the magnification properties of the imaging lens. To create a bump-on-tail or a double bump-on-tail distribution (Fig. 2.6, inset), the spatially-incoherent beam is split (one or more times) using a Mach-Zehnder interferometer, attenuated in the bump arm(s), and then recombined on the input face of the crystal. For SBN, the nonlinear index change ∆ n = γ Eapp hIi/(1 + hIi), where Eapp is an electric field applied across the crystalline c-axis and γ = n0 r33 (1 + hI0 i) is a constant depending on the base index of refraction n0 , the electro-optic coefficient r33 , Fig. 2.6 Experimental Setup. 532nm laser light is made partially spatially incoherent by a ground-glass diffuser and separated into a superposition of two or three beams (for three, a dot-lined interferometer arm is added). A, attenuator; M, mirror; L, lens; BS, beam-splitter; POL, linear polarizer; SBN:60, nonlinear photorefractive crystal; FP, focal Fourier plane of the lens L3; CCD, digital detector.

L1 Laser Diffuser

CCD1 A FP

L4 CCD2

L3 POL SBN:60

2 Photonic Plasma Instabilities and Soliton Turbulence in Spatially Incoherent Light

and the spatially-hom*ogeneous incident light intensity hI0 i [27]. In the experiments, the beams have a relative angle of 0.3◦ (between any adjacent two beams), the intensity ratio is fixed at 3:2, and the strength of the nonlinearity (wave coupling) is controlled by varying the applied voltage (similar results occur at other angles and intensities). To observe the interaction, light exiting the crystal is directly imaged in both position (x) space and momentum (k) space, the latter by performing an optical Fourier transform. For comparison and calibration, we performed a single-beam MI experiment with the main hki = 0 hump (not shown). In this case, the background distribution is Gaussian with a correlation length lc = 176µ m, and no intensity modulations appeared until the voltage reached 0.9kV . Using n0 = 2.3 and r33 = 255pm/V , this corresponds to a nonlinear index change of ∆ n = 8 × 10−4 . Above this threshold, two symmetric momentum peaks appear at k/k0 = ±5.6 × 10−3 . This is the same behavior as in [4] but quantitatively calibrated to our initial input conditions and particular crystal.

2.5.2 Single Bump-on-Tail Instability All-optical examples of a wave-kinetic bump-on-tail instability are shown in Figs. 2.7, 2.8 and 2.9. Figs. 2.7(a-f) show the behavior for weak interaction. In this case, the photorefractive nonlinearity is turned on by applying a 0.7kV voltage bias across the crystal, below the 0.9kV bias necessary to trigger single-beam MI. As shown in Figs. 2.7(c,f), nonlinear modes are excited precisely in the expected region of positive slope, growing until there is no more driving gradient (a process known as quasilinear flattening, see Section 2.3). Remarkably, the momentum-space distribution is changed [Figs. 2.7(e,f)] while the position-space intensity shows no observable variations [Fig. 2.7(d)]. The nature of the instability depends on the spectral geometry of the system. For a single-humped distribution [3], or one with widely-separated peaks [39], strong nonlinearity is required to see any significant dynamics. Here, the spectral peaks overlap, giving an unstable condition with |gR | ¿ |gI |. The resulting momentum exchange, along with the resonance from Eq. (2.11), suggests that there is an underlying phase matching among the modes. Indeed, recent work with incoherent light in a medium with instantaneous (vs. inertial) nonlinearity shows an analogous velocity locking [40]. This demonstration, combined with similar momentum exchange observed in collisions of coherent vector solitons [41, 42], implies that the BOT dynamics should occur for true Kerr media as well. In the incoherent case considered here, the dynamics depends on the statistics of the interacting beams (Fig. 2.8). Local correlation measurements can reveal details of the speckle-wave coupling, but a simpler measure can be obtained from the visibility

ν = ( f (k1 ) − f (k01 ))/( f (k1 ) + f (k01 ))

Dmitry V. Dylov and Jason W. Fleischer

(a)

(d)

(g)

(b)

(e)

(h)

(c)

(f)

(i)

Power Spectrum (a.u.)

100 mm

1.0

0.5

0.0

-3

k / k0 10 - 3

(j)

-3

(l)

(k)

0.0 kV 0.5 kV 0.7 kV 0.9 kV

0.5 0.9

-4

k / k0 10 - 3

0.7 kV kV

0.0

(m)

0.9 kV 1.3 kV 1.7 kV 2.1 kV 2.3 kV

-4

k / k0 10 - 3

Fig. 2.7 Experimental output pictures of bump-on-tail and double bump-on-tail instability. (a,d,g,j,l) Intensity in position (x) space; (b,c,e,f,h,i,k,m) power spectrum in momentum (k) space. (a,b,c): crystals exit face after linear propagation of double-hump distribution (no applied voltage); (d,e,f): same, but nonlinear propagation (applied voltage of 0.7kV , in the weak-coupling regime); (g,h,i) same, but in the strong-coupling regime (1.6kV ). The blue and red curves in (c,f,i) show holographic readouts of single-beam propagation for the straight (blue) and angled (red) distributions, respectively. (j,k) Weak-coupling regime when the distribution is triple-humped. (l,m) Same, but in strong-coupling regime.

2 Photonic Plasma Instabilities and Soliton Turbulence in Spatially Incoherent Light

of the angled hump, as shown in Fig. 2.8(d). The efficiency of the flattening depends on the relative gain |gR | of the unstable modes. Using

η = ( f NL (k01 ) − fLin (k01 ))/( fNL (k0 )) + fLin (k0 ))

Power Spectrum (a.u.)

as a measure of efficiency, Fig. 2.8(e) shows interaction behavior that is relatively insensitive to nonlinear coupling strength but highly sensitive to beam statistics [14]. If the beam is too incoherent, then attempts at spectral energy transfer are de-phased. If the beam is too coherent, then the system loses its statistical nature (and thus its wave-kinetic properties). More rigorously, the first condition states that the angular separation between the beams must be greater than the spectral width of the distribution, while the second condition states that if the relative bandwidth is too small, then there are too few quasi-particles (speckles) in resonance with the growing waves [8, 9]. As a result, there is an optimal correlation length, for a given intensity ratio and angle given by ∂ gR /∂ k = 0, for efficient dynamical coupling. For stronger nonlinearity, the system enters a regime of strong wave coupling, significantly distorting the original distribution in k-space and creating modulations in x-space (Figs. 2.7(g-i)). These modulations are different from those arising from MI, however, as the spectrum in Figs. 2.7(h,i) shows a range of modal excitation (between the original humps), rather than the symmetric high-k side lobes charac-

(a) 1.0

(b)

linear

5 4 23

(c)

0.5 kV

0.7 kV

0.5

0.0

-3

(d)

(e)

-3

Efficiency

0.02

Gaussian (num) Lorentzian (num) Experiment Theory

0.055 0.050

linear 0.5 kV 0.7 kV

0.04

Visibility

0.045 0.040 0.035 0.030 0.025

0.00 1

120

160

200

240

Correlation Length, µm

Fig. 2.8 Dynamical coupling as a function of correlation length and nonlinearity. (a-c) Power spectra at 0kV (linear), 0.5kV and 0.7kV , respectively. (d) Visibility of the angled hump. (e) ”Efficiency” of nonlinear flattening. Curves and bars in (a-d) are numbered for correlation lengths of 243µ m (1), 206µ m (2), 176µ m (3), 142µ m (4), and 109µ m (5). Note the dependence of efficiency on the underlying distribution.

Dmitry V. Dylov and Jason W. Fleischer

teristic of MI (e.g. [3, 43]). Using our reference correlation length lc = 176µ m, as in Fig. 2.7(a-f), we observe that the required nonlinearity for modulations is 1.1kV , stronger than the one needed for single-beam MI. That is, the presence of a second statistical beam further suppresses the growth of modulations. Moreover, the appearance of modulations coincides with a breakdown of the quasilinear plateau and a resumption of wave growth in the unstable, non-equilibrium region [Fig. 2.7(i)]. The higher threshold can be understood by returning to the strong-coupling condition κ I0 ≈ β h∆ k2 i obtained in Section 2.2. It is clear that for a given spectral width, additional intensity lowers the required value of κ for instability [3, 44]. However, the presence of a second beam increases the effective bandwidth due to cross-beam interaction, potentially requiring a higher value of κ . A simple estimate can be ¡ ¢ obtained ¡ by considering¢ the variance of two Gaussian beams exp −k2 /∆ k2 + A exp −(k − δ k)2 /∆ k2 , which is ∆ k2 + δ k2 A/(1 + A)2 . For A = 2/3 and δ k ∼ ∆ k, as in the experiments, there is an increase in threshold nonlinearity from κ to (31/25)κ . Given the measured single-beam MI threshold of 0.9kV , the predicted double-beam threshold of 1.12kV matches the observed value.

2.5.3 Holographic Readout of Dynamics The different behaviors above and below the modulation threshold are the result of different nonlinear dynamics within the initial distribution. Experimentally, we can observe this by taking advantage of the slow photorefractive response time of SBN and recording a volume hologram of the interactions. Subsequently, we can block one of the beams and use the other as a probe of the coupling, observing the energy transfer that would have happened if the other beam were present [41]. These holographic reconstructions are shown in Figs. 2.7(c,f,i). For linear propagation [Fig. 2.7(c)], each beam maintains its Gaussian form, as there is no nonlinear intensity interaction to induce an index change. By contrast, there are significant changes in the nonlinear cases. For weak coupling [Fig. 2.7(f)], light originally in the perturbative bump (shown in red) is seen to flow towards lower momentum states, while light from the equilibrium distribution (shown in blue) scatters in the opposite direction. For strong coupling [Fig. 2.7(i)], the momentum transfer is asymmetric. The thermal light is unchanged, while the non-thermal distribution looks bimodal, with half the intensity in the original angled hump and half centered at hki = 0, beyond the initial instability range of positive slope. At this point, it is useful to revisit the plasma correspondence and interpret the scattering dynamics from a quasi-particle (speckle) perspective. From this viewpoint, the instability mechanism is essentially a resonant process, in which smallscale wavepackets generate and interact with large-scale modulations [41, 14]. The coupling threshold αλD ≈ 1 then separates the dynamics between regimes of weak and strong spatial turbulence. Indeed, weak (quasilinear) turbulence theory in plasma is characterized by the formation of a k-space plateau and the bidirectional transfer of momentum between the thermal and non-thermal distributions

2 Photonic Plasma Instabilities and Soliton Turbulence in Spatially Incoherent Light

[8, 45]. In the theory of strong turbulence, the thermal, non-resonant distribution is unchanged but the resonant distribution is greatly affected by wave-wave interactions [9]. In this case, there should be a direct transfer of momentum towards large scales (hki = 0), a stimulated scattering process known as Langmuir condensation in plasma physics [9, 10]. All of this is consistent with the observations in Figs. 2.7 (f) and (i). To the authors’ knowledge, these internal dynamics have not been observed in material plasma.

2.5.4 Multiple Bump-on-Tail Instability and Long-Range Turbulence Spectra Above the threshold αλD ∼ 1, the dynamics of the single BOT instability suggested that there is a turbulent breakdown of the quasi-linear plateau over the resonant range of k-vectors. However, the wavenumber region is too small to conclude that there is a self-similar spectrum between the humps. To extend the range, we add a third hump to the system, as described in Fig. 2.6. Experimental results for the weak-coupling case are shown in Figs. 2.7(j) and 2.7(k). The output intensity remained uniform, up to unavoidable striations in the crystal, while the energy spectrum underwent significant redistribution due mode coupling. As the nonlinear interaction strength (applied voltage) was increased, the spectral bumps were observed to flatten. In all cases, the profile flattening was sequential, with lower momenta reaching a plateau first. This observation agrees with the simulations in Fig. 2.5 and supports the general conclusion that the final state of the system depends on the temporal sequence of wave diffusion [46, 6]. To our knowledge, this is the first demonstration of a multiple bump-on-tail instability and its associated competition of growth rates. Similar behavior should occur in any wave-kinetic system obeying Eq. (2.4), such as material plasma and cold atoms at finite temperature. More complex behavior occurs for higher nonlinearity (Figs. 2.7(l) and 2.7(m)). In the strong coupling regime [14, 15, 1, 2, 3], modulations start to appear in intensity and momentum transfer continues beyond the plateau (zero-gradient) limit (though the inertial approximation is still valid [14, 4]). As before, wave-wave coupling is the dominant process of energy/momentum exchange [10, 9, 6]. A closer examination of this spectrum, shown in Fig. 2.9, reveals a self-similar profile with a k−2 fall-off. This algebraic spectrum holds for the entire wavenumber range between the first and last peaks, despite the fact that the central hump provided an initial region of stability (∂ f0 /∂ k < 0). The intensity waves present in the strong-coupling regime (Fig. 2.7(l)) are suggestive of solitons and, indeed, an ensemble of solitons can give the observed power spectrum [47]. For N solitons occupying a space of length L in 1D, having random phases and positions and maintaining total energy ES , the Wigner spectrum:

Dmitry V. Dylov and Jason W. Fleischer

(b)

(a)

1.0

a.u a.u.) .) Power Spectrum ((a.u

1.0

0.5

-4

k / k0, 10 – 3

0.5

~ k –2

log10[ k / k0, 10 – 3 ]

Fig. 2.9 Detailed look at the asymptotic spectral profile of the turbulent state during multiple BOT instability. (a) Comparison of input (solid) and output (dashed) profiles. (b) Comparison in log space, showing algebraic spectrum in interaction region.

h fk i ∝

1 L

Z Nmax Nmin

h i−2 dN N℘(N) cosh(kN/ES ) ,

(2.37)

where ℘(N) is the probability for the system to be in the N-soliton state and Nmin ≤ N ≤ Nmax due to soliton merging and turbulent redistribution. Choosing Nmax ∼ ES /k (using the condition of close packing) and replacing the cosh−2 term in (2.37) by R ES /k a step function yields h fk i ∝ L1 Nmin dN N℘(N). For the case when all states are occupied uniformly [℘(N) =const], h fk i ∝ k−2 . Interestingly, the equipartition spectrum k−2 observed experimentally in Section 2.5.4, is robust and appears in several different contexts of strong wave coupling. For example, dynamics with phase-dependent coupling — e.g., four-wave mixing — can give an effective wave collision term that leads asymptotically to a k−2 spectrum [48, 49, 50]. For the phase-independent coupling here, the intensityinduced interactions are enough to drive the dynamics. Indeed, similar conservation arguments on the photonic plasmons (speckles), rather than number ¡ ¢of solitons, also leads to a Rayleigh-Jeans distribution [48, 49, 50] f k = T / k2 − µ , where the effective temperature T ∝ lc−2 and the effective chemical potential µ is given by the average propagation constant (energy eigenvalue) of the waves. Note, however, that there must be a sufficient density of modes to achieve the equipartition. For example, the presence of (incoherent) solitons — e.g. from modulation instability — is not enough to guarantee equipartition. There must be enough interaction and propagation distance (evolution time) to go beyond soliton clustering [11] and cascade the interactions [12, 13]. Here, we encourage the wave-mixing cascade by seeding a quasi-thermal background distribution with additional non-equilibrium humps.

2 Photonic Plasma Instabilities and Soliton Turbulence in Spatially Incoherent Light

2.6 Discussion and Conclusions In conclusion, we have treated the nonlinear propagation of statistical light as a photonic plasma of interacting speckles. A general Bohm-Gross dispersion relation was derived, allowing the identification of both a plasma frequency and a Debye length. These determined the nonlinear propagation constant and scale of wave dephasing, respectively. This approach unified previous observations using partially coherent light, such as nonlinear diffraction and incoherent modulation instability, and predicted a new class of optical phenomena. As representative examples, we considered single and multiple bump-on-tail instabilities. Optical methods of measurement, such as holography, allowed observation of dynamical behavior that had been predicted, but not observed, in material plasma. This included equal and opposite momentum exchange for weak nonlinear coupling and evidence for wave condensation for strong coupling. In the latter regime, wave-wave interactions caused the humped power spectrum to merge into a single-peaked profile, with an algebraic k−2 spectrum in the inertial range. This profile, and its associated intensity modulations, is the hallmark signature of optical Langmuir turbulence. The results extend plasma dynamics beyond their fluid context and show clearly that there is much potential for controlling correlation dynamics and optical energy distributions using plasma-type wave phenomena. Acknowledgements We thank P.H. Diamond, C. Sun, and L.I. Dylova for very valuable discussions. This work was supported by the NSF, DOE and AFOSR.

References 1. A.A. Vedenov, L.I. Rudakov, Dokl. Akad. Nauk SSSR 159, 767 (1964). DOI OSTI-ID4648594 2. A.A. Vedenov, A.V. Gordeev, L.I. Rudakov, Plasma Physics 9(6), 719 (1967). URL http://stacks.iop.org/0032-1028/9/719 3. M. Soljacic, M. Segev, T. Coskun, D.N. Christodoulides, A. Vishwanath, Phys. Rev. Lett. 84(3), 467 (2000). DOI 10.1103/PhysRevLett.84.467 4. D. Kip, M. Soljacic, M. Segev, E. Eugenieva, D.N. Christodoulides, Science 290(5491), 495 (2000). DOI 10.1126/science.290.5491.495. URL http://www.sciencemag.org/cgi/content/abstract/290/5491/495 5. U. Frisch, Turbulence: The Legacy of A.N. Kolmogorov (Cambridge University Press, Cambridge, 1995) 6. B.B. Kadomtsev, Plasma Turbulence, vol. 1 (Academic Press, London, New York, 1965) 7. S. Jia, W. Wan, J.W. Fleischer, Opt. Lett. 32(12), 1668 (2007). URL http://ol.osa.org/abstract.cfm?URI=ol-32-12-1668 8. N.A. Krall, A.W. Trivelpiece, Principles of Plasma Physics (McGraw-Hill, New York, 1973, 1973) 9. M.V. Goldman, Rev. Mod. Phys. 56(4), 709 (1984). DOI 10.1103/RevModPhys.56.709 10. P.A. Robinson, Rev. Mod. Phys. 69(2), 507 (1997). DOI 10.1103/RevModPhys.69.507 11. Z. Chen, S.M. Sears, H. Martin, D.N. Christodoulides, M. Segev, Proceedings of the National Academy of Sciences of the United States of America 99(8), 5223 (2002). DOI 10.1073/pnas.072287299. URL http://www.pnas.org/content/99/8/5223.abstract

Dmitry V. Dylov and Jason W. Fleischer

12. V.N. Tsytovich, preprint, Academy of Sciences, USSR, [SOV PHYS USPEKHI, 1973, 15 (5), 632650] (150) (1969) 13. S.I. Popel, V.N. Tsytovich, S.V. Vladimirov, Physics of Plasmas 1(7), 2176 (1994). DOI 10.1063/1.870617. URL http://link.aip.org/link/?PHP/1/2176/1 14. D.V. Dylov, J.W. Fleischer, Phys. Rev. Lett. 100(10), 103903 (2008). DOI 10.1103/PhysRevLett.100.103903. URL http://link.aps.org/abstract/PRL/v100/e103903 15. D.V. Dylov, J.W. Fleischer, Phys. Rev. A (Atomic, Molecular, and Optical Physics) 78(6), 061804 (2008). DOI 10.1103/PhysRevA.78.061804. URL http://link.aps.org/abstract/PRA/v78/e061804 16. W. Wan, S. Jia, J.W. Fleischer, Nat Phys 3, 2007/01//print (2007). DOI 10.1038/nphys486. URL http://dx.doi.org/10.1038/nphys486 17. R. Dong, C.E. R¨uter, D. Kip, O. Manela, M. Segev, C. Yang, J. Xu, Phys. Rev. Lett. 101(18), 183903 (2008). DOI 10.1103/PhysRevLett.101.183903. URL http://link.aps.org/abstract/PRL/v101/e183903 18. D.V. Dylov, J.W. Fleischer, Opt. Lett. 34(17), 2673 (2009). URL http://ol.osa.org/abstract.cfm?URI=ol-34-17-2673 19. A.V. Mamaev, M. Saffman, A.A. Zozulya, Phys. Rev. Lett. 76(13), 2262 (1996). DOI 10.1103/PhysRevLett.76.2262 20. T. Bohr, M.H. Jensen, G. Paladin, A. Vulpiani, Dynamical Systems Approach to Turbulence, Cambridge Nonlinear Science Series, vol. 8 (Cambridge University Press, Cambridge, England, 1998, 1998) 21. D.N. Christodoulides, E.D. Eugenieva, T.H. Coskun, M. Segev, M. Mitchell, Phys. Rev. E 63(3), 035601 (2001). DOI 10.1103/PhysRevE.63.035601 22. E. Wigner, Phys. Rev. 40(5), 749 (1932). DOI 10.1103/PhysRev.40.749 23. J.E. Moyal, Mathematical Proceedings of the Cambridge Philosophical Society 45(01), 99 (1949). DOI 10.1017/S0305004100000487 24. B. Hall, M. Lisak, D. Anderson, R. Fedele, V.E. sem*nov, Phys. Rev. E 65(3), 035602 (2002). DOI 10.1103/PhysRevE.65.035602 25. L. Helczynski, D. Anderson, R. Fedele, B. Hall, M. Lisak, IEEE J. Sel. Top. Quantum Electron. 8(3), 408 (2002). DOI 10.1109/JSTQE.2002.1016342 26. V.V. Shkunov, D.Z. Anderson, Phys. Rev. Lett. 81(13), 2683 (1998). DOI 10.1103/PhysRevLett.81.2683 27. D.N. Christodoulides, T.H. Coskun, M. Mitchell, M. Segev, Phys. Rev. Lett. 78(4), 646 (1997). DOI 10.1103/PhysRevLett.78.646 28. S.V. Bulanov, M. Yamagiwa, T.Z. Esirkepov, D.V. Dylov, F.F. Kamenets, et al., Plasma Physics Reports 32(4), 263 (2006) 29. S.V. Bulanov, D.V. Dylov, T.Z. Esirkepov, F.F. Kamenets, D.V. Sokolov, Plasma Physics Reports 31(5), 369 (2005) 30. R. Fedele, D. Anderson, Journal of Optics B: Quantum and Semiclassical Optics 2(2), 207 (2000). URL http://stacks.iop.org/1464-4266/2/207 31. W. Martienssen, E. Spiller, American Journal of Physics 32(12), 919 (1964). DOI 10.1119/1.1970023. URL http://link.aip.org/link/?AJP/32/919/1 32. A. Picozzi, Opt. Express 15(14), 9063 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-14-9063 33. G.E. Vekstein, American Journal of Physics 66(10), 886 (1998). DOI 10.1119/1.18978. URL http://link.aip.org/link/?AJP/66/886/1 34. S.L. Musher, A.M. Rubenchik, V.E. Zakharov, Physics Reports 252(4), 177 (1995). DOI 10.1016/0370-1573(94)00071-A 35. C. Sun, D.V. Dylov, J.W. Fleischer, Opt. Lett. 34(19), accepted (2009) 36. M. Lisak, D. Anderson, L. Helczynski-Wolf, P. Berczynski, R. Fedele, V. sem*nov, Physica Scripta T113, 56 (2004). URL http://stacks.iop.org/1402-4896/T113/56 37. D. Anderson, B. Hall, M. Lisak, M. Marklund, Phys. Rev. E 65(4), 046417 (2002). DOI 10.1103/PhysRevE.65.046417 38. V. Arunasalam, American Journal of Physics 36(7), 601 (1968). DOI 10.1119/1.1975025. URL http://link.aip.org/link/?AJP/36/601/1

2 Photonic Plasma Instabilities and Soliton Turbulence in Spatially Incoherent Light

39. D. Anderson, L. Helczynski-Wolf, M. Lisak, V. sem*nov, Phys. Rev. E 69(2), 025601 (2004). DOI 10.1103/PhysRevE.69.025601 40. S. Pitois, S. Lagrange, H.R. Jauslin, A. Picozzi, Phys. Rev. Lett. 97(3), 033902 (2006). DOI 10.1103/PhysRevLett.97.033902. URL http://link.aps.org/abstract/PRL/v97/e033902 41. C. Anastassiou, M. Segev, K. Steiglitz, J.A. Giordmaine, M. Mitchell, M. feng Shih, S. Lan, J. Martin, Phys. Rev. Lett. 83(12), 2332 (1999). DOI 10.1103/PhysRevLett.83.2332 42. C. Anastassiou, J.W. Fleischer, T. Carmon, M. Segev, K. Steiglitz, Opt. Lett. 26(19), 1498 (2001). URL http://ol.osa.org/abstract.cfm?URI=ol-26-19-1498 43. D.N. Christodoulides, M.I. Carvalho, J. Opt. Soc. Am. B 12(9), 1628 (1995). URL http://josab.osa.org/abstract.cfm?URI=josab-12-9-1628 44. Z. Chen, J. Klinger, D.N. Christodoulides, Phys. Rev. E 66(6), 066601 (2002). DOI 10.1103/PhysRevE.66.066601 45. A.N. Kaufman, Journal of Plasma Physics 8(01), 1 (1972). DOI 10.1017/S0022377800006887 46. N.J. Fisch, J.M. Rax, Physics of Fluids B: Plasma Physics 5(6), 1754 (1993). DOI 10.1063/1.860809. URL http://link.aip.org/link/?PFB/5/1754/1 47. A.S. Kingsep, L.I. Rudakov, R.N. Sudan, Phys. Rev. Lett. 31(25), 1482 (1973). DOI 10.1103/PhysRevLett.31.1482 48. S. Dyachenko, A.C. Newell, A. Pushkarev, V.E. Zakharov, Physica D 57(1–2), 96 (1992). DOI 10.1016/0167-2789(92)90090-A 49. V.M. Malkin, Phys. Rev. Lett. 76(24), 4524 (1996). DOI 10.1103/PhysRevLett.76.4524 50. A. Picozzi, Opt. Express 15(14), 9063 (2007). DOI 10.1364/OE.15.009063. URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-14-9063

Chapter 3

Gap-Acoustic Solitons: Slowing and Stopping of Light Richard S. Tasgal, Roman Shnaiderman, and Yehuda B. Band

Abstract Solitons are paradigm localized states in physics. We consider here gapacoustic solitons (GASs), which are stable pulses that exist in Bragg waveguides, and which offer promising new avenues for slowing light. A Bragg grating can be produced by doping the waveguide with ions, and imprinting a periodic variation in the index of refraction with ultraviolet light. The Bragg grating in an optical waveguide reflects rightward-moving light to the left, and vice versa, and creates a gap in the allowed frequency spectrum of light. Nonlinearities, though, add complications to this simple picture. While low intensity light cannot propagate at frequencies inside the band gap, more intense fields can exist where low-intensity fields cannot. An optical gap soliton is an intense optical pulse which can exist in a Bragg waveguide because the intensity and nonlinearity let it dig a hole for itself inside the band gap, in which it can then reside. Far from the center of the pulse, the intensity is weak, and drops off exponentially with distance from the center. The optical gap soliton structure can be stable, and can have velocities from zero (i.e., stopped light) up to the group-velocity of light in the medium. When one also considers the system’s electrostrictive effects, i.e., the dependence of the index of refraction on the density of the material, which is a universal light-sound interaction in condensed matter, one obtains GASs. These solitons share many of the properties of standard gap solitons, but they show many fascinating new characteristics. GASs have especially interesting dynamics when their velocities are close to the speed of sound, in which range they interact strongly with the acoustic field. GASs which are moving Richard S. Tasgal Departments of Chemistry and Electro-Optics, and the Ilse Katz Center for Nano-Science, BenGurion University of the Negev, Beer-Sheva 84105, Israel e-mail: [emailprotected] Roman Shnaiderman Departments of Chemistry and Electro-Optics, and the Ilse Katz Center for Nano-Science, BenGurion University of the Negev, Beer-Sheva 84105, Israel e-mail: [emailprotected] Yehuda B. Band Departments of Chemistry and Electro-Optics, and the Ilse Katz Center for Nano-Science, BenGurion University of the Negev, Beer-Sheva 84105, Israel e-mail: [emailprotected]

O. Descalzi et al. (eds.), Localized States in Physics: Solitons and Patterns, DOI 10.1007/978-3-642-16549-8_3, © Springer-Verlag Berlin Heidelberg 2011

Richard S. Tasgal, Roman Shnaiderman, and Yehuda B. Band

at supersonic velocities may experience instabilities which leave the GAS whole, but bring the velocity abruptly to almost zero. Furthermore, GASs may be made to change velocity by collision with acoustic pulses. Moving GASs may be retarded by the phonon viscosity, as well as by interaction with high wave number (Brillouin) acoustic waves. Thus, the opto-acoustic interactions provide the basis for a set of tools with which light in the form of a GAS can be slowed down and controlled. In contrast with other forms of slow or stopped light, GASs can exist at room temperature, in relatively unexotic materials. This makes the GAS an attractive form in which to create and work with slow and stopped light.

3.1 Introduction One of the paradigm examples of localized states in physics is the soliton, a pulse that gets its stability from a balance of dispersion and nonlinearity. The gap-acoustic soliton (GAS) is an optical and acoustic structure that can exist in an optical waveguide with a Bragg grating. Figure 3.1 is a schematic illustration of a fiber waveguide with a periodically varying refractive index with light and sound waves propagating within it. A Bragg grating can be produced by doping the waveguide with ions (e.g., germanium), and imprinting a periodic variation in the index of refraction with ultraviolet light [1]. As we shall see, GASs are good systems in which to realize slow light. GASs can be viewed from three perspectives: (1) as an extension of optical gap solitons to a regime where their interaction with sound waves is important, (2) as a new application of electrostriction and Brillouin scattering, or (3) as a means to produce slow light. The study of solitons has a long history. There is a narrow definition, which applies only to completely integrable systems. In this strict sense, solitons are connected to the inverse scattering method [2]. A broader definition of a soliton is a pulse that is stable due to a balance of dispersion and nonlinearity [1, 2]. The first solitons to be discovered were localized shallow-water waves, by John Scott Russell in 1834 [3], known as Korteweg–de Vries (KdV) solitons [4]. The KdV equation was shown to be completely integrable by the inverse scattering method, and that the pulses are solitons in the stricter sense was demonstrated much later [2, 5]. The first optical soliton discovered was in the nonlinear Schr¨odinger (NLS) equation. The NLS equation was found to be completely integrable and to support solitons, in the strict sense, in Ref. [6]. Independently, Ref. [7] showed that there are solitons in the broad sense which can be realized in optical fibers. In the NLS equation, linear (small amplitude) continuous wave (cw) solutions exist along a parabolic curve in the space (k, ω ) of wave number and frequency that is concave upwards (there is a maximum wave number). The soliton solutions exist in the space above the cw dispersion curve. The first gap soliton article did not use the phrase “gap soliton,” but rather referred to the equations as the massive Thirring model (MTM) [8], with particle physics in mind rather than optics. The solutions discovered were solitons even in

3 Gap-Acoustic Solitons: Slowing and Stopping of Light

periodically varying refractive index fiber

light waves sound waves Fig. 3.1 Schematic illustration of a fiber with a periodically-varying refractive index. Light and sound waves propagate in the fiber. Photons are shown as wavy lines with arrows indicating the direction of motion and phonons are shown as a solid line with double-sided arrows.

the strictest sense—the system was shown to be integrable by the inverse scattering method, and the pulse solutions (solitons) were shown to correspond to poles of the transmission coefficient [9]. The frequencies of the solitons are inside the gap between the continuous wave solutions of the linear system appropriate for lowintensity waves. However, the soliton frequencies are not all between the maximum of the lower cw band and the minimum of the upper cw band; for this reason, interpreting the band gap more narrowly, some authors prefer the term “Bragg soliton” for solitons with frequencies that are either above the minimum of the upper cw band or below the maximum of the lower cw band—see, e.g., Ref. [10]. Independently of the mathematical discovery of the gap solitons, a qualitative description and prediction of the still theoretical optical gap solitons was made in Ref. [11]. Exact analytic forms for optical gap solitons were found for a nonlinearity with self-phase modulation in addition to cross-phase modulation: Ref. [12] found the solutions in the exact middle of the band gap, and Ref. [13] found the full family of gap soliton solutions. This is not a completely integrable system, and the pulses are solitons in the broader sense but not the narrower sense. As a result, the pulses are not guaranteed to have the stability of the MTM solitons. The stability of gap solitons beyond the completely integrable MTM limit (optical gap solitons have self-phase modulation, so are not MTM), was not immediately clear. Ref. [12] showed one direct numerical simulation of a gap soliton collision, in which the individual gap solitons were stable, and the solitons emerged from a collision intact but perturbed. Ref. [14] performed variational model calculations of optical gap solitons, which showed some regions where excited modes exist, and other regions with instabilities. References [15, 16, 17] showed rigorously that op-

Richard S. Tasgal, Roman Shnaiderman, and Yehuda B. Band

tical gap solitons are stable in the top half of the frequency band gap, and unstable in most of the bottom half of the band gap. Reference [18] generalized the optical gap soliton equations to include dependence of the index of refraction on the density of the material, and acoustic waves. In other words, acoustic waves and their interaction with light through electrostriction were included in the model. New generalized “gap-acoustic soliton” (GAS) solutions were also found. The GASs are similar to optical gap solitons, but they exhibit many intriguing novel dynamical properties, especially when the soliton velocities are small. Reference [19] looked in detail at the system in the case that the physical parameters of bulk fused silica, and found that electrostriction will have much large, not merely perturbative, influences on the GASs when velocities are as slow as two orders of magnitude less than the group velocity of light; solitons need not be, as is the speed of sound, five orders of magnitude slower than the speed of light, for acoustic effects to be strongly felt. Dependence of the index of refraction on the density of the material is a universal property of materials [20], and interaction of light with sound waves is ubiquitous. Interaction between light and high wavenumber acoustic waves—approximately twice the wave numbers of the light—is called Brillouin scattering [21], and interaction between light and low wavenumber acoustic waves is generally referred to as electrostriction [22]. Distinctions in nomenclature notwithstanding, the two effects have the same physical source. Reference [19] derived the Brillouin scattering (short acoustic wavelength) interaction together with the electrostrictive (long acoustic wavelength) interactions in a unified manner. There have been significant research efforts in recent decades towards the achievement of slow light (see, e.g., Ref. [23]). One way to achieve slow light is to use electromagnetically-induced transparency to reduce group-velocities without large absorption [23]. Another form of slow light—which we concentrate on here—is the optical gap soliton, which moves slower than the group velocity or even at velocity zero [8, 9, 11, 12, 13, 15, 16, 17, 24, 25, 26, 27, 28, 29]. Optical gap solitons may exist in a nonlinear waveguide with a Bragg grating. The Bragg grating creates a band gap for light that is in-phase with the grating. The nonlinearity allows a pulse of light in the waveguide to dig a hole for itself in the forbidden region. This structure may be stable, balancing the nonlinearity against the Bragg-grating-induced dispersion, i.e., it will be a soliton. This optical soliton may have a velocity which is slow or even zero. It is slow or stopped light. To date, the slowest experimentally realized optical gap solitons had velocity c/6 [29]. When the soliton velocity is comparable to the sound velocity, the interaction between the light and sound can be strong because they can propagate together. The outline of this paper is as follows. Section 3.2 gives a derivation of the equations for the dynamics of the optical gap soliton system, along with all the acoustic interactions that the system supports. Section 3.3 details the general properties of this system, and Sec. 3.4 gives the soliton solutions. Section 3.5 goes over the stability properties of the solitons, and retardation effects to obtain slow light. Finally, Sec. 3.6 contains a summary and conclusion.

3 Gap-Acoustic Solitons: Slowing and Stopping of Light

3.2 Derivation of the Equations We begin with general equations for two fields, light and sound. Light is governed by Maxwell’s equations. Sound can be described by a wave equation for the density change in the medium. In addition, light and sound interact via electrostriction. For optical gap solitons in a fiber, the light is of approximately one color, but the direction can be either forward or backward. The electromagnetic field can thus be broken down into two separate slowly-varying envelopes about fast-varying carrier waves, one for forward-moving light and one for backward-moving light. The acoustic fields that interact with this light can be of high wave number or low wave number. The high wave number acoustic fields can be either forward- or backwardmoving. The low wave number acoustic field is centered at wave number zero, with some spread to both positive and negative values. The acoustic field for this system can then be broken down into three slowly-varying envelopes, two for the high wave number waves, and one for low wave number.

3.2.1 Electromagnetic Field Equations with Phonon Perturbations The starting point of the derivation is Maxwell’s equations. We begin considering an isotropic medium without free charges, currents, or magnetic polarization. Bragg and Brillouin scattering will be covered as extensions of this, by dropping the assumption of isotropy. The electromagnetic field, and the linear and nonlinear polarization of the medium satisfy the equations, ∇ · (E + 4π Plinear + 4π PNL ) = 0

(3.1a)

∇·B = 0

(3.1b)

1 ∂ B c ∂t 1 ∂ ∇×B = − (E + 4π Plinear + 4π PNL ) , c ∂t ∇×E = −

(3.1c) (3.1d)

The dependence of polarization P = Plinear + PNL on the electromagnetic field E, B, has a part which is linear in the electromagnetic field, with an additional dependence on the density of the material, E + 4π Plinear ≡ D = n2 (ω , w) E ,

(3.2a)

where the expression on the right hand side, relating electric displacement to electric field via a frequency-dependent index of refraction, holds in frequency space and in real space for monochromatic fields. We have indicated a dependence of the index of refraction on the density of the material w. There is also a contribution to the polarization due to a third-order Kerr nonlinearity:

Richard S. Tasgal, Roman Shnaiderman, and Yehuda B. Band

PNL =

χs (E · E)E . 3

(3.2b)

Fourier transform over the time dimension to get the equation in frequency-space. In an isotropic medium, Coulomb’s law (3.1a) and Ampere’s law (3.1d) take the form 0 = n(ω )2 ∇ · E(x, ω ) + 4π ∇ · PNL (x, ω ) , ω 0 = ∇ × B(x, ω ) + i [n(ω )2 E(x, ω ) + 4π PNL (x, ω )] . c

(3.3a) (3.3b)

Inserting these into the curl of Faraday’s law (3.1c) gives the wave equation, · ¸ n(ω )2 ω 2 0 = ∇2 + E(x, ω ) c2 +

4π ω 2 4π ∇ · PNL (x, ω )] . PNL (x, ω ) + ∇ [∇ c2 n(ω )2

(3.4a)

A Fourier transform in the spatial dimensions gives the wave equation in momentum space, · ¸ n(ω )2 ω 2 0 = k2 − E(k, ω ) c2 ½ ¾ 4π ω 2 c2 − 2 PNL (k, ω ) − k [k · P (k, ω )] . (3.4b) NL c n(ω )2 ω 2 If the nonlinear polarization is transverse, which will be the case with the Kerr nonlinearity (3.2b), and transverse electric field, the last terms on the right-hand sides of Eqs. (3.4) vanish. The basic optical gap soliton has one (nontrivial) spatial dimension, and takes the system to have light of one polarization, so we reduce the generality of the mathematical model and obtain, · 2 ¸ ∂ n(ω )2 ω 2 4π ω 2 0= + E(z, ω ) + 2 PNL (z, ω ) . (3.5a) 2 2 ∂z c c · ¸ n(ω )2 ω 2 4π ω 2 2 0= k − E(k, ω ) − PNL (k, ω ) . (3.5b) c2 c2 If we consider the wave equation (3.5) in the vicinity of frequency ω0 and wave number k0 = n(ω0 )ω0 /c, complete the square for the quadratic equation, Taylor expand in the small terms, and truncate, we obtain,

3 Gap-Acoustic Solitons: Slowing and Stopping of Light

· 0 = (k0 + δ k)2 −

n(ω0 + δ ω )2 (ω0 + δ ω )2 c2

¸ 4π (ω0 + δ ω )2 PNL (k0 + δ k, ω0 + δ ω ) − E(k0 + δ k, ω0 + δ ω ) , c2 E(k0 + δ k, ω0 + δ ω ) s " # n(ω0 + ω ) (ω0 + ω ) 4π PNL 0 = ∓(k0 + δ k) + 1+ c [n(ω0 + ω )]2 E

(3.6a)

E(k0 + δ k, ω0 + δ ω ) ,

(3.6b) n(ω0 + ω ) (ω0 + ω ) = ∓(k0 + δ k)E(k0 + δ k, ω0 + δ ω ) + E c 2π (ω0 + ω ) + PNL (k0 + δ k, ω0 + δ ω ) + . . . (3.6c) n(ω0 + ω )c µ ¶ µ ¶ n(ω0 ) ω0 d n(ω ) ω = ∓δ kE(k0 + δ k, ω0 + δ ω ) + ∓ k0 E + ωE c dω c ω0 +

2πω0 /c PNL (k0 + δ k, ω0 + δ ω ) + . . . n(ω0 )

(3.6d)

Let us now Fourier transform back to real space, and include a nonuniformity in the index of refraction, which is also a function of the material density, µ ¶ ∂ ∂ n(ω0 , z,W ) ω0 2π ³ω0´2 0 = ik00 E(z,t) ± i E + ∓ k0 E + PNL (z,t) + (3.7a) ... ∂t ∂z c k0 c µ ¶ ∂ ∂ ω0 ω0 dn 2π ³ω0 ´2 = ik00 E(z,t) ± i E + ∆ n(z) + W E+ PNL (z,t) (3.7b) +... ∂t ∂z c c dW k0 c Here k0 = ±n(ω0 )ω0 /c is the phase velocity, and k00 = (d/d ω )[n(ω )ω /c]ω =ω0 is the reciprocal of the group velocity. When arguments of the index of refraction are implicit, they are based on an average value at a baseline material density W . ∆ n(z) is the spatially-varying part of the refractive index. The result is an equation for a slowly-varying envelopes about a carrier wave with wave vector (k0 , ω0 ). Equation (3.7) applies generally to any quasi-monochromatic electromagnetic field with any nonlinearity. For the optical gap soliton, there is one frequency of light in the system, and the light may be traveling forward or backward. The electric field E may then be written as two slowly-varying envelopes (SVEs) about carrier waves with frequencies ω = ω0 and wave numbers k = ±k0 = ±n(ω0 )ω0 /c. The acoustic fields that can interact with these light fields are those centered at wave numbers k = 0 and ±2k0 . If the speed of sound (which we can refer to as βsound ) is constant—which to a good approximation it is—then for the carrier waves of the acoustic waves, the frequencies of the acoustic waves are simply the speed of sound (βsound ) times the wave numbers. We also allow the index of refraction to have a small component at half the wavelength of the light, which will yield Bragg scattering from the periodic grating set up in the material,

Richard S. Tasgal, Roman Shnaiderman, and Yehuda B. Band

E(z,t) = U(z,t) exp[i(k0 z − ω0t)] +V (z,t) exp[−i(k0 z + ω0 t)] + U ∗ (z,t) exp[−i(k0 z − ω0 t)] +V ∗ (z,t) exp[i(k0 z + ω0t)] ,

(3.8a)

W (z,t) = Wu (z,t) exp[2ik0 (z − βsoundt)] +Wv (z,t) exp[−2ik0 (z + βsoundt)] + Wu∗ (z,t) exp[−2ik0 (z − βsoundt)] +Wv∗ (z,t) exp[2ik0 (z + βsoundt)] + W0 (z,t) , ∆ n(z) = ∆ n cos(2k0 z) .

(3.8b) (3.8c)

Substituting the fields in terms of SVEs [Eqs. (3.8)] into the general dynamical equations for light [Eq. (3.7b)], while taking the nonlinearity to be Kerr [Eq. (3.2b)], and separating the different frequency and wave number components, gives 2π (ω0 /c)2 (χs |U|2 + χx |V |2 )u k0 +χes [W0U + exp(−2ik0 βsoundt)WuV + exp(2ik0 βsoundt)Wv∗V ] , (3.9a)

0 = ik00 Ut + iUz + κ V +

2π (ω0 /c)2 (χx |U|2 + χs |V |2 )v k0 +χes [W0V + exp(2ik0 βsoundt)Wu∗U + exp(−2ik0 βsoundt)WvU ] , (3.9b)

0 = ik00 Vt − iVz + κ ∗U +

where

ω0 ∆ n , c 2 ω0 dn χes = . c dW κ=

(3.10a) (3.10b)

This assumes that the speed of sound βsound is small enough so that the frequencies 2k0 βsound are within the frequency spread of the SVEs, U and V . These are the equations for the dynamics of the SVEs of light.

3.2.2 Acoustic Wave Equations with Electrostrictive Perturbations To complete the dynamical system, we need equations for the density of the material—that is, acoustic waves. In silica glass, the speed of sound has a very weak dependence on frequency, and acoustic waves are also subject to viscosity [30]. Dependence of the index of refraction on the density of the material creates electrostriction, a force (pressure gradient) attracting the material to regions of higher light intensity. The equation for evolution of the density of a material of this system is [22, 31] 0=

∂2 ∂ Λ 2 W (x, y, z,t) − βsound ∇2W − Γs ∇2W + ∇2 hE(x, y, z,t)2 i , 2 ∂t ∂t 2

(3.11)

3 Gap-Acoustic Solitons: Slowing and Stopping of Light

where W (x, y, z,t) is the density of the material, E(x, y, z,t) is the amplitude of the electric field, ∇2 = ∂ 2 /∂ x2 + ∂ 2 /∂ y2 + ∂ 2 /∂ z2 is the Laplacian, βsound is the speed of sound, Γs is a phonon viscosity coefficient, and Λ is an electrostrictive coefficient. We will focus on single-mode waveguides, in which any transverse dynamics are trivial, in the sense that that the transverse confinement affects only the values of the coefficients, but qualitative terms will be the same as one-dimensional infinite plane waves [32].) This reduces the system to 1+1-dimensions, 0=

∂2 ∂2 ∂3 Λ ∂2 2 W (z,t) − βsound W − Γs W+ hE(z,t)2 i . 2 2 2 ∂t ∂z ∂ t∂ z 2 ∂ z2

(3.12)

Since we will be dealing with optical gap solitons, the light in the system is approximately monochromatic and may be moving forward or backward, as expressed by Eq. (3.8a). Electrostrictive response times are on the order of 10−9 s [22]. This is several (∼ 6) orders of magnitude slower than the temporally fast-varying terms (∝ U 2 ,V 2 ,U ∗2 ,V ∗2 ) for visible or near infra-red light, so these may be dropped from the averaged square field in the phonon equation (3.12), 2 0 = Wtt − βsound Wzz − ΓsWtzz £ 2 ¤ +Λ |U| + |V |2 +UV ∗ exp(2ik0 z) +U ∗V exp(−2ik0 z) zz .

(3.13)

where we have denoted partial derivatives by subscripts. Since U(z,t) and V (z,t) are SVEs, the phonons’ source terms will be centered around wavenumbers k = 0, 2k0 , and −2k0 . Thus light in the optical gap solitons will interact by electrostriction only with phonons around those same wave numbers, which is consistent with Eq. (3.8b). Fourier transform the phonon equation (3.13) to momentum space, 2 0 = −ω 2W (k, ω ) − iω k2ΓsW + k2 βsound W − k2Λ F {|U|2 + |V |2 }(k, ω )

− k2Λ F {UV ∗ }(k − 2k0 , ω ) − k2 Λ F {U ∗V }(k + 2k0 , ω ) .

(3.14)

Since U and V are SVEs, F {|U|2 + |V |2 }(k, ω ) will only be significant in the vicinity of k ≈ 0, F {UV ∗ }(k − 2k0 , ω ) will only be significant at k ≈ 2k0 , and F {U ∗V }(k + 2k0 , ω ) will only be significant at k ≈ −2k0 . We substitute the expression for W in Eq. (3.8b) into the general phonon equation (3.14), and separate into the different (and, in k-space, non-overlapping) regions: 2 0 = −ω 2W0 (k, ω ) − iω k2ΓsW0 + k2 βsound W0 − k2Λ F {|U|2 + |V |2 }(k, ω(3.15a) ), 2 0 = (ω − ω0 )2Wu (k, ω ) + i(ω − ω0 )(k − 2k0 )2ΓsWu − (k − 2k0 )2 βsound Wu 2 ∗ +(k − 2k0 ) Λ F {UV }(k − 2k0 , ω − ω0 ) , (3.15b) 2 0 = (ω − ω0 )2Wv (k, ω ) + i(ω − ω0 )(k + 2k0 )2 ΓsWv − (k + 2k0 )2 βsound Wv 2 ∗ +(k + 2k0 ) Λ F {U V }(k + 2k0 , ω ) . (3.15c)

Here Wu (k, ω ) = W (k−2k0 , ω − ω0 ), Wv (k, ω ) = W (k +2k0 , ω − ω0 ), and W0 (k, ω ) = W (k, ω ) are SVEs of the density W .

Richard S. Tasgal, Roman Shnaiderman, and Yehuda B. Band

3.2.2.1 Slowly-Varying Phonon Field We take the phonon equation (3.15a), which is for the region near the origin in (k, ω )-space, or the slowly-varying part of the phonon field, and inverse Fourier transform it to real space, 2 0 = W0,tt − βsound W0,zz − ΓsW0,tzz + Λ (|U|2 + |V |2 )zz .

(3.16)

This is the most useful form of the governing equations for low wave number (long wavelength) acoustic waves.

3.2.2.2 Brillouin Scattering—Phonon Fields at k ≈ 2k0 Consider Eq. (3.15b) for the part of the phonon field with wave numbers close to k = 2k0 . We complete the square, expand the root into a Taylor series, and drop higher-order terms: 0 = [ω + 2i(k0 + k/2)2Γs + ωu ±2(k0 + k/2)βsound {1 − (1/2)[(k0 + k/2)Γs /βsound ]2 }]Wu (k, ω ) −1 ∓(k0 + k/2)Λ βsound F {UV ∗ }(k, ω + ω0 ) + . . . (3.17) Dropping wavenumber dependence of the damping, higher-order dispersion, a selfsteepening-like term (in the sense that it comes from going from a second- to a first-order differential equation), and the quadratic or higher terms in the phonon viscosity, since phonon viscosity is generally a small perturbation, we obtain ¤ £ −1 0 = ω + 2ik02Γs ± kβsound Wu (k, ω ) ∓ k0 Λ βsound F {UV ∗ }(k, ω ∓ 2k0 βsound ) + . . . (3.18) where ω0 = 2k0 βsound . We now inverse Fourier transform this to real space, 0 = iWu,t + i(2k02Γs )Wu ∓ iβsoundWu,z ∓

k0Λ exp[∓2ik0 βsoundt](UV ∗ ) + . . . (3.19) βsound

The positive sign is the relevant solution for the field Wu , 0 = iWu,t + iβsoundWu,z + i(2k02Γs )Wu +

k0 Λ exp(2ik0 βsoundt)UV ∗ . (3.20a) βsound

The corresponding equation for the Brillouin field moving in the opposite direction (k = −2k0 ) is 0 = iWv,t − iβsoundWv,z + i(2k02Γs )Wv +

k0 Λ exp(2ik0 βsoundt)UV ∗ . (3.20b) βsound

3 Gap-Acoustic Solitons: Slowing and Stopping of Light

3.2.3 The Bragg-Brillouin-Kerr System Let us collect the definitions of the SVEs of the electromagnetic and phonon fields, E(z,t) = U(z,t) exp[i(k0 z − ω0t)] +V (z,t) exp[i(−k0 z − ω0 t)] + U ∗ (z,t) exp[−i(k0 z − ω0 t)] +V ∗ (z,t) exp[i(k0 z + ω0 t)] , (3.21a) W (z,t) = W0 (z,t) +Wu (z,t) exp[2ik0 (z − βsoundt)] +Wv (z,t) exp[−2ik0 (z + βsoundt)] + Wu∗ (z,t) exp[−2ik0 (z − βsoundt)] +Wv∗ (z,t) exp[2ik0 (z + βsoundt)] , (3.21b) and their dynamical equations, 2π (ω0 /c)2 (χs |U|2 + χx |V |2 )U k0 +χes [W0U + exp(−2ik0 βsoundt)WuV + exp(2ik0 βsoundt)Wv∗V ] , (3.22a)

0 = ik00 Ut + iUz + κ V +

2π (ω0 /c)2 (χx |U|2 + χs |V |2 )V k0 +χes [W0V + exp(2ik0 βsoundt)Wu∗U + exp(−2ik0 βsoundt)WvU] , (3.22b) 2 0 = W0,tt − βsound W0,zz − ΓsW0,tzz + Λ (|U|2 + |V |2 )zz . (3.22c) k Λ 0 0 = iWu,t + iβsoundWu,z + i(2k02Γs )Wu + exp(2ik0 βsoundt)UV ∗ , (3.22d) βsound k0 Λ 0 = iWv,t − iβsoundWv,z + i(2k02Γs )Wv + exp(2ik0 βsoundt)U ∗V . (3.22e) βsound 0 = ik00 Vt − iVz + κ U +

3.3 Lagrangian, Hamiltonian, and Conserved Quantities The Bragg-Brillouin-Kerr system (3.22) can be derived from a Lagrangian density in the limit in which phonon viscosity is nil (Γs = 0), L =

i 0 ∗ i i i k (U Ut −UUt∗ ) + k00 (V ∗Vt −VVt∗ ) + (U ∗Uz −UUz∗ ) − (V ∗Vz −VVz∗ ) 2 0 2 2 2 i 2π (ω0 /c)2 h χs +κ U ∗V + κ ∗UV ∗ + (|U |4 + |V |4 ) + χx |U|2 |V |2 k0 2 χes 2 2 + (r − βsound rz2 ) + χes (|U|2 + |V |2 )rz 2Λ t χes βsound i ∗ + [(Wu∗Wu,t −WuWu,t ) + (Wv∗Wv,t −WvWv,t∗ )] k0 Λ 2 2 χes βsound i ∗ ∗ + [(Wu∗Wu,z −WuWu,z ) − (Wv∗Wv,z −WvWv,z )] k0 Λ 2 +χes exp(2ik0 βsoundt) (UV ∗Wu∗ +U ∗VWv∗ ) +χes exp(−2ik0 βsoundt) (U ∗VWu +UV ∗Wv ) ,

(3.23a)

Richard S. Tasgal, Roman Shnaiderman, and Yehuda B. Band

for which we have introduced a potential for the slowly-varying phonon field r(z,t) ≡

Z z

W0 (z0 ,t)dz0 ,

(3.23b)

where z0 is an arbitrary constant. The system has a Hamiltonian and three conserved quantities, corresponding to conservation of momentum P, conservation of the number of photons N (also sometimes called energy), and conservation of mass M (slightly abstracted, such that divergences on an infinite domain are avoided), Z ∞ ½ i i dz − (U ∗Uz −UUz∗ ) + (V ∗Vz −VVz∗ ) − κ U ∗V − κ ∗UV ∗ H= 2 2 −∞ i 2π (ω0 /c)2 h χs − (|U|4 + |V |4 ) + χx |U|2 |V |2 k0 A 2 χes 2 2 + (r + βsound rz2 ) − χes (|U|2 + |V |2 )rz 2Λ t · ¸ 2 χes βsound i i ∗ ∗ − (Wu∗Wu,z −WuWu,z ) − (Wv∗Wv,z −WvWv,z ) k0 Λ 2 2 ∗ −χes exp(−2ik0 βsoundt) (U VWu +UV ∗Wv ) −χes exp(2ik0 βsoundt) (UV ∗Wu∗ +U ∗VWv∗ )} , (3.24a) i P= (U ∗Uz −UUz∗ ) + (V ∗Vz −VVz∗ ) 2 −∞ 2 · ¸¾ χes βsound i ∗ ∗ ∗ ∗ + r r + (W W −W W +W W −W W ) dz(3.24b) z t u,z u u,z v v,z v v,z Λ k00 k0 2 u Z ∞½ i

N= M=

Z ∞

(|U|2 + |V |2 )dz ,

(3.24c)

rt dz .

(3.24d)

−∞ Z ∞ −∞

If phonon viscosity is included, then the number of photons N and the material mass M are still constant, but the momentum P and the energy H decay according to the formulas Z

d Γ χes ∞ H= W0,t W0,z dz , dt Λ −∞ Z ∞ d Γ χes P=− (W0,t )2 dz . dt Λ k00 −∞

(3.25a) (3.25b)

3 Gap-Acoustic Solitons: Slowing and Stopping of Light

3.3.1 Dimensionless Variables To give a clearer and more systematic picture of the dynamics, we rewrite the equations in terms of dimensionless variables, s 2π (ω0 /c)2 u≡ U, (3.26a) κ k0 s 2π (ω0 /c)2 v≡ V, (3.26b) κ k0 1 w0 ≡ W0 , (3.26c) κ 1 wu ≡ Wu , (3.26d) κ 1 wv ≡ Wv , (3.26e) κ κ τ ≡ 0 t, (3.26f) k0 ζ ≡ κ z. (3.26g) The governing Eqs. (3.22) take the form 0 = iuτ + iuζ + (1 + κBrill )v + ( χs |u|2 + χx |v|2 + χes w)u ,

(3.27a)

∗ ivτ − ivζ + (1 + κBrill )u + (χx |u|2 + χs |v|2 + χes w)v , w0,ττ − Γ w0,τζ ζ − βs2 w0,ζ ζ + λ (|u|2 + |v|2 )ζ ζ ,

(3.27b)

0= 0=

(3.27c)

λ k0 /κ exp[2i(k0 /κ )βs τ ] uv∗ ,(3.27d) βs λ k0 /κ 0 = iwv,t − iβs wv,z + i[2(k0 /κ )2Γ ]wv + exp[2i(k0 /κ )βs τ ]u∗ v , (3.27e) βs 0 = iwu,t + iβs wu,z + i[2(k0 /κ )2Γ ]wu +

with

κBrill (ζ , τ ) = exp[2i(k0 /κ )βs τ ]wu + exp[−2i(k0 /κ )βs τ ]w∗v ,

(3.28)

and normalized coefficients

βs = βsound k00 , Γ = Γs κ k00 , k0 (k00 )2 λ = Λ χes . 2π (ω0 /c)2

(3.29a) (3.29b) (3.29c)

Richard S. Tasgal, Roman Shnaiderman, and Yehuda B. Band

In ordinary optical materials, χx = 2 χs . We have elected to eliminate one fewer variable than is possible, keeping χx , to make clearer the effects of large electrostriction compared to instantaneous Kerr effect.

3.4 Gap-Acoustic Solitons Let us begin by looking at Eqs. (3.27) without the Brillouin fields wu , wv , 0 = iuτ + iuζ + v + (χs |u|2 + χx |v|2 )u + χes w u , 2

(3.30a)

0 = ivτ − ivζ + u + ( χx |u| + χs |v| )v + χes w v ,

(3.30b)

w0,τζ ζ − βs2 w0,ζ ζ

(3.30c)

0 = w0,ττ − Γ

+ λ (|u| + |v| )ζ ζ ,

We found a family of solutions for gap-acoustic solitons for Eqs. (3.30) with zero phonon viscosity (Γ = 0), p

i γ (1 + β ) α sin Q sech(ζ˜ sin Q − Q) exp[i θ (ζ˜ ) − i τ˜ cos Q] (, 3.31a) 2 p i v(ζ , τ ) = − γ (1 − β ) α sin Q sech(ζ˜ sin Q + Q) exp[i θ (ζ˜ ) − i τ˜ cos Q] (3.31b) , 2 λ 4|α |2 γ sin2 Q w(ζ , τ ) = 2 , (3.31c) βs − β 2 cosh(2ζ˜ sin Q) + cos Q u(ζ , τ ) =

where

θ (ζ˜ ) = 4|α |2 γ 2 β [χs + λ χes /(βs2 − β 2 )] tan−1{tanh[ζ˜ sin Q] tan(Q/2)}(3.32a) , α = {χx + χs γ 2 (1 + β 2 ) + 2λ γ 2 /(βs2 − β 2 )}−1/2 , τ˜ ≡ γ (τ − β ζ ) , ζ˜ ≡ γ (ζ − β τ ) , 2 −1/2

γ ≡ (1 − β )

(3.32b) (3.32c) (3.32d) (3.32e)

and α must be real-valued. In the quiescent limit (β → 0), these are also solutions for non-zero phonon viscosity (Γ > 0). The solitons Eqs. (3.31)-(3.32) have two essential intrinsic parameters, β , the velocity, and Q, which takes values 0 < Q < π . The soliton parameter Q resembles a similar parameter in the family of the ordinary gap solitons. The soliton’s full width at half maximum intensity is [cosh−1 (2 + cos Q)/(γ sin Q)]. Frequency in the rest frame is γ cos Q. Frequency in the frame moving with the soliton is not generally equal to cos Q because group velocity in a medium is normally less than the speed of light in vacuum. The soliton velocity (β ) may have any value up to the group velocity of light in the medium (|β | < 1), except for a range of slightly supersonic gap solitons, |β | 6∈ [βs , βcr ], where

3 Gap-Acoustic Solitons: Slowing and Stopping of Light



 s µ ¶2 1 χ + χ χ + χ 8 λ χ x s x s es  βcr2 = βs2 + + βs2 − − . 2 χx − χs χx − χs χx + χs

(3.33)

In fused silica, the critical velocity is 10% greater than the speed of sound, βcr = 1.10βs [19]. Bright supersonic as well as subsonic solitons exist if the critical velocity βcr is less than the speed of light in the medium. (The equations suggest existence of a dark soliton [33] in the supersonic region βs < β < βcr , but we choose to limit this paper to bright solitons.) The closer the soliton velocity is to the speed of sound, the larger is the percentage of energy in the phonon field. Figure 3.2 shows a moderately supersonic soliton.

0.5 0 −0.5

0 −1 −2 −3 −4

−3

−2

−1

Fig. 3.2 Supersonic gap-acoustic soliton. The solitons’s frequency is in the middle of the band gap (soliton parameter Q = π /2), and it’s velocity is βs = 0.25, which is 125% of the speed of sound, βs = 0.2. The self- and cross-phase modulation coefficients are χs = 1, χx = 2, and the electrostrictive coefficients are χes = 1, λ = 0.1. The first part of the figure shows the amplitude of the envelope u of the forward-moving electromagnetic wave, the second the envelope v of the backward-moving wave, and the third part the acoustic field (material density). Solid lines are for the magnitudes of the amplitudes, dashed lines for the real parts, and dotted lines are for the imaginary parts.

The gap-acoustic solitons (3.31)-(3.32) reduce to standard gap solitons [13] in the limit of zero electrostriction (λ = 0). There are resemblances to solitons in the Zakharov system [34, 35, 36], in that both contain dispersive equations coupled to a nondispersive equation, interaction with the non-dispersive field changes the amplitude of the soliton, and the dispersive field takes a profile the same shape as the soliton intensity. Below the speed of sound, the accompanying phonon pulse is a positive density variation, and above the speed of sound, the phonon pulse is a depression.

Richard S. Tasgal, Roman Shnaiderman, and Yehuda B. Band

Note that Eqs. (3.30) do not admit optical solitons without an acoustic component; purely acoustic pulses are possible. In the case of zero phonon viscosity Γ = 0, these have the form u = v = 0, while the phonon field w is a combination of two arbitrary functions, w(ζ , τ ) = w+ (ζ − τ ) and w− (ζ + τ ), which represent forwardand backward-moving acoustic waves. The soliton’s (quasi-)conserved quantities, number of photons, phonons, momentum, and Hamiltonian, are obtained by substituting the soliton formulas (3.31)(3.32) into Eqs. (3.24) to obtain MGAS =

λ βs2 − β 2 2

4|α |2 Q ,

NGAS = 4|α | Q , PGAS = β γ (4|α |2 ) sin Q µ ¶ λ χes 3 2 2 + β γ (4|α | ) χs + 2 (sin Q − Q cos Q) βs − β 2 λ χes (sin Q − Q cos Q) , + β γ (4|α |2 )2 2 (βs − β 2 )2 HGAS = 4γ |α |2 {sin Q + γ −2 (sin Q − Q cos Q) −[χs (1 + β 2 − 4γ 2 β 2 ) + χx γ −2 ](sin Q − Q cos Q) λ χes +4|α |2 2 (sin Q − Q cos Q) . (βs − β 2 )2

(3.34a) (3.34b)

(3.34c)

(3.34d)

If a GAS is the only field present in the system (i.e., dispersive radiation can be neglected), then the decay of a GAS’s quasi-conserved quantities can be calculated by inserting the GAS formulas (3.31)-(3.32) into Eqs. (3.25) [37], d Γ χes PGAS = − γβ dτ λ

Z ∞ −∞

= −2Γ λ χes β γ d Γ χes HGAS = − γ dτ λ

(w0,ζ )2 d ζ

Z ∞ −∞

= −2Γ λ χes γ 3

¶ ¶2 µ 4γ | α | 2 1 3 sin Q − sin Q − Q cos Q ,(3.35a) βs2 − β 2 3

(w0,ζ )2 d ζ

¶2 µ ¶ 4γ | α | 2 1 3 sin Q − sin Q − Q cos Q , (3.35b) βs2 − β 2 3

It is in fact impossible for both NGAS and MGAS to remain constant while PGAS and HGAS decay according to Eqs. (3.35). This prooves that phonon viscosity causes a moving GAS to emit dispersive radiation. That phonon viscosity retards the soliton and that it causes emission of phonons from the slowing soliton is confirmed by direct numerical simulation, as illustrated by Fig. 3.3. Slowing of a GAS might also be achieved in a fiber loop with a mechanism for damping out emitted sound waves.

3 Gap-Acoustic Solitons: Slowing and Stopping of Light

Fig. 3.3 Gap-acoustic soliton decelerating due to phonon viscosity. The top shows evolution of the light intensity, and the bottom shows the material density. The soliton begins with dimensionless velocity β = 0.19, compared to sound velocity βs = 0.2. The soliton parameter is Q = π /2, frequency in the middle of the band gap. The electrostrictive coefficients are χes = 1, λ = 0.001, the phonon viscosity is Γ = 0.04, and the self- and cross-phase modulation coefficients are χs = 1, χx = 2.

Brillouin scattering—interaction of the light fields u, v with the high wave number acoustic fields wu , wv —can be calculated explicitly in the approximation that the Brillouin fields are small perturbations to the GAS. This is the same as neglecting the effect of the Brillouin fields on the GAS. We can show that a moving GAS will emit acoustic waves preferentially backwards, carrying off some of the GAS’s momentum, thus retarding it. The effect is relatively small, except when the soliton velocities are close to the group velocity of light in the medium. This retardation can add to the retardation effect on the GAS by phonon viscosity.

3.5 Soliton Stability and Instability To analyze stability of gap-acoustic solitons, we carried out full numerical simulations of the partial differential equations (3.30) using a split-step fast Fourier transform scheme, which treats the linear part of the equations in momentum space, and the nonlinear part in real space [1]. The simulations were carried out systematically for three values of the soliton coefficient Q: Q = π /3 (in the middle of the top half of the band gap), which, for gap solitons without electrostriction (λ = 0) [11, 12, 13], is well inside the stable region; Q = π /2 (in the middle of the band gap), which is stable but close to the instability border; and Q = 2π /3 (in the middle of the bottom

Richard S. Tasgal, Roman Shnaiderman, and Yehuda B. Band

half of the band gap), which has an oscillatory instability [15, 16, 17]. We calculated using ten different values of the electrostrictive coefficient, ranging over four orders of magnitude, λ = 0.0001 to 1, and the limit λ → ∞. The speed of sound was held at βs = 0.2. This is much faster than physically realistic (unless one were to consider light that is slow in the sense of the group velocity being much less than the phase velocity [23]). Choosing here to take the speed of sound relatively fast and studying a range of electrostrictive coefficients allows us to illustrate qualitative properties of the system that would be not easily demonstrated if simulations were limited to physically realizable cases. (A detailed analysis of the dynamics in a waveguide made of bulk fused silica can be found in Ref. [19]. In this case, the large disparity—five orders of magnitude—between the group velocity of light and the speed of sound makes a thorough study of the system computationally extremely expensive, rendering some effects practically indetectable and other effects huge.) The initial gap soliton velocity was taken at ten distinct values, from zero to twice the speed of sound β = 2βs = 0.4, with special emphasis close to the speed of sound. For consistency in the stability analyses, all the direct numerical simulations had initial light amplitudes 1% greater than those of the exact soliton solutions. In addition to this systematic coverage of part of the parameter space, we ran many simulations at scattered values of all the free parameters. Like gap solitons without electrostriction [15, 16, 17], gap-acoustic solitons are subject to oscillatory instabilities, which can grow until the soliton is destroyed, as illustrated in Fig. 3.4. In this case, when the oscillations grow too large, they destroy the soliton, which then goes into dispersive radiation moving to the left and to the right. Electrostriction, however, decreases the rate of growth of the oscillatory instability. The larger is the electrostrictive coefficient, the slower is the instability, as is visible in Fig. 3.5, which, over a series of runs with a range of electrostrictive coefficients, shows the growth rate of the instability to be smaller for larger electroctrictive coefficients λ . Additionally, the closer the velocity is to the speed of sound, the greater is the damping of the oscillatory instability. Figure 3.6 shows evolution of the peak light intensity for four different soliton velocities (all subsonic, so as not to introduce the supersonic instability, which is detailed below). The growth rate of gap-acoustic solitons’ instability is smaller the closer is the soliton velocity to the speed of sound. Common to both trends is that a larger phonon field has a stronger damping effect on the oscillatory instability. In contrast, without electrostriction, the dependence of the instability on soliton velocity is quite weak, and with no special importance to the speed of sound. Next, we consider solitons which are known to be stable in the absence of electrostriction. Among our simulations, that is the runs with soliton parameters Q = π /3 and Q = π /2. All solitons that are stable with zero electrostriction were found to be stable with electrostriction and velocities up to the speed of sound. Above the speed of sound, a new and distinct (“supersonic”) instability appears. It is associated with the downward slope of the soliton momentum with respect to velocity, which is due to the decreasing importance in the supersonic region of the acoustic contribution to the momentum as a function of soliton velocities. The su-

3 Gap-Acoustic Solitons: Slowing and Stopping of Light

Fig. 3.4 Destruction of a gap-acoustic soliton by the oscillatory instability. The graph shows light intensity (|u|2 + |v|2 ) and phonon fields (w). The soliton is described by parameter Q = 2 π /3, the velocity is zero (β = 0), the electrostrictive coefficients are χes = 1, λ = 0.0005, the speed of sound is βs = 0.2, and self- and cross-phase modulation are χs = 1, χx = 2.

peak

6 4 2

λ=0.0001

peak

6 4 2

λ=0.001

peak

3.1 3.05

λ=0.01

peak

0.95

λ=0.1

0.945 0.94 0

100

Fig. 3.5 Evolution of the peak powers {maxζ (|u(ζ , τ )|2 + |v(ζ , τ )|2 )} with time, for gap-acoustic solitons with four different electrostrictive coefficients, λ = 0.0001, 0.001, 0.01, and 0.1, and χes = 1. The solitons are unstable, Q = 2 π /3, β = 0, with self- and cross-phase modulation χs = 1, χx = 2, and speed of sound βs = 0.2.

personic instability goes away when the soliton velocity is high enough for the momentum in the electromagnetic part of the GAS to outweigh the momentum in the acoustic part of the GAS [19]. This supersonic instability is qualitatively different than the oscillatory instability, and is unknown for gap solitons without electrostriction. (A non-oscillatory and strongly velocity-dependent instability was found in Ref. [15]. This instability exists in a region that is already unstable because of two

Richard S. Tasgal, Roman Shnaiderman, and Yehuda B. Band

peak

6 4 2

ρ=0

peak

6 4 2

ρ=0.1

peak

6 4 2

ρ=0.15

peak

6 4 2

ρ=0.19

0 0

100

Fig. 3.6 Evolution of the peak powers [maxζ (|u(ζ , τ )|2 + |v(ζ , τ )|2 )] with time, for unstable gapacoustic solitons at by four different subsonic velocities, β = 0, 0.1, 0.15, and 0.19, where the dimensionless speed of sound is β = 0.2. The solitons have electrostrictive coefficients, χes = 1, λ = 0.0005, soliton parameter Q = 2 π /3, and self- and cross-phase modulation χs = 1, χx = 2.

oscillatory instabilities, so may not be clearly realizable experimentally. It is not related to acoustic waves and the speed of sound plays no role in these dynamics.) Figures 3.7-3.8 show supersonic gap-acoustic soliton simulations, with the same parameters except for the initial soliton velocity. In almost all cases—displayed and not—the gap-acoustic solitons retained their integrity throughout the instability. The closer was the supersonic soliton’s velocity to the speed of sound, the sooner the supersonic instability took effect. The changes in velocity were abrupt, and were accompanied by emission of phonons. The solitons sometimes changed speed and direction a few times before eventually settling to a stable subsonic GAS. Figures 3.9-3.11 have the same parameters as Figs. 3.7-3.8, but with an order of magnitude larger electrostrictive coefficient. When the electrostrictive coefficient was larger (and the phonon field larger), the supersonic instabilities tended to be stronger in the sense that they happened sooner, and in that the soliton was more often destroyed. In some instances, as in Fig. 3.7, a soliton that was close to the speed of sound made a smooth transition to subsonic (necessarily passing through a non-solitonic configuration). The GASs tend to end with velocities much slower than the speed of sound may because the momentum of the soliton is larger at slightly subsonic velocities than at many supersonic velocities. Figure 3.12 shows the momentum and energy in a soliton at fixed Q-parameter over a range of velocities. After onset of the supersonic instability, there is generally only enough momentum (and energy) for the resulting subsonic GAS to have a velocity up to a small percentage of the speed of sound.

3 Gap-Acoustic Solitons: Slowing and Stopping of Light

Fig. 3.7 Gap-acoustic soliton subject to the supersonic instability. The graph shows light intensity (|u|2 + |v|2 ) and phonon fields (W ). The soliton parameter is (Q = π /3), with initially supersonic velocity (β = 0.3, compared to a speed of sound βs = 0.2). The electrostrictive coefficients are χes = 1, λ = 0.001, and self- and cross-phase modulation are χs = 1, χx = 2.

Fig. 3.8 Gap-acoustic soliton subject to the supersonic instability. The graph shows light intensity (|u|2 + |v|2 ) and phonon fields (W ). The soliton parameter is (Q = π /3), with initially hypersonic velocity (β = 0.4, compared to a speed of sound βs = 0.2). The electrostrictive coefficients are χes = 1, λ = 0.001, and self- and cross-phase modulation are χs = 1, χx = 2.

The oscillatory instability and the supersonic instability can compete. For example, Fig. 3.13 shows a gap-acoustic soliton which first slows down sharply, going

Richard S. Tasgal, Roman Shnaiderman, and Yehuda B. Band

Fig. 3.9 Gap-acoustic soliton subject to the supersonic instability. The graph shows light intensity (|u|2 + |v|2 ) and phonon fields (W ). The soliton parameter is (Q = π /3), with initially supersonic velocity (β = 0.25, compared to a speed of sound βs = 0.2). The electrostrictive coefficients are χes = 1, λ = 0.01, and self- and cross-phase modulation are χs = 1, χx = 2.

Fig. 3.10 Gap-acoustic soliton subject to the supersonic instability. The graph shows light intensity (|u|2 + |v|2 ) and phonon fields (W ). The soliton parameter is (Q = π /3), with initially supersonic velocity (β = 0.3, compared to a speed of sound βs = 0.2). The electrostrictive coefficients are χes = 1, λ = 0.01, and self- and cross-phase modulation are χs = 1, χx = 2.

from supersonic to zero velocity, and after which the oscillatory instability grows until it destroys the soliton.

3 Gap-Acoustic Solitons: Slowing and Stopping of Light

Fig. 3.11 Gap-acoustic soliton subject to the supersonic instability. The graph shows light intensity (|u|2 + |v|2 ) and phonon fields (W ). The soliton parameter is (Q = π /3), with initially hypersonic velocity (β = 0.4, compared to a speed of sound βs = 0.2). The electrostrictive coefficients are χes = 1, λ = 0.01, and self- and cross-phase modulation are χs = 1, χx = 2.

−2

−1

Fig. 3.12 Plots of the GAS momentum and energy over a range of soliton velocities, holding other parameters constant—Q = π /2, βs = 0.2, χes = 1, λ = 0.01, χs = 1 and χx = 2.

Larger electrostrictive coefficients damp the oscillatory instability and increase the supersonic instability, so changing the electrostriction can change the dominant type of instability. The dimensionality of the parameter space (χs /χx , λ , Γ , βs , β , Q) is too large, the behavior too varied, the sensitivity to initial conditions to strong,

Richard S. Tasgal, Roman Shnaiderman, and Yehuda B. Band

Fig. 3.13 Gap-acoustic soliton experiencing two instabilities, supersonic and oscillatory. The top and bottom parts of the figure show the light intensity and the material density, respectively. The soliton begins with dimensionless velocity β = 0.25, compared to sound velocity βs = 0.2. The soliton parameter is Q = 2 π /3, the electrostrictive coefficients are χes = 1, λ = 0.0002, and selfand cross-phase modulation coefficients are χs = 1, χx = 2.

and the computational cost of numerical simulations too high to obtain a complete simple picture of the ultimate results following a GAS instability. There are no breathers or localized excited states as a small perturbation about (i.e., on top of) the gap-acoustic solitons. Any oscillation of a localized mode generates waves in the acoustic field which move at velocity plus or minus the speed of sound. The acoustic field will carry away energy, dissipating the oscillations. Small oscillations have energy proportional to the amplitude of the oscillation, and the energy radiated away is also proportional to the square of the amplitude of the oscillation. Therefore, small oscillations about a stable GAS decay exponentially.

3.6 Summary and Conclusions In this work, we formulated a set of equations to describe propagation of light in a nonlinear waveguide with a Bragg grating, with the light coupled to sound waves by electrostriction. Light waves’ dispersion curve has a band gap in the vicinity of the resonance of the Bragg grating. Forward- and backward-moving light in the vicinity of the band gap can interact with acoustic waves of low wave numbers (in which case the interaction is generally referred to as electrostriction) or high wave numbers, twice the wave numbers of the light (in which case the interaction is called Brillouin scattering).

3 Gap-Acoustic Solitons: Slowing and Stopping of Light

There is a localized structure in this system—a “gap-acoustic soliton”, for the case when Brillouin scattering may be neglected; there is an exact analytic form of the solitons for the case of vanishing phonon viscosity, as well as for zero velocity solitons. Gap-acoustic solitons have frequencies in the band gap, as do standard gap solitons (without electrostriction). They exist at all velocities up to the speed of light in the medium, except for exactly the speed of sound (near which the phonon component of the soliton is large, and at which there is a singularity). Coupling of the light to the acoustic field via electrostriction changes the stability properties of the soliton. Solitons which would experience an oscillatory instability without electrostriction experience a damping of the instability, the larger the electrostrictive coefficient and/or the closer the velocity is to the speed of sound, the larger is the damping of the oscillatory instability. Electrostriction introduces a new “supersonic” instability for gap-acoustic solitons moving faster than the speed of sound. The closer the soliton is to the speed of sound, the faster the supersonic instability takes effect. The supersonic instability may cause an abrupt change in the velocity, or sometimes destruction of the soliton. The soliton may experience several changes in direction or some complex dynamics before going to a stable subsonic soliton or being destroyed. Phonon viscosity slows the gap-acoustic soliton and causes emission of significant phonon radiation. If the soliton is subsonic, the soliton velocity will decrease exponentially, and if the soliton is supersonic, phonon viscosity will slow the soliton to the speed of sound in finite time. Solitons cannot exist with speeds between the critical velocity and the speed of sound. But if the phonon viscosity makes the soliton pass through the velocity gap quickly, a soliton will emerge as a similar but subsonic gap-acoustic soliton. At velocities close to the group-velocity of light in the waveguide, Brillouin scattering can make the soliton emit acoustic radiation, which carries away momentum, and acts as an additional retardation mechanism for the moving soliton. Since electrostriction, like the Kerr effect, is present to some extent in virtually all materials, any understanding of physically realistic optical gap solitons should entail a grasp of the effects of electrostriction. The results herein suggest that an initially fast-moving gap-acoustic soliton can be retarded by the effects of the nonzero phonon viscosity. A practical means of doing this would be in a recirculating loop. Either alternatively or complementarily, once a gap-acoustic soliton is slowed to on the order of the speed of sound, the gap-acoustic solitons supersonic instability will do the work of slowing the soliton to soliton to significantly below the speed of sound. In addition, the dynamics of gap-acoustic solitons open the door to new means of controlling light by sound.

Richard S. Tasgal, Roman Shnaiderman, and Yehuda B. Band

References 1. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 2006). 2. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, 1981). 3. J. S. Russell, Report of the Eighteen Meeting of the British Association for the Advancement of Science (John Murray, London, 1849) 18, 37. 4. D. J. Korteweg and G. de Vries, Philos. Mag. Ser. 5, 422 (1895). 5. C. S. Gardner, J. M. Greene, M.D. Kruskal and R. M. Miura, Phys. Rev. Lett. 19 1095, (1967). 6. V. E. Zakharov and A. B. Shabat Sov. Phys. JETP 34, 62 (1972). 7. A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 171 (1973). 8. W. E. Thirring, Annals Phys. 3, 91 (1958). 9. E. A. Kuznetsov and A. V. Mikhailov, Teor. Mat. Fiz. 30, 193 (1977); D. J. Kaup and A. C. Newell, Lett. Nuovo Cimento 20, 325 (1977). 10. B. J. Eggleton, C. Martijn de Sterke, and R. E. Slusher, J. Opt. Soc. Am. B 16, 587 (1999). 11. W. Chen and D. L. Mills, Phys. Rev. Lett. 58, 160 (1987). 12. D. N. Christodoulides and R. I. Joseph, Phys. Rev. Lett. 62, 1746 (1989). 13. A. B. Aceves and S. Wabnitz, Phys. Lett. A 141, 37 (1989). 14. B. A. Malomed and R. S. Tasgal, Phys. Rev. E 49, 5787 (1994). 15. I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, Phys. Rev. Lett. 80, 5117 (1998); 16. I. V. Barashenkov and E. V. Zemlyanaya, Comp. Phys. Comm. 126, 22 (2000); 17. A. De Rossi, C. Conti, and S. Trillo, Phys. Rev. Lett. 81, 85 (1998). 18. R. S. Tasgal, Y. B. Band, and B. A. Malomed, Phys. Rev. Lett. 98, 243902 (2007). 19. R. S. Tasgal, R. Shnaiderman, and Y. B. Band, J. Opt. Soc. Am. B, in press. 20. J. D. Jackson, Classical Electrodynamics (Wiley, NY, 1975) 21. M. Born and E. Wolf, Principles of Optics (Pergamon, NY, 1980). 22. R. W. Boyd, Nonlinear Optics (Academic, NY, 2003). 23. M. D. Lukin and A. Imamoglu, Nature 413, 273 (2001); R. W. Boyd and D. J. Gauthier, Progress in Optics 43, 497 (2002). 24. A. B. Aceves, Chaos 10, 584 (2002). 25. C. M. de Sterke, and J. E. Sipe, Progr. Opt. 33, 203 (1994). 26. “Focus Issue: Bragg Solitons and Nonlinear Optics of Periodic Structures,” Opt. Exp. 3, 384 (1998). 27. W. C. K. Mak, B. A. Malomed, and P. L. Chu, J. Mod. Opt. 51, 2141 (2004). 28. W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 68, 026609 (2003). 29. J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, Nature Phys. 2, 775 (2006). 30. E. Rat, M. Foret, G. Massiera, R. Vialla, M. Arai, R. Vacher, and E. Courtens, Phys. Rev. B 72, 214204 (2005); R. Vacher, E. Courtens, and M. Foret, Phys. Rev. B 72, 214205 (2005). A. Devos, M. Foret, S. Ayrinhac, P. Emery, and B. Ruffl, Phys. Rev. B 77, 100201 (2008). 31. I. L. Fabelinskii, Molecular Scattering of Light (Plenum, NY, 1968). 32. P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, Phys. Rev. B 19, 4986 (1979); D. T. Hon, Opt. Lett. 5, 516 (1980); R. M. Shelby, M. D. Levenson, and P. W. Bayer, Phys. Rev. B 31, 5244 (1985); K. Smith and L. F. Mollenauer, Opt. Lett. 14, 1284 (1989); E. L. Buckland and R. W. Boyd, Opt. Lett. 21, 1117 (1996); E. L. Buckland and R. W. Boyd, Opt. Lett. 22, 676 (1997); A. Fellegara, A. Melloni, and M. Martinelli, Opt. Lett. 22, 1615 (1997); P. J. Hardman, P. D. Townsend, A. J. Poustie, and K. J. Blow, Opt. Lett. 21, 393 (1996); E. M. Dianov, A. V. Luchnikov, A. N. Pilipetskii, and A. N. Starodumov, Opt. Lett. 15, 314 (1990). 33. J. Feng and F. K. Kneubuhl, IEEE J. Quantum Electron 29, 590 (1992). 34. V. E. Zakharov, Zh. Eksp. Teor. Fiz. 62, 1745 (1972) [Sov. Phys. JETP 35, 908 (1972)]. 35. A. S. Davidov, Phys. Scr. 20, 387 (1979); L. Stenflo, Phys. Scr. 33, 156 (1986). 36. H. Hadouaj, B. A. Malomed, and G. A. Maugin, Phys. Rev. A 44, 3925–3931 (1991); H. Hadouaj, B. A. Malomed, and G. A. Maugin, Phys. Rev. A 44 3932–3940 (1991); G. A. Maugin, H. Hadouaj, and B. A. Malomed, Phys. Rev. B 45, 9688–9694 (1992). 37. V. I. Karpman, Phys. Scripta 20, 462 (1979); V. I. Karpman and V. V. Solovev, Physica D 3, 487 (1981).

Chapter 4

Optical Wave Turbulence and Wave Condensation in a Nonlinear Optical Experiment Jason Laurie, Umberto Bortolozzo, Sergey Nazarenko and Stefania Residori

Abstract We present theory, numerical simulations and experimental observations of a 1D optical wave system. We show that this system is of a dual cascade type, namely, the energy cascading directly to small scales, and the photons or wave action cascading to large scales. In the optical context the inverse cascade is particularly interesting because it means the condensation of photons. We show that the cascades are induced by a six-wave resonant interaction process described by weak turbulence theory. We show that by starting with weakly nonlinear randomized waves as an initial condition, there exists an inverse cascade of photons towards the lowest wavenumbers. During the cascade nonlinearity becomes strong at low wavenumbers and, due to the focusing nature of the nonlinearity, it leads to modulational instability resulting in the formation of solitons. Further interaction of the solitons among themselves and with incoherent waves leads to the final condensate state dominated by a single strong soliton. In addition, we show the existence of the direct energy cascade numerically and that it agrees with the wave turbulence prediction.

Jason Laurie Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom e-mail: [emailprotected] Umberto Bortolozzo INLN, Universit´e de Nice Sophia-Antipolis, CNRS, 1361 route des Lucioles 06560 Valbonne, France, e-mail: [emailprotected] Sergey Nazarenko Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom e-mail: [emailprotected] Stefania Residori INLN, Universit´e de Nice Sophia-Antipolis, CNRS, 1361 route des Lucioles 06560 Valbonne, France, e-mail: [emailprotected]

O. Descalzi et al. (eds.), Localized States in Physics: Solitons and Patterns, DOI 10.1007/978-3-642-16549-8_4, © Springer-Verlag Berlin Heidelberg 2011

J. Laurie, U. Bortolozzo, S. Nazarenko and S. Residori

4.1 Introduction The idea to create a state of optical wave turbulence (OWT) has been the subject of a large number of theoretical papers over the last thirty years [1, 2, 3, 4, 5]. Indeed there are some far-reaching fluid analogies in the dynamics of nonlinear light, for example vortex-like solutions [15, 7] and shock waves [8]. In the case of weakly interacting random waves, the dynamics and statistics of the optical field are predicted to share strong similarities with the system of random waves on water surface [9]. Indeed, OWT was theoretically predicted to exhibit dual cascade properties similar to 2D fluid turbulence, namely the energy cascading directly, from low to high frequencies, and the photons cascading inversely, toward the low energy states. The mechanism for optical interactions is provided by the Kerr effect, which is routinely used in nonlinear optics and permits photon wave-mixing. When the nonlinearity is small, OWT can be described by wave turbulence theory (WTT) [9] which possesses classical attributes of general turbulence theory, particularly predictions of the Kolmogorov-like cascade states, which in the WTT context are called Kolmogorov-Zakharov (KZ) spectra. It appears that OWT has two KZ states: one describing the direct energy cascade from large to small scales, and the second one - an inverse cascade of wave action toward larger scales. It is the inverse cascade that can provide the mechanism for condensation of light, i.e. formation of a coherent phase out of initially incoherent wave field. Furthermore, it was theoretically predicted that in the course of the inverse cascade the nonlinearity will grow, which will eventually lead to the breakdown of WTT at a some low wavenumber k and the formation of coherent structures, i.e. solitons/collapses for the focusing nonlinearity or a quasi-uniform condensate and vortices [5] for the de-focusing case. Experimentally, the problem of realizing the OWT state is that the nonlinearity is usually very weak and it is a challenge to make it overpower the dissipation. Here we show that the OWT regime can be implemented in a liquid crystal system [10], where optical solitons have been previously reported [11]. Our experiment is based on the propagation of an enlarged light beam inside a liquid crystal layer acting as the nonlinear medium. The principal direction of the light propagation plays the role of time and the 2D field evolution is described by a nonlinear Schr¨odinger equation (NLSE). Starting with weakly nonlinear waves with randomized phases, we observe the formation of an inverse cascade of photons towards the lowest wavenumbers. We show that the cascade is induced by a six-wave resonant interaction process, and it is characterized by increasing nonlinearity along the cascade. At low wavenumbers the nonlinearity becomes strong and, due to the focusing nature of the nonlinearity, it leads to modulational instability developing into solitons. Further interaction of the solitons among themselves and with incoherent waves leads to the final condensate state dominated by a single strong soliton. Furthermore, it was theoretically predicted, and numerically observed in some WT systems, that in the course of the inverse cascade the nonlinearity will grow, which will eventually lead to the breakdown of the WTT description at low wavenumbers and to the formation of coherent structures [1, 3, 5, 12, 13, 2]. In optics, these can be solitons or collapses for focusing nonlinearity or a quasi-uniform condensate

4 Optical Wave Turbulence and Wave Condensation in a Nonlinear Optical Experiment

and vortices for the defocusing case. The final thermalized state was studied extensively theoretically in various settings for non-integrable Hamiltonian systems starting with the pioneering paper by Zakharov et al [14], see also [15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. The final state with a single soliton and small-scale noise was interpreted as a statistical attractor, and an analogy was pointed out to the over-saturated vapor system, where the solitons are similar to droplets and the random waves are like molecules [24]. There, small droplets evaporate while the big ones gain size from the free molecules, resulting in the decrease in the number of droplets. In our work we put emphasis on turbulence, i.e. on a transient non-equilibrium process leading to thermalization rather than the thermal equilibrium itself.

4.2 Experimental setup The experimental apparatus is shown in Fig.4.1a. It consists of a liquid crystal cell, inside which a laminar shaped beam propagates, and with the input beam prepared in such a way, as to have an initial condition of weak and random waves. The liquid crystal (LC) cell is made by sandwiching a nematic layer (E48) of thickness d = 50 µ m between two 20 × 30 mm2 glass windows and is schematically depicted in Fig.4.1b. On the interior, the glass walls are coated with Indium-Tin-Oxide (ITO) transparent electrodes. We have pre-treated the ITO surfaces with polyvinyl-alcohol, polymerized and then rubbed, in order to align all the molecules parallel to the confining walls. When a voltage is applied across the cell, liquid crystal molecules tend to reorient in such a way as to become parallel to the direction of the electric field. By applying a 1kHz electric field with rms voltage V0 = 2.5 V we preset the molecular director to an average tilt angle Θ .

Fig. 4.1 Schematic representation of a) the experimental setup, b) the liquid crystal cell. A laminar shaped input beam propagates inside the liquid crystal (LC) layer; random space modulations are imposed at the entrance of the cell by means of a spatial light modulator (SLM). A voltage V is applied to the LC in order to favor the molecular reorientation towards the optical field E.

J. Laurie, U. Bortolozzo, S. Nazarenko and S. Residori

The LC layer behaves as a positive uniaxial medium, with nk = nz = 1.7 the extraordinary and n⊥ = 1.5, the ordinary refractive indices [25]. LC molecules tend to turn more along the applied field and the refractive index n(Θ ) follows the distribution of the tilt angle θ . When a linearly polarized beam is injected into the cell, the LC molecules reorient following the direction of the incoming beam polarization. The input light comes from a diode pumped solid state laser, λ = 473 nm, polarized along y and shaped as a thin laminar Gaussian beam of 30 µ m thickness. The beam evolution inside the cell is monitored with an optical microscope and a CCD camera. The light intensity is kept very low, Iin = 30 µ W /cm2 to ensure the weakly nonlinear regime. A SLM at the entrance plane of the cell is used to produce suitable intensity masks for injecting random phased fields with large wavenumbers.

4.3 Theoretical Background Theoretically, the evolution is described by a propagation equation for the input beam coupled to a relaxation equation for the LC dynamics

∂ ψ ∂ 2ψ + 2 + k02 n2a aψ = 0, ∂z ∂x 2 ∂ a 1 ε0 n2a 2 − 2 a+ |ψ | = 0, 2 ∂x 4K lξ

2iq

(4.1) (4.2)

where ψ (x, z) is the complex amplitude of the input beam propagating along “time axis” zˆ, x the coordinate across the beam, a the liquid crystal reorientation angle, na = ne − no the birefringence p of the LC, k0 the optical wavenumber, ε0 the vacuum permittivity and lξ = π K/2∆ ε (d/V0 ) the electrical coherence length of the ¡ ¢ LC [42], with K the elastic constant, q2 = k02 n2o + n2a /2 and ∆ ε the dielectric anisotropy. Note that lξ fixes the typical dissipation scale, limiting the extent of the inertial range in which the OWT cascade develops. In other contexts, see e.g. [11, 27, 28], such a spatial diffusion of the molecular deformation has been denoted as a nonlocal effect. In our experiment, for V0 = 2.5 V we have lξ = 9 µ m. By considering that a typical value of K is of the order of ∼ 10 pN, we can derive a typical dissipation length scale of the order of ∼ 10 µ m.

4.3.1 Evolution Equation Our system is modeled by two coupled equations, one describing the evolution of the complex amplitude of the input beam ψ (x, z), equation (4.1) and the second for the liquid crystal reorientation angle a(x, z), equation (4.2). However, it is convenient for us to construct a single evolutionary equation for the complex wave amplitude

4 Optical Wave Turbulence and Wave Condensation in a Nonlinear Optical Experiment

ψ (x, z). The system of equations (4.1) and (4.2) can be formally re-written, so that one eliminates the variable a(x, z). In order to achieve this, one must invert the operator applied to a(x, z) in equation (4.2). Subsequently, this procedure yields a single equation for the complex amplitude of the beam ψ (x, z), ∂ ψ ∂ 2 ψ k02 n4a ε0 2iq + 2 + ψ ∂z ∂x 4K

1 ∂2 − 2 2 lξ ∂ x

!−1 |ψ |2 = 0.

(4.3)

Equation (4.3) models the full dynamics of the complex wave function ψ (x, z), to the same extent as the system described by equations (4.1) and (4.2). On inspection of equation (4.3) we find that the operator applied to the nonlinear term is rather complicated and would not yield a scale invariant nonlinear interaction coefficient convenient for the application of WTT. The experimental setup is located in the long-wave limit, klξ ¿ 1 due to the nature of the LC. Therefore, we can overcome the problem of the nonlinear operator by expanding in terms of the small parameter klξ .

4.3.2 Long-Wave Model We derive this long-wave model by Taylor expanding the nonlinear operator in equation (4.3) in the limit klξ ¿ 1. Taking the expansion up to the order of O((klξ )4 ), one can derive an evolutionary equation for the wave function ψ (x, z) that is of the form of a modified 1D NLSE. Note, that we took the expansion of the nonlinear operator up to the second order, this is because if we take the leading order, we would have identically the 1D NLSE, which is a completely integrable system. So the resultant non-integrable model for the evolution of ψ (x, z) in the long-wave limit is, ¶ 4 2 2µ 2 2 ∂ψ ∂ 2 ψ ε 0 na l ξ k 0 2 2 ∂ |ψ | 2iq =− 2 − ψ |ψ | + lξ ψ . ∂z ∂x 4K ∂ x2

(4.4)

Equation (4.4) hold the property that it conserves the energy,

" ¯2 µ ¶ # 2 2 ε0 n4a lξ2 k02 ¯ ∂ ψ ∂ | ψ | ¯ ¯ |ψ |4 − lξ2 dx, ¯ ∂ x ¯ − 8K ∂x

Z ¯¯

(4.5)

where H is defined as the Hamiltonian of the system and satisfies 2iq∂ ψ /∂ z = δ H/δ ψ ∗ . Also equation (4.4) conserves the total number of photons, or wave action, Z

|ψ |2 dx.

(4.6)

J. Laurie, U. Bortolozzo, S. Nazarenko and S. Residori

4.3.3 The Fjørtoft Argument As already mentioned, our system possesses two conserved quantities (energy H, and total wave action N), and as such will support a dual cascade regime [1]. This is analogous to 2D turbulence, where entrophy cascades towards high wavenumbers and energy towards low wavenumbers. It is worth to point out that, as opposed to 2D turbulence, WTT assumes weak nonlinearity of the system, and as such implies smallness of wave amplitudes. This weak nonlinearity assumption also implies that the linear energy is dominant over the nonlinear energy. In the case of weak nonlinearity, one can determine the direction of each cascade, by applying a Fjørtoft style argument [29]. Suppose that the forcing and damping occur over certain intervals of wavenumber space, damping near k = 0 to absorb the inverse cascade of particles, excitation in a small interval at intermediate wavenumber around k = k0 , and finally high frequency dissipation for k > kD . Then, transfer of wave action an energy takes places and fluxes can be defined as Z k ∂ nk 0

dk0 ∂t Z k ∂n 0 Pk = ωk0 k dk0 , ∂t 0 Qk =

(4.7) (4.8)

representing, respectively, the flux of particle towards low wavenumbers and the flux of energy towards high wavenumbers. The Fjørtoft reasoning goes as follows: Assume that the system has reached a steady state, therefore the total amount of energy flux Pk , andRwave action flux Qk , contained within the system must be zero, i.e. R Pk dk = 0, and Qk dk = 0 respectively. Then, let the system be forced at a specific intermediate scale k f , with both energy and wave action fluxes being generated into the system at rates Pf and Q f . Moreover, let there exist two sinks, one at the high wavenumber limit k+ À k f , with energy and wave action being dissipated at rates P+ and Q+ , and one at the low wavenumber limit k− ¿ k f , dissipated at rates P− and Q− . Therefore, in between forcing and dissipation there exist two distinct inertial ranges where neither forcing or dissipation has an effect. In the weakly nonlinear regime, the energy flux is related to the wave action flux by Pk ≈ ωk Qk = k2 Qk . Therefore, in a steady state system, the energy and wave action balance implies that Pf = P− + P+ ,

(4.9)

Q f = Q− + Q+ ,

(4.10)

and roughly speaking, from weak nonlinearity we have

4 Optical Wave Turbulence and Wave Condensation in a Nonlinear Optical Experiment

Pf ≈ k2f Q f ,

(4.11)

2 P− ≈ k− Q− ,

(4.12)

P+ ≈

2 k+ Q. ,

(4.13)

Subsequently, the balance equations (4.9) and (4.10) can be written as 2 2 k2f Q f ≈ k− Q− + k+ Q+ ,

(4.14)

Q f = Q− + Q+ ,

(4.15)

respectively. One can then rearrange equations (4.14) and (4.15), so that we can predict at what rates the wave action and energy fluxes are dissipated at the two sinks at dissipative scales k− and k+ . From equations (4.14) and (4.15) we find that,

Q+ = Q− =

2 k2f − k− 2 − k2 k+ − 2 k2f − k+ 2 − k2 k− +

Qf ,

(4.16)

Qf .

(4.17)

If we consider the region around large scales, k− ¿ k f < k+ , then we have from 2 Q i.e. that energy is mostly absorbed at the region equation (4.16): k2f Q f ≈ k+ + around k+ . Moreover, considering the region around small scales k− < k f ¿ k+ , we have from (4.17) that Q f ≈ Q− i.e. that wave action is mostly absorbed at regions around k− , this is illustrated in Fig. 4.2. Therefore, if we force at an intermediate scale, we should have the majority of the energy flowing towards high wavenumbers and the majority of the wave action flowing towards small wavenumbers. This determines the dual cascade picture of 1D OWT.

P− Q− Wave Action Cascade

Qf Pf Dissipation

Forcing

k−

Fig. 4.2 The dual cascade regime in 1D OWT.

Energy Cascade Q+

P+ Dissipation k+

J. Laurie, U. Bortolozzo, S. Nazarenko and S. Residori

4.3.4 Hamiltonian Formulation The Hamiltonian formulation provides a very convenient way of describing systems with waves. It enables us separate the linear and nonlinear wave dynamics and to mathematically describe their wave interactions. The most desirable way of describing the wave dynamics is to represent the Hamiltonian system in terms of the wave action variable a(k, z). The wave action variable a(k, z) is the Fourier coefficient in the Fourier representation of the complex beam amplitude variable ψ (x, z), defined by its Fourier transform. Hamiltonian (4.5) can be written in terms of the wave action variable and can be expressed in the form, Z

H = H2 + H4 =

ωk ak a∗k dk +

12 W1234 δ34 a1 a2 a∗3 a∗4 dk1234 ,

(4.18)

12 = δ (k +k −k −k ), dk we use the notation that a(k1 , z) = a1 , δ34 1 2 3 4 1234 = dk1 dk2 dk3 dk4 ∗ and to denote the complex conjugate. The linear wave frequency is ωk = k2 , and the four-wave interaction coefficient W1234 for the nonlinear interaction of waves is

W1234 =

ε0 n4a lξ4 k02 16K

(k1 k4 + k2 k3 + k2 k4 + k1 k3 − 2k3 k4 − 2k1 k2 ) −

ε0 n4a lξ2 k02 8K

. (4.19)

Hamiltonian (4.18) can be easily split into terms of differing orders of interaction (or nonlinearity) with respect to the wave action variable ak . Hamiltonian (4.18), can be split into a quadratic term, H2 - this corresponds to the linear wave dynamics of the propagation of a wave with linear frequency ωk . But in addition, we have a quadric term H4 , which corresponds to the nonlinear wave dynamics. In the limit ε0 n4a lξ2 k02 /2K → 0, equation (4.4) becomes the linear Schr¨odinger equation which has linear wave solutions ψ (x, z) ∼ bk exp(−iωk z + ikx) with “frequencies” ω = k2 and constant complex amplitudes bk . For weak nonlinearity the amplitude bk become weakly dependent on “time” z. Applicability of WTT needs to be checked, this is achieved by verify that the linear dynamics do indeed dominate in the system. The ratio of the linear term and the leading nonlinear term in the long-wave model is J=

4Kk2 , ε0 n4a k02 lξ2 I

(4.20)

where I = |ψ |2 is the input intensity. For the weak nonlinear regime to be reached, we require that the nonlinearity parameter 1 ¿ J. For nonlinear wave mixing to occur, waves must be in a state of resonance, this means that they must satisfy a resonance condition on the conservation of wavenumbers and frequencies. These conditions together are known as the resonant manifold condition, where both k + k1 − k2 − k3 = 0,

ωk + ω1 − ω2 − ω3 = 0,

(4.21)

4 Optical Wave Turbulence and Wave Condensation in a Nonlinear Optical Experiment

are satisfied (for four-wave interactions). For a one-dimensional system with dispersion ∼ k2 there exists no non-trivial solution to the resonance condition (4.21). However, WTT provides a near-identity transformation that allows one to eliminate unnecessary lower orders of nonlinearity in the system if the corresponding order of wave interactions are absent.

4.3.5 Canonical Transformation The near-identity transformation allows one to eliminate “unnecessary” lower orders of nonlinearity in the system if corresponding order of the wave interaction process is zero, i.e. there exist no non-trivial solutions to the resonant manifold condition of that order [9]. In our case, there can be no four-wave resonances (there are no non-trivial solution for the resonance conditions in for ω ∼ kx if x > 1). There are also no five-wave resonances because the original terms in the Hamiltonian are of the even orders. However, there are non-trivial solutions of the six-wave resonant conditions. Thus, one can use the near-identity transformation to convert our system into one with the lowest order interaction Hamiltonian to be of degree six. A trick for finding a shortcut derivation of such a transformation is found in [9]. It relies on the fact that the time evolution operator is a canonical transformation. Taking the Taylor expansion of a(k, z) around a(k, 0) = c(k, 0) we get a desired transformation, that is by its derivation, canonical. The coefficients of each term can be calculated from an auxiliary Hamiltonian Haux - this is a generic Hamiltonian with arbitrary interaction coefficients that once found, determines the canonical variable ck . A similar procedure was done in Appendix A3 of [9] to eliminate the cubic Hamiltonian in cases when the three-wave interaction is nil, and here we apply a similar approach to eliminate the quadric Hamiltonian. The transformation is represented as µ ¶ µ ¶ ∂ c(k, z) z2 ∂ 2 c(k, z) ak = c(k, 0) + z + +··· (4.22) ∂z 2 ∂ z2 z=0 z=0 The transformation is canonical for all z, so for simplicity we set z = 1. Subsequently, Hamiltonian (4.18) can be transformed into, Z

H = H2 + H6 =

ωk ck c∗k dk +

123 ∗ ∗ ∗ T123456 δ456 c1 c2 c3 c4 c5 c6 dk123456 .

The explicit formula for T123456 stemming from the transformation is

(4.23)

J. Laurie, U. Bortolozzo, S. Nazarenko and S. Residori

T123456

3 6 1 =− ∑ ∑ 18 i, j,k=1,i6= j6=k6=i p,q,r=4,p6=q6=r6= p ! Wi+ j−ppi jWq+r−kkqr ¢ . +¡ ωq+r−k + ωk − ωq − ωr

Ã ¡

Wp+q−iipqW j+k−rr jk ¢+ ω j+k−r + ωr − ω j − ωk (4.24)

The six-wave interaction coefficient T123456 is formed from a coupling of two four-wave interactions W1234 , i.e. the six-wave interaction is produced from two coupled four-wave interactions that occur simultaneously, a simple graphic showing this is depicted in Fig. 4.3. We find that the six-wave interaction coefficient T123456 is the sum of two terms of different scalings. However, in the long-wave limit, klξ ¿ 1, T123456 tends towards a k-independent constant T123456 =

ε02 n8a lξ6 k04

. (4.25) 64K 2 This implies that the six-wave interaction coefficient goes to zero in the long-wave limit. Four Wave Interaction W1234

Six Wave Interaction T123456 k1 k4

k1 k3 k4

k2 k3

Fig. 4.3 Graphic to show how the six wave interactions arises from the coupling of two four wave interactions

4.3.6 The Kinetic Wave Equation The kinetic wave equation (KE) describes the evolution of the wave action density nk = hck c∗k i (the averaging is over the random phases) of wave packets in Fourier space. In order to derive the kinetic equation, we must express the wave action Hamiltonian (4.23) into an equation for the wave action density nk , by applying a random phase approximation (RPA). The dynamical equation for the variable ck can be determined from Hamiltonian (4.23) by the relation i∂ ck /∂ z = δ H/δ c∗k ,

4 Optical Wave Turbulence and Wave Condensation in a Nonlinear Optical Experiment

∂ ck − ωk ck = ∂z

Z k12 Tk12345 c∗1 c∗2 c3 c4 c5 δ345 dk12345 .

(4.26)

By multiplying equation (4.26) by c∗k , subtracting the complex conjugate, and averaging we arrive at µZ ¶ ∂ hck c∗k i ∂ nk k12 = = 6Im Tk12345 Jk12345 δ345 dk12345 , (4.27) ∂z ∂z k12 = hc∗ c∗ c∗ c c c i. where Jk12345 δ345 k 1 2 3 4 5 To compute the average applied to Jk12345 , one must use a RPA. Taking Jk12345 to (0) (0) the zeroth order Jk12345 by assuming a Gaussian wave field, implies Jk12345 can be written as a product of three pair correlators,

£ ¡ ¢ (0) Jk12345 = n1 n2 n3 δ3k δ41 δ52 + δ51 δ42 ¡ ¢ ¡ ¢¤ + δ4k δ31 δ52 + δ51 δ32 + δ5k δ31 δ42 + δ41 δ32 .

(4.28)

However, due to the symmetry of Tk12345 , this makes the right hand side of the (1) KE zero. To find a nontrivial answer we need to obtain a first order addition Jk12345 (1)

to Jk12345 . To calculate Jk12345 one takes the “time” derivative of Jk12345 , using the equation of motion (4.26) and inserts the zeroth order approximation for the tenth correlation function (this is similar to equation (4.28), but a product of five pair (1) correlators involving ten wavevectors). Jk12345 can then be expressed as Ak12345 (1) Jk12345 = Bei∆ ω z + , (4.29) ∆ω ³ ´ ∗ where Ak12345 = 3Tk12345 nk n1 n2 n3 n4 n5 n1 + n11 + n12 − n13 − n14 − n1 and ∆ ω = k 5 ωk + ω1 + ω2 − ω3 − ω4 − ω5 . The first term of (4.29) is a fast oscillating function, its contribution to the integral (4.27) decreases with z and is negligible at z larger than 1/ωk , and as a result we will ignore the contribution arising from this term. The second term is substituted in equation (4.27),the relation Im(∆ ω ) ∼ −πδ (∆ ω ) is applied because of integration around the pole, and the KE is derived, Z

∂ nk = 18π |Tk12345 |2 fk12345 δ (k + k1 + k2 − k3 − k4 − k5 ) ∂z ×δ (ωk + ω1 + ω2 − ω3 − ω4 − ω5 )dk1 dk2 dk3 dk4 dk5 , with T123456 being the six-wave interaction coefficient for the system and µ ¶ 1 1 1 1 1 1 fk12345 = nk n1 n2 n3 n4 n5 + + − − − . nk n1 n2 n3 n4 n5

(4.30)

(4.31)

J. Laurie, U. Bortolozzo, S. Nazarenko and S. Residori

The KE has important exact power law solutions. These solutions are known as KZ solutions, namely from the discovery by Zakharov and their analogies to the Kolmogorov spectrum seen in classical turbulence theory. Moreover, the KE also contains thermodynamical equilibrium solutions, that correspond to the relaxation of energy and wave action within the system to equilibria. These solutions are limiting cases of the more generalized Rayleigh-Jeans distribution, nk =

T , ωk + µ

(4.32)

where T is the temperature and µ is a chemical potential. One can predict the scaling of the KZ solutions by considering a dimensional analysis argument upon an inertial interval in k-space where the fluxes of both energy and wave action are constant. We have seen that in the long-wave model, equation (4.25) implies that the six-wave interaction coefficient scales as T123456 ∼ |k|0 .

(4.33)

Thus, the scaling of the KE in terms of wavenumber k and wave action density nk is, n˙k ∼ |k|2 n5k .

(4.34)

For the direct cascade of energy, the KZ spectrum is realized when there exists a constant, non-zero energy flux Pk that is scale independent, Pk =

Z k

ωk n˙k dk ∼ |k|0 .

(4.35)

Equations (4.34) and (4.35) gives the scaling of nk for the direct cascade to be, nk = C|k|−1 ,

(4.36)

where C is an arbitrary constant. Similarly, the wave action cascade implies a constant wave action flux flowing towards low wavenumbers, i.e. the wave action flux Qk scales as Qk =

Z k

n˙k dk ∼ |k|0 ,

(4.37)

consequently generating an inverse KZ wave action spectrum of 3

nk = C|k|− 5 .

(4.38)

The inverse cascade spectrum is of a finite capacity type, in a sense that only a finite amount of the cascading invariant (wave action in this case) is needed to fill the infinite inertial range. (Indeed, the integral of nk ∼ k−3/5 converges at k = 0). In these cases the turbulent systems have a long transient (on its way to the final thermal equilibrium state) in which the scaling is of the KZ type. This is because the initial condition serves as a huge reservoir of the cascading invariant. Note that

4 Optical Wave Turbulence and Wave Condensation in a Nonlinear Optical Experiment

the situation here is not specific for WT only and it is valid generally for turbulence. For example, it is valid for Navier-Stokes turbulence, i.e. the Kolmogorov-Obukhov spectrum, which is also finite capacity.

4.3.7 Modulational Instability and the Creation of Solitons Closeness of equation (4.4) to integrability means that we should expect not only random waves but also soliton-like coherent structures. In the inverse cascade setup, solitons appear naturally. Indeed, the wave turbulent description (equation (4.30)) breaks down when the inverse cascade reaches some low k’s. Modulational instability (MI) develops at these scales, which results in the filamentation of light and its condensation into coherent structures - solitons. The inverse cascade of photons is a very important process in the creation of solitons. It provides the means, via nonlinear wave interaction to allow wave action to reach lower wavenumbers. Once the intensity at these wavenumbers passes a certain threshold, MI can take over. MI of a wave packet occurs when the nonlinearity of the wave packet increases such that the linear dynamics (defined by the linear frequency ωk = k2 ) become Bogoliubov modified, that is to say that the wave packet no longer propagates linearly. As a consequence, the wave packet’s dynamics are no longer determined by the linear dispersion, but now by the Bogoliubov dispersion relation or frequency. In focusing nonlinear states, this Bogoliubov frequency can become imaginary when the wave intensity is high enough, resulting in an instability of exponential growth of the wave envelope. This process was first discovered in the context of water waves, where it is originally known as the Benjamin-Feir instability [30]. One can characterize MI by deriving the Bogoliubov frequency [31], - the nonlinear wave frequency that takes into account the first order effect of weak nonlinearity. To derive such a relation, one must expand the wave function ψ (x, z) around a condensate. The description of the condensate can be calculated by assuming an x-independent solution of the evolution equation, i.e. ψ (x, z) = ψc (z). Looking for a small perturbation around this condensate, this expansion can be written in either physical or Fourier space and is given by either

ψ = ψc (1 + φ )

ak = ψc (δ (k) + φk ) .

(4.39)

where |φ |, |φk | ¿ 1. The condensate is defined as the k = 0 mode, which corresponds to an xindependent state. We can determine its dynamics by considering an x-independent solution ψc (z) to equation (4.4). We find that ψc (z) = ψ0 exp(−iωc z), with ωc = −ε0 n4a lξ2 k02 |ψ0 |2 /8q. ψc (z) describes the background rotation of the condensate with frequency ωc . Substituting ansatz (4.39) into system (4.4) and linearizing to the first order in φk , gives a nonlinear evolution equation for the perturbation φk . Finally, one

J. Laurie, U. Bortolozzo, S. Nazarenko and S. Residori

makes the assumption of a single monochromatic plane wave solution for φk of the form φk = A exp (ikx − iΩk z) + c.c., where A is a complex constant, and c.c. means complex conjugate. Equating the exponentials, one can derive the Bogoliubov formula or the MI condition for a perturbation upon a condensate. vÃ ! u u |ψ0 |2 ε0 n4a lξ4 k02 |ψ0 |2 ε0 n4a lξ2 k02 t 2qΩk = 1+ k4 − k2 , (4.40) 2K 2K where |ψ0 |2 is the average density of the condensate, and Ωk is the frequency of the waves upon the condensate. To obtain the frequency of the original wave function ψ (x, z), we must add back the condensate shift (or the frequency of the condensate) to the Bogoliubov frequency Ωk . Therefore, the Bogoliubov dispersion relation for a weakly nonlinear wave packet is ωk = ωc + Ωk , where ωc is the condensate frequency shift and Ωk is the MI condition for φk for the weakly nonlinear wave packet upon the condensate, equation (4.40).

4.4 Numerical Method We numerically solve equation (4.4) using a standard pseudo-spectral method with periodic boundary conditions. We de-alias on half the wavenumbers to remove any aliasing errors when computing the cubic nonlinear term. We set a resolution of 2048 points in physical space and apply a fourth order Runge-Kutta method to solve in “time” z. We set the “time step” to be smaller than the CFL condition and the smallest linear time of evolution to ensure the simulation is properly resolved. We compute the energy of each term in equation (4.5) to ensure that the linear energy is greater than the nonlinear and moreover, that the leading nonlinear term dominates the sub-leading one, i.e. that the long-wave limit is satisfied. Weak nonlinearity is verified by parameter J (see equation (4.20)), but not too weakly nonlinear that we experience “frozen turbulence” [32]. To increase stability of the numerical scheme, we integrate the linear terms exactly and apply the time stepping algorithm to the nonlinear terms only by using integrating factors.

4.5 Experimental and Numerical Results Both experimental and numerical setups are configured for decaying OWT, where an initial condition is defined and allowed to develop absent of any forcing or numerical dissipation. Experimentally, we have injected photons at intermediate spatial scales, where intermediate means around a wavenumber k = k0 in between k = 0 and dissipation scale occurring at k = kD . At this purpose, the intensity of the input beam is modulated with a patterned intensity mask, and in order to impose an initial condition close to the random phase approximation required by the theory, we have

4 Optical Wave Turbulence and Wave Condensation in a Nonlinear Optical Experiment

randomize the phases by a phase modulator. This is made by creating through the SLM a random distribution of diffusing spots with the average size ∼ 35 µ m, which is relatively larger than the liquid crystal electrical coherence length lξ fixing the dissipative scale. The numerical initial condition is more idealized and strictly localized at a small-scale range: we excite five wavenumbers with constant amplitude around |k f | ∼ 1.5 × 102 mm−1 , and the phase of ψk is random and independent at each k. Moreover, we have applied a Gaussian filter in physical space to achieve a beam profile comparable to that of the experiment.

100

z = 0 mm

10-1

-3/5

nk 10-2 10-3

|Ik|2

z = 0 mm

100

k-1/5

z = 63 mm

10 k [mm-1]

10-1

z = 63 mm 10

102 k [mm-1]

Fig. 4.4 a) Numerical spectrum of a) the wave action nk , and b) the light intensity, Nk = |Ik |2 at distances z = 0 mm and z = 63 mm.

The numerical wave action spectrum is shown in Fig. 4.4a at two different distances, we see at z = 0 mm the peak from the initial condition at high wavenumbers, then at z = 63 mm we see evidence of an inverse cascade, as the majority of the wave action is situated towards low wavenumbers. However, at low wavenumbers we do not see the spectrum matching our theoretical KZ prediction of nk ∼ |k|−3/5 , this is because of high nonlinearities towards low wavenumbers causing a breakdown of the WTT. The higher nonlinearity towards low wavenumbers causes soliton formation out of the weakly nonlinear waves, and as a result we observe a flattening of the wave action spectrum. However, it remains to be found why the inverse wave action spectrum disagrees with the WTT prediction in the same run, and whether we could confidently say that what we see is a KZ spectrum. Note that even in weak wave turbulence, the KZ spectrum may not be realized, e.g. if the interaction of scales is nonlocal. In the other words, the presence of wave turbulence (claimed in this paper) does not automatically imply the presence of KZ spectra (for which we see an indication but not a solid proof). Experimentally, we measure the light intensity I(x, z) = |ψ |2 and not the phases of ψ and, therefore, the spectrum nk is not directly accessible. Instead, we measure the spectrum of intensity N(k, z) = |Ik (z)|2 . The scaling for Nk in the inverse cascade state is easy to obtain from nk ∼ |k|−3/5 and the random phase condition, this gives Nk ∼ |k|−1/5 . Numerical and experimental spectra of the light intensity are shown in Figs. 4.4b and 4.5 respectively. In both cases one can see an inverse cascade excitation of the lower k states, and good agreement with the WT prediction Nk ∼ |k|−1/5 .

J. Laurie, U. Bortolozzo, S. Nazarenko and S. Residori

|Ik |2

k -1/5 z=4.2 mm

z=0 mm

k [mm-1]

Fig. 4.5 Experimental spectrum of the light intensity, Nk = |Ik |2 at two different distances z.

Note that the numerical spectrum of the light intensity shows a better agreement with the expected KZ scaling than the numerical wave action spectrum. A possible explanation is that averaging over phases can provide a wash-out effect of the WTT breakdown phenomena occurring at low wavenumbers, thus restoring the scaling predicted in the approximation of weak nonlinearities. 200

z=0 mm

I [a.u.]

z=1.9 mm

150

100

0.12

0.25

0.72

0.5

x [mm]

Fig. 4.6 Intensity distribution I(x,z) showing the beam evolution during propagation; a) linear case, b) weakly nonlinear case. c) Two intensity profiles I(x) recorded at z = 0 and z = 1.9 mm in the weakly nonlinear regime, V = 2.5 V .

Experimentally, the inverse cascade can be seen directly by inspecting the light pattern in the x − z plane under the beam propagation evolution. Two magnified images of the intensity distribution I(x, z) showing the beam evolution during propagation are displayed in Fig.4.6. For comparison, in Fig.4.6a and b, respectively, we show the beam evolution in the linear and in the weakly nonlinear (wave turbulence) regime. In Fig.4.6a the initial condition is periodic with an uniform phase and no voltage is applied to the liquid crystal layer (V = 0). We see that the linear propagation is characterized by the periodic recurrence of the pattern with the same period, a phase slip occurring at every Talbot distance, this one being determined by p2 /λ with p the period of the initial condition and λ the laser wavelength [33]. In Fig.4.6b the voltage on the liquid crystal cell is switched on, V = 2.5 V rms, and the initial condition is periodic with the same period as in Fig.4.6a but with random phases.

4 Optical Wave Turbulence and Wave Condensation in a Nonlinear Optical Experiment

In this case, the initial period of the pattern is becoming larger as the light beam propagates forward along z. If a random phase distribution would have been chosen in the linear case, then, a speckle pattern would have developed along propagation [34], therefore destroying the initial modulation and preventing a direct comparison of the linear and weakly nonlinear case. While the linear propagation leads to Talbot intensity carpets [35], with the initial intensity distribution reappearing periodically along the beam propagation direction z, the weak nonlinearity leads to wave interaction, so that, as the beam propagates, the different spatial frequencies components mix-up and the periodic occurrence of the Talbot carpet is broken. In Fig.4.6c we show two intensity profiles taken in the nonlinear case at different stages of the beam propagation. The inverse cascade is accompanied by a smoothing of the intensity profiles and amplification of low wavenumbers as the beam propagates forward inside the nematic layer. b)

x [mm]

1.0

I (gray values)

4000

7.5 mm

3000 2000 1000

0.5 0

z [mm] 8

4.5 mm -0.4

-0.2

0.3 mm 0

0.2

0.4

x [mm]

Fig. 4.7 a) Experimental results for intensity distribution I(x,z) for the same parameter values for Fig.4.6. Area marked by the dashed line is shown at a higher resolution (using a larger magnification objectif). b) Linear intensity profiles I(x) taken at different propagation distances, z = 0.3, 4.5 and 7.5 mm

Fig. 4.8 Numerical results for intensity distribution I(x, z). The frame on the left is a magnified section of the initial propagation of the beam.

Example zooms of the intensity distribution I(x, z) showing the beam evolution during propagation in the experiment and in the numerics are displayed in Fig. 4.7 and Fig. 4.8 respectively. The experimental data shown in Fig. 4.7 have

J. Laurie, U. Bortolozzo, S. Nazarenko and S. Residori

been recorded for the same parameter values as for Fig.4.6. In the high resolution inset on Fig. 4.7 one can visually observe that the typical scale increases along the beam which corresponds to an inverse cascade process. Further, in both Fig. 4.7 and Fig. 4.8 one can see formation of coherent solitons out of the random initial field, such that one strong soliton is dominant at the largest distance z (the soliton peak intensity in Fig.4.7 is ∼ 800 times greater than the initial light intensity).

√ Fig. 4.9 The k − ω spectrum of the wave field at z = 2.1 m. ω∗ = 1/256qlξ2 and k∗ = 1/ 128lξ . The Bogoliubov dispersion relation is shown by the solid line.

Separating the random wave and the coherent soliton components can be done via performing an additional Fourier transform with respect to ”time” z over a finite z-window. Such numerically obtained (k, ω )-plot is shown in Fig. 4.9. There, the incoherent wave component is distributed around the wave dispersion relation, which is Bogoliubov-modified by the condensate (equation (4.40)) and shown by a solid line in Fig. 4.9. This distribution is narrow for large k which corresponds to weak nonlinearity, and it gets wider toward low k, which corresponds to a growth of nonlinearity and breakdown of the WT applicability conditions. For these low k values one can see pieces of slanted lines (under the dispersion curve). Each of these lines corresponds to a coherent soliton, whose speed is equal to the inclination slope. We observe that the formation of solitons is seen in the (k, ω )-plot as straight line “peeling” with a gradient tangential to the dispersion curve. Experimental realization of the (k, ω )-plot will be performed in next future by employing a higher depth resolution camera.

4.5.1 Direct cascade of energy Finally, we numerically investigated the direct energy cascade. We found that the direct cascade prediction is an infinite capacity spectrum. This means that unlike

4 Optical Wave Turbulence and Wave Condensation in a Nonlinear Optical Experiment

the inverse cascade, one cannot realize the KZ prediction in a decaying simulation. Therefore, we ran a numerical simulation of equation (4.4), but with the additional terms that correspond to additive forcing +iF, and dissipation −iD. We forced with constant amplitude and random phases at three wavenumbers at the low wavenumber region and applied hyper and hypo-viscosities of the form D = ν1 ∂ 8 ψ /∂ x8 + ν2 ∂ −8 ψ /∂ x−8 . We ran the simulation, and then averaged the wave action spectrum once the simulation reached a steady state. We show the direct wave action spectrum with the WTT KZ prediction in Fig. 4.10. We see a good agreement with the KZ spectrum for about a decade in wavenumber space. There is some slight noise at high wavenumbers, due to the simulation not completely reaching a true steady state. This regime is not accessible in the experiment because the system is not large enough to let the cascade to develop over a significant wavenumber interval. Further experiments are in progress in order to increase the size of the system and to observe the direct cascade.

101 0

-1

10 nk

10-1 -2

10-3

101

102 -1

k [mm ] Fig. 4.10 The numerical direct wave action spectrum of energy with the predicted KZ spectrum of nk = k−1 .

4.6 Conclusions In conclusion, we have presented an experimental implementation of the 1D OWT regime, accompanied by numerical simulations and theory. We observe an inverse cascade of photons toward the states with lower wavenumbers in both the experiment and numerics, the predicted intensity spectrum is seen clearly in the experiment plot, with a reasonable trend in the numerical plot. The wave action spectrum in the numerical plot is vaguely seen over a short region, we argue that decaying OWT isn’t the ideal setup for seeing the KZ spectrum, because of the high lev-

J. Laurie, U. Bortolozzo, S. Nazarenko and S. Residori

els of nonlinearity at lower wavenumbers and the presence of solitons where WTT breaks down. Furthermore, we have shown that after the initial inverse cascade to low wavenumbers, we see the development of solitons by MI, then the further merging of these solitons into one dominate soliton at later times. Finally, we also verified numerically the KZ prediction for the direct cascade, with a good agreement for about a decade in wavenumber space.

4.7 Acknowledgements This work has been partially supported by the Royal Society’s International Joint Project grant Vortices, turbulence and photon condensation in a nonlinear optical experiment.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

S. Dyachenko, A.C. Newell, A. Pushkarev, and V.E. Zakharov, Physica D 57, 96 (1992) V. Zakharov, F. Dias, and A. Pushkarev, Physics Reports, 398, 1 (2004) S. Nazarenko, and V. Zakharov, Physica D 201, 203 (2005) C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, Phys. Rev. Lett. 95, 263901 (2005) S. Nazarenko, and M. Onorato, Physica D 219, 1 (2006) F.T. Arecchi, G. Giacomelli, P.L. Ramazza, and S. Residori, Phys. Rev. Lett. 67, 3749 (1991) G.A. Swartzlander, Jr. and C.T. Law, Phys. Rev. Lett. 69, 2503 (1992). C. Barsi, W. Wan, C. Sun, and J.W. Fleischer, Opt. Lett. 32, 2930 (2007). V. E. Zakharov, V. S. Lvov, and G. Falkovich, Kolmogorov Spectra of Turbulence, (SpringerVerlag, 1992). U. Bortolozzo, J. Laurie, S. Nazarenko, and S. Residori, J. Opt. Soc. Am. B 26, 2280 (2009). M. Peccianti, C. Conti, and G. Assanto, Phys. Rev. E 68, 025602 (2003). A.J. Majda, D.W. McLaughlin, and E.G. Tabak, J. Nonlinear Sci. 6, 944 (1997). D. Cai, A.J. Majda, D.W. McLaughlin, and E.G. Tabak, Physica D, 152, 551, (2001). V.E. Zakharov, A.N. Pushkarev, V.F. Shvets, and V.V. Yan’kov, JETP Lett. 48, 83 (1988). B. Rumpf, and A.C. Newell, Phys. Rev. Lett. 87, 5 (2001). B. Rumpf, and A.C. Newell, Physica D, 184, 162 (2003). B. Barviau, B. Kibler, A. Kudlinski, A. Mussot, H. Millot, and A. Picozzi, Opt. Express 17, 7392 (2009). A. Eisner and B. Turkington, Physica D, 213, 85 (2006). R. Jordan, and C. Josserand, Phys. Rev. E, 61, 1 (2000). K.Ø. Rasmussen, T. Cretegny, and P.G. Kevrekidis, Phys. Rev. Lett. 84, 17 (2000). R. Jordan, B. Turkington, and C.L. Zirbel, Physica D, 137, 353-378 (2000). A. Picozzi, S. Pitois, and G Millot, Phys. Rev. Lett. 101, 093901 (2008). S. Pitois, S. Lagrange, H.R. Jauslin, and A. Picozzi, Phys. Rev. Lett. 97, 033902 (2006). V.I. Petviashvili, and V.V. Yan’kov, Reviews of Plasma Physics, 14, (1987). I.C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley, New York, 1995). P.G. De Gennes and J. Prost, The Physics of Liquid Crystals, (Oxford Science Publications, Clarendon Press, second edition, 1993). M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, Nature 432, 733 (2004).

4 Optical Wave Turbulence and Wave Condensation in a Nonlinear Optical Experiment 28. 29. 30. 31. 32. 33. 34. 35.

C. Conti, M. Peccianti, and G. Assanto, Opt. Lett. 31, 2030 (2006). R. Fjortoft, Tellus 5, 225 (1953). T.B. Benjamin and J.E. Feir, J. Fluid Mech., 27, 417-430 (1967). N.N. Bogoliubov, J. Phys. U.S.S.R. 11, 23 (1947). A.N. Pushkarev, Eur. J. Mech. B/Fluids 18, 345-352 (1999). H.F. Talbot, Phil. Mag. 9, 401 (1836). J. W. Goodman, Statistical Optics (John Wiley, New York, 1985). M. Berry, I. Marzoli, and W. Schleich, Phys. World, June 2001, 1 (2001).

Part II

Localized structures in pattern forming systems

Chapter 5

Localized Structures in the Liquid Crystal Light Valve Experiment Umberto Bortolozzo, Marcel G. Clerc, Ren´e G. Rojas, Florence Haudin and Stefania Residori

Abstract We will review the conditions for the appearance of coherent or localized states in a nonlinear optical feedback system, with particular reference to the Liquid Crystal Light Valve (LCLV) experiment. The localized structures here described are of dissipative type, that is, they represent the localized solutions of a pattern-forming system. We will show that different types of localized states are observed in the system and can be selected depending on the control parameters: round localized structures that interact forming bound-states, triangular localized structures, characterized by the presence of phase singularities, localized peaks, appearing above a structured background. Then, we will discuss the nonvariational behaviors of such coherent states, like the bouncing of round localized structures and the chaotic front propagation for the triangular ones. We will present the full model equations for the LCLV system as well as a one-dimensional spatially forced Ginzburg-Landau equation, which is the simplest model accounting for the phenomenology observed in the experiment. We will show how, by using a properly intensity/phase modulated input beam, we can either induce a large pinning effect or control the dynamics of large Umberto Bortolozzo INLN, Universit´e de Nice Sophia-Antipolis, CNRS, 1361 route des Lucioles 06560 Valbonne, France e-mail: [emailprotected] Marcel G. Clerc Departamento de F´ısica, Facultad de Ciencias F´ısicas y Matem´aticas, Universidad de Chile, Casilla 487-3, Santiago, Chile, e-mail: [emailprotected] Ren´e G. Rojas Instituto de F´ısica, Pontificia Universidad Cat´olica de Valpara´ıso, Casilla 4059, Valpara´ıso, Chile e-mail: [emailprotected] Florence Haudin INLN, Universit´e de Nice Sophia-Antipolis, CNRS, 1361 route des Lucioles 06560 Valbonne, France e-mail: [emailprotected] Stefania Residori INLN, Universit´e de Nice Sophia-Antipolis, CNRS, 1361 route des Lucioles 06560 Valbonne, France, e-mail: [emailprotected]

O. Descalzi et al. (eds.), Localized States in Physics: Solitons and Patterns, DOI 10.1007/978-3-642-16549-8_5, © Springer-Verlag Berlin Heidelberg 2011

U. Bortolozzo, M. G. Clerc, F. Haudin and S. Residori

arrays of localized structures, addressing each site independently from the others. Finally, the propagation properties of localized structures will be presented.

5.1 Introduction Non equilibrium processes often lead in nature to the formation of spatially periodic and extended structures, so-called patterns [2, 2]. The appearance of a pattern from a hom*ogeneous state takes place through the spontaneous breaking of one or more of the symmetries characterizing the system [4]. In some cases, it is possible to localize a pattern in a particular region of the available space, so that we deal with localized instead of extended structures. From a theoretical point of view, localized structures in out of equilibrium systems can be seen as a sort of dissipative solitons [19]. Experimentally, during the last years localized patterns or isolated states have been observed in many different fields. Examples are domains in magnetic materials [5], chiral bubbles in liquid crystals [6], current filaments in gas discharge experiments [7], spots in chemical reactions [8], oscillons in granular media [9, 10], localized fluid states in surface waves [9] and in thermal convection [12], solitary waves in nonlinear optics [13, 14, 15, 16, 17, 18, 48, 20]. All these localized states can be considered to belong to the same general class of localized structures, that is, they are patterns that extend only over a small portion of a spatially extended system. In optics, solitary waves have first been predicted to appear in bistable ring cavities [13], then, they have been largely studied not only for their fundamental properties but also in view of their potential applications as elementary bits of information [21, 22, 23, 24]. Sometimes named as cavity solitons, optical localized structures have been observed in photorefractive media [25], in lasers with saturable absorber [26], in Liquid-Crystal-Light-Valves (LCLVs) with optical feedback [14, 15, 16, 17, 18], in Na vapors [27] and more recently in semiconductor micro-cavities [20]. Here, we will review the conditions for the appearance of coherent or localized states in a nonlinear optical feedback system, with particular reference to the Liquid Crystal Light Valve (LCLV) experiment [28]. We will show that different types of localized states are observed in the system and can be selected depending on the control parameters: round localized structures that interact forming bound-states [29], triangular localized structures, characterized by the presence of phase singularities [30], localized peaks, appearing above a structured background [31]. Then, we will discuss the nonvariational behaviors of such coherent states, like the bouncing of round localized structures [32] and the chaotic front propagation for the triangular ones [33]. We will present the full model equations for the LCLV system as well as a one-dimensional spatially forced Ginzburg-Landau equation, which is the simplest model accounting for the phenomenology observed in the experiment [34] and for the tilted snaking bifurcation diagram [35]. Then, we will show how, by using a properly intensity/phase modulated input beam, we can either induce a large pin-

5 Localized Structures in the Liquid Crystal Light Valve Experiment

ning range or control the dynamics of large arrays of localized structures, addressing each site independently from the others [36]. Finally, we will present the propagation properties of localized structures [37].

5.2 The Liquid Crystal Light Valve Experiment 5.2.1 Description of the setup The experimental setup, shown in Fig.8.3, consists of a LCLV with optical feedback, as it was originally designed by the Akhmanov group [38]. The LCLV is composed of a nematic liquid crystal film sandwiched in between a glass window and a photoconductive plate over which a dielectric mirror is deposed. Coating of the bounding surfaces induces a planar anchoring of the liquid crystal film (nematic director n parallel to the walls). Transparent electrodes covering the two confining plates permit the application of an electric field across the liquid crystal layer. The photoconductor behaves like a variable resistance, which decreases for increasing illumination. The feedback is obtained in the following way: the light which has passed through the liquid-crystal layer, and has been reflected by the dielectric mirror inside the LCLV, is sent back onto the photoconductor of the LCLV. This way, the light beam experiences a phase shift which depends on the liquid crystal reorientation and, on its turn, modulates the effective voltage that locally is applied to the liquid crystals. The feedback loop is closed by an optical fiber bundle and is designed in such a way that diffraction and polarization interference are simultaneously present [28]. The presence of diffraction leads to the spontaneous generation p of self-organized patterns, which display a typical spatial period scaling as ∼ λ |L|, where λ is the laser wavelength and L is the optical free propagation length in the feedback loop [39]. On the other hand, the presence of polarization interference leads to bistability between different spatial states. Setting L = 0 eliminates diffraction effects, so that in this case the system exhibits bistability between hom*ogeneous states and front propagation [25]. To obtain localized structures the optical free propagation length is usually fixed to L = −8 cm. For p this value of L the transverse size of a single localized structure, which scales as λ |L|, is about 250 µ m. At the linear stage for the pattern formation, a negative propagation distance selects the first unstable branch of the marginal stability curve, as for a focusing medium [41]. Moreover, an input and feedback polarizer are inserted in such a way to form with the liquid crystal director an angle of 45◦ and −45◦ , respectively. For this parameter setting and close to the point of Fr´eedericksz transition, there is coexistence between a periodic pattern and a hom*ogeneous solution. The Fr´eedericksz transition point is attained for an applied r.m.s. voltage V0 of approximately 3 V , with a frequency of 5 kHz [25]. By increasing V0 , successive branches of bistability are excited. Most of the experimental observations here reported were obtained close to one of the points of nascent bistability onto the

U. Bortolozzo, M. G. Clerc, F. Haudin and S. Residori

different bistable branches. There, the bistable behavior observed is similar to the one observed close to the Fr´eedericksz transition point [25]. The input beam has a Gaussian profile with a transverse size of approximately 2 cm, whereas a diaphragm before the LCLV selects a central active zone with a diameter of 1 cm. The input intensity Iin usually varies in between 0.3 and 1 mW /cm2 . As shown in Fig.8.3, the setup includes also a spatial light modulator (SLM) connected to a personal computer (PC) and inserted in the optical path of the input beam Iin . The SLM is a twisted nematic liquid crystal display that can be used either without polarizers or in between crossed polarizers providing, respectively, phase or intensity modulations that are used to control the spatial profile of the input beam.

feedback loop LCLV

Iin Iw

SLM

A C Q U I S I T I O N

C O N T R O L

liquid cristals ITO electrodes photoconductor dielectric mirror glass plate Fig. 5.1 Experimental setup: the LCLV is illuminated by a plane wave collimated beam (red line in the central picture); the beam reflected by the LCLV (green line) is sent back to the photoconductor through a beam-splitter, a mirror and, finally, an optical fiber bundle. In the upper left inset is shown an enlarged picture of the LCLV. A schematic representation is displayed below: V0 is the voltage applied, Iin and Iw are the input and feedback intensity, respectively. A small portion of the beam (white line), is extracted from the feedback loop through a beam-splitter and sent to the acquisition line that is composed by a lens, a mirror and a computer interfaced CCD camera. In the bottom right inset is shown an enlarged image of the SLM that is computer interfaced and used to control the spatial profile of the input beam.

5 Localized Structures in the Liquid Crystal Light Valve Experiment

5.2.2 The optical feedback: model equations The theoretical model for the LCLV feedback system was previously derived in [32] and consists of two coupled equations, one for the average director tilt θ (r⊥ ,t), 0 ≤ θ ≤ π /2, and one for the feedback light intensity Iw . The average director tilt θ (r⊥ ,t) accounts for the average orientation angle of the liquid crystal molecules with respect to the longitudinal direction of the nematic layer, r⊥ denotes the transversal direction of the liquid crystal layer. For θ = 0 (θ = π /2) all the molecules are parallel (orthogonal) to the confining walls, which corresponds, respectively, to a planar and a homeotropic alignment of the liquid crystals [42]. When one applies an electric voltage V0 along the longitudinal direction of the nematic layer, all the molecules in the bulk reorient in such a way to align with the direction of the applied field, because of their positive dielectric anisotropy. Hence, liquid crystal molecules are under the influence of two opposite torques, the elastic restoring torque and the electric torque. The equation for the average director tilt around equilibrium reads as

τ∂t θ = l 2 ∇2⊥ θ − θ + θc (V ),

(5.1)

where l is the electrical coherence length, τ the local relaxation time and θc (V ) the equilibrium average director tilt. There is a critical value of the voltage–VFT –for which the electric force overcomes the elastic one, so that the molecules reorient. This process is called Fr´eedericksz transition [42]. In the LCLV, we must take into account the response of the photoconductor. In the absence of optical feedback, the response of th

Localized States in Physics: Solitons and Patterns - PDF Free Download (2024)

References