Extreme Photonics & Applications (NATO Science for Peace and Security Series B: Physics and Biophysics)

Extreme Photonics & Applications NATO Science for Peace and Security Series This Series presents the results of scien...

Author: Sofia Paredes | Trevor J. Hall | Sergey V. Gaponenko

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Extreme Photonics & Applications

NATO Science for Peace and Security Series This Series presents the results of scientific meetings supported under the NATO Programme: Science for Peace and Security (SPS). The NATO SPS Programme supports meetings in the following Key Priority areas: (1) Defence Against Terrorism; (2) Countering other Threats to Security and (3) NATO, Partner and Mediterranean Dialogue Country Priorities. The types of meeting supported are generally "Advanced Study Institutes" and "Advanced Research Workshops". The NATO SPS Series collects together the results of these meetings. The meetings are coorganized by scientists from NATO countries and scientists from NATO's "Partner" or "Mediterranean Dialogue" countries. The observations and recommendations made at the meetings, as well as the contents of the volumes in the Series, reflect those of participants and contributors only; they should not necessarily be regarded as reflecting NATO views or policy. Advanced Study Institutes (ASI) are high-level tutorial courses intended to convey the latest developments in a subject to an advanced-level audience Advanced Research Workshops (ARW) are expert meetings where an intense but informal exchange of views at the frontiers of a subject aims at identifying directions for future action Following a transformation of the programme in 2006 the Series has been re-named and re-organised. Recent volumes on topics not related to security, which result from meetings supported under the programme earlier, may be found in the NATO Science Series. The Series is published by IOS Press, Amsterdam, and Springer, Dordrecht, in conjunction with the NATO Public Diplomacy Division. Sub-Series A. B. C. D. E.

Chemistry and Biology Physics and Biophysics Environmental Security Information and Communication Security Human and Societal Dynamics

http://www.nato.int/science http://www.springer.com http://www.iospress.nl

Series B: Physics and Biophysics

Springer Springer Springer IOS Press IOS Press

Extreme Photonics & Applications

edited by

Trevor J. Hall

Centre for Research in Photonics at the University of Ottawa Ontario, Canada and

Sergey V. Gaponenko

B.I. Stepanov Institute of Physics National Academy of Sciences of Belarus Minsk, Belarus technical editor

Sofia A. Paredes Centre for Research in Photonics at the University of Ottawa Ontario, Canada

Published in cooperation with NATO Public Diplomacy Division

Proceedings of the NATO Advanced Study Institute on Laser Control & Monitoring in New Materials, Biomedicine, Environment, Security & Defense Ottawa, Ontario, Canada 24 November – 5 December 2008

Library of Congress Control Number: 2009937941

ISBN 978-90-481-3633-9 (PB) ISBN 978-90-481-3632-2 (HB) ISBN 978-90-481-3634-6 (e-book)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

Printed on acid-free paper

All Rights Reserved © Springer Science + Business Media B.V. 2010 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

Contents Preface ................................................................................................................... xi Ultrashort Laser Pulse Generation ...................................................................... 1 Olivier Chalus, Jens Biegert 1

2

Filamentation .................................................................................................... 1 1.1 Introduction ............................................................................................. 1 1.2 Filamentation experiment ....................................................................... 3 1.3 Numerical simulation of filamentation ................................................... 6 1.4 Measurement of ultrashort pulse ............................................................ 7 1.5 Self-compression by filamentation ......................................................... 8 Ultrashort pulse source in the mid-IR ............................................................ 11

Femtosecond Dynamics and Control: From Rydberg Molecules to Photochemistry and Photobiology .......................................................... 19 Helen H. Fielding 1 2 3 4 5 6

Introduction .................................................................................................... 19 1.1 Coherent control ................................................................................... 20 Young’s double slit experiment in an atom .................................................... 20 Controlling Rydberg electron wave packet localization in H2 ....................... 24 Femtosecond dynamics of the S1 excited state of benzene ............................ 28 A new experiment for investigating the photochemistry of biological chromophores in the gas-phase ................................................. 32 Summary ......................................................................................................... 33

Coherent Control for Molecular Ultrafast Spectroscopy ................................ 37 Tiago Buckup, Marcus Motzkus, Jürgen Hauer 1 2

3

Introduction .................................................................................................... 37 Pump-depletion-probe spectroscopy on carotenoids ...................................... 39 2.1 The complex deactivation network of carotenoids ............................... 39 2.2 Pump-depletion-probe experimental setup ........................................... 41 2.3 Results and discussion .......................................................................... 41 Enhancement of vibrational coherence and population transfer .................... 43 3.1 Tailored excitation of vibrational wavepackets .................................... 44 3.2 Experimental setup ............................................................................... 45 3.2.1 Tailored degenerate four-wave mixing setup .......................... 45 3.2.2 Tailored pump-probe setup ...................................................... 45

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3.3

4

Results ................................................................................................... 46 3.3.1 Enhancement of vibrational coherence ................................... 46 3.3.2 Excited-state population and vibrational coherence ................ 47 3.4 Discussion ................................................................................................ 49 Conclusions .................................................................................................... 51

Photonic Integration Enables Single-Beam Nonlinear Spectroscopy for Microscopy and Microanalytics ............................................................ 57 Bernhard von Vacano, Marcus Motzkus 1 2 3 4 5

Introduction .................................................................................................... 58 Creation of broadband laser pulses in photonic crystal fibres ....................... 59 Characterization, compression and shaping of broadband pulses for microspectroscopy .................................................................................... 63 Single-beam nonlinear spectroscopy with broadband shaped pulses .............................................................................................................. 66 Summary and outlook ..................................................................................... 71

Applications of Coherent Raman Scattering .................................................... 75 Alexei V. Sokolov 1 2 3

Background: Atomic coherence and EIT ....................................................... 75 Novel light sources utilizing maximal quantum coherence in molecular gasses and solids ........................................................................ 76 FAST CARS for rapid identification of chemical and biological unknowns ................................................................................ 84

Use of Ketoprofenate and Xanthonate Photocages for Antiviral Release ........................................................................................................... 95 Juan C. Scaiano, Jessie A. Blake, May Griffith 1 2 3 4 5

Introduction .................................................................................................... 95 Ketoprofen photochemistry ............................................................................ 96 Ketoprofenate based photocage ...................................................................... 98 HSV-1 treatment ............................................................................................. 98 Xanthonate based photocage .......................................................................... 99

Ultrasensitive Laser Analysis of Nanostructures: Theoretical Background and Experimental Performance .......................................... 107 Sergey V. Gaponenko 1 2 3

Introduction .................................................................................................. 107 Spontaneous emission and scattering of photons in terms of quantum electrodynamics ......................................................................... 108 Field enhancement in metal-dielectric structures ......................................... 109

CONTENTS

4 5 6

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Density of states effects on emission and scattering of photons .................. 113 Experimental performance of enhanced photoluminescence ....................... 117 Conclusions .................................................................................................. 119

Laser–Matter Interaction in Transparent Materials: Confined Micro-explosion and Jet Formation ......................................................... 121 Ludovic Hallo, Candice Mézel, Antoine Bourgeade, David Hébert, Eugene G. Gamely, Saulius Juodkazis 1 2 3

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6 7

Introduction .................................................................................................. 122 Experimental evidence of void formations in solid dielectrics .................... 123 Laser–matter interactions inside a bulk of a solid at high intensity ......................................................................................................... 124 3.1 Delivery of the laser beam to the focal area inside a solid: Limitations imposed by the self-focusing ................................ 125 3.2 Laser interaction zone ......................................................................... 126 3.3 Absorbed energy density .................................................................... 127 3.4 Ionization thresholds ........................................................................... 128 3.5 Absorbed energy density, electron energy and pressure in the focal region ............................................................................... 129 Hydrodynamic process ................................................................................. 130 4.1 Shock wave formation ........................................................................ 130 4.2 Shock wave expansion and stopping .................................................. 130 4.3 Rarefaction wave: Formation of void ................................................ 131 3D Maxwell and hydrodynamic modeling ................................................... 132 5.1 Discussion on Equations of State (EOS) parameters .......................... 134 5.2 Computations with Quotidian Equations of State (QEOS) ................ 134 5.3 Computations with SESAME ............................................................. 135 5.4 Parametric study ................................................................................. 137 5.5 EOS validation and design .................................................................. 140 5.6 EOS study using an ultra-short low energy laser ............................... 142 Application to deposition of nano-objects .................................................... 142 Conclusions .................................................................................................. 144

Influence of the Cut-Off Wavelength on the Supercontinuum Generation in a Highly Non-linear Photonic Crystal Fiber ................... 147 Rim Cherif, Mourad Zghal, Luca Tartara, Vittorio Degiorgio 1 2 3 4 5

Introduction .................................................................................................. 147 Experimental set-up and fiber properties ..................................................... 149 Experimental results ..................................................................................... 151 Comparison between experiment and simulation ........................................ 154 Conclusions .................................................................................................. 157

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Symmetry and the Local Field Response in Photonic Crystals .................... 161 Jeffrey F. Wheeldon, Henry P. Schriemer 1 2 3

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Introduction .................................................................................................. 161 Periodic symmetry and the photonic crystal point group ............................. 163 2.1 Group theory fundamentals ................................................................ 163 2.2 Transformation operators and the eigenvalue problem ...................... 165 Site symmetry and point subgroups of the photonic crystal ........................ 171 3.1 Transformation operators of the site symmetry groups ...................... 171 3.2 Space group: Crystallographic orbits and Wyckoff positions ............ 173 3.3 Eigenmode symmetry transformations at secondary axes .................. 178 Expressing polarization singularities with group theory .............................. 181 4.1 The local polarization state ................................................................ 181 4.2 Symmetry transformation and the local field ..................................... 183 4.3 Singularities at special Wyckoff positions ......................................... 187 Conclusions ................................................................................................... 190

On the Photonic Dispersion of Periodic Superlattices Made of Left-Handed Materials .......................................................................... 193 Solange B. Cavalcanti, Ernesto Reyes-Gómez, Alexys Bruno-Alfonso, Carlos A. A. de Carvalho, Luiz E. Oliveira 1 2 3 4 5

Introduction ................................................................................................... 194 Periodic systems ........................................................................................... 196 Quasiperiodic sequences .............................................................................. 201 Oblique incidence ......................................................................................... 203 Conclusions .................................................................................................. 205

Slow Light Propagation and Disorder-Induced Localization in Photonic Crystal Waveguides .................................................................................... 209 Mark Patterson, Stephen Hughes, Sylvain Combríé, Nguyen-Vi-Quynh Tran, Alfredo De Rossi, Renaud Gabet, Yves Jaouën 1 2 3 4 5 6 7 8

Introduction ................................................................................................... 209 Theory ........................................................................................................... 213 Experimental device ..................................................................................... 215 Transmission spectra .................................................................................... 216 Time–frequency reflectance maps ................................................................ 217 Localization .................................................................................................. 219 Connection to slow-light effects in metamaterial waveguides ..................... 220 Conclusion .................................................................................................... 221

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Silicon Photonic Waveguide Structures and Devices: From Fundamentals to Implementations in Spectroscopy and Biological Sensing ................. 225 Pavel Cheben, Adam Densmore, Jens H. Schmid, Dan-Xia Xu, André Delâge, Mirosław Florjańczyk, Siegfried Janz, Boris Lamontagne, Jean Lapointe, Edith Post, Martin Vachon, Philip Waldron 1 2

3

4 5

Silicon photonics: Promises and challenges ................................................. 225 Subwavelength structures in silicon waveguides ......................................... 229 2.1 Subwavelength grating (SWG) principle ............................................ 229 2.2 Subwavelength grating waveguides for fiber-chip coupling .............. 230 2.3 Antireflective and highly reflective waveguide facets ....................... 231 Silicon planar waveguide spectrometers ...................................................... 232 3.1 Arrayed waveguide grating spectrometer ........................................... 232 3.2 Interrogating a fibre Bragg grating sensor with SOI AWG spectrometer ........................................................................................ 234 3.3 Fourier-transform waveguide spectrometer ........................................ 235 Silicon photonic biological sensors .............................................................. 238 4.1 Optical sensors for label-free detection of biological molecules ....... 238 4.2 Evanescent field silicon photonic wire waveguide biosensors ........... 239 Conclusions .................................................................................................. 242

Index .................................................................................................................... 247

Preface This book arises from the 2008 NATO Advanced Study Institute in Laser Control & Monitoring in New Materials, Biomedicine, Environment, Security and Defense which took place in Ottawa, Canada, from November 24th to December 5th, 2008. Leading experts in the manipulation of light offered by recent advances in laser physics, nano-science, and applications, were invited to give lectures in their fields of expertise and participate in discussions on current research, applications, and new directions. The sum of their contributions to this book is a primer for the state of scientific knowledge and the issues within the subject of photonics taken to the extreme frontiers: molding light at the ultra-finest scales, which represents the beginning of the end to limitations in optical science for the benefit of 21st Century technological societies. Laser light is an exquisite tool for physical and chemical research. In the past 5 years physicists have developed pulsed lasers with such short durations that one laser shot takes the time of one molecular vibration or one electron rotation in an atom. The new laser methods make it possible to look inside an individual atom or molecule and observe its internal electronic structure. The technique of quantum control exploits lasers to prepare the state of atoms and molecules and manipulate the interference of competing pathways enabling inter alia the study of physical processes and new chemical reactions. In parallel, advances in micro- and nano-structured photonic materials allow the precise manipulation of light on its natural scale of a wavelength. Photonic crystals: – periodic dielectrics with a period on the scale of a wavelength: the analog for photons to the semiconductor crystals for electrons – allow the manipulation of the photonic density of state and, specifically, the introduction of a photonic ‘bandgap’, i.e. a frequency band over which light cannot propagate within the material. Related metamaterials – materials composed of sub-wavelength nanostructures – permit the manipulation of the dispersive properties of materials. Plasmonics exploits plasmon resonances (collective oscillations of an electron gas) that enable the tight sub-wavelength confinement of light. Often in combination, metamaterials have allowed the experimental confirmation of bizarre new effects such as slow light and negative refraction. These advances open a vista on a new era in which it is possible to engineer materials to control photons as precisely as it is now possible to control electrons using quantum confinement in semiconductors. Ultimately, one can expect techniques to be developed that exploit the control of electrons, photons and their interaction. In the context of Information and Communications Technology, one can envisage new devices for which the question: ‘are they electronic or photonic devices’ is undecidable. Laser light is not only a tool for matter research and material technology. Because of its high coherence and adjustable wave structure, laser light now finds

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a wide field of applications from nanophysics to material processing. Thin film transistors, light crystals and optical waveguides give just a few examples where femtosecond lasers lead to absolutely new materials and processing technologies. Recent advances of femtosecond laser techniques have opened new horizons for laser-based diagnostics of environment. The femtosecond and laser techniques have provided a major technological breakthrough in remote diagnostics of atmosphere and brought to life the first in the world mobile terawatt laser for atmospheric research. For many years coherent laser light has helped to efficiently diagnose the dynamics of explosion and combustion processes. There is a strong demand in search for directions, methods and ideas, which can lead to the development of new methods and technologies for global monitoring and remote control of atmosphere and sea/air pollution, and for rapid detection and localization of chemical and biological hazards as well as warfare agents. Optical coherence tomography and laser phototherapy have become routine in medicine, though just 5 years ago they were only research projects. Applications of highly coherent laser light and recently developed ultra-short lasers to biomaterials create new and very promising techniques of constructing and manipulating living cells and tissues. Today lasers can properly align the cells, engineer tissues, imprint biostructures, and even operate on cells. The objectives of the Advanced Study Institute (ASI) were to: (i) build a creative learning environment by bringing together worldwide recognized scientists, strongly motivated and internationally trained researchers who have already made a major contribution to the institute topics; (ii) highlight and explore important aspects of fundamental science and applications in the field of nanoscience, biomedicine, environment and security and defense, where the methods of laser control can lead to developing new technologies and diagnostic tools as well as creating new materials; (iii) provide participants with an opportunity for a deeper integration into the international research community and foster cooperation and the broad exchange of knowledge and ideas; (iv) create the environment to enter into new national/international collaborations and broaden existing ones, including collaborations with non-NATO countries. The ASI was held in Ottawa, the National Capital of Canada, a city with a major concentration of industry, government and academic organizations active in laser related science, engineering and technology. The Centre for Research in Photonics at the University of Ottawa (CRPuO) was honoured to have been granted the lead in the organization of the Institute and to play a prominent role in the planning and execution of the meeting. The University of Ottawa, the oldest and largest bilingual university in North America, is a place where languages, cultures and national and international perspectives come together to create a unique educational experience in the heart of Canada’s capital. Our bilingualism, multiculturalism and research excellence set us apart from other universities. At the University of Ottawa we recognize that innovation is increasingly the result of international cooperation in science and technology. This targeted academic and socio-economic transformation is achieved in part by identifying key partners and

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opportunities at the international level and by participating in collaborative projects and technology transfer. Featuring eminent scientists from 15 different countries, the NATO Advanced Study Institute aimed to translate cutting-edge research into practice in the real world. This international forum brought together students, scientists, and local technology companies to tackle key areas of growth, specifically the use of laser technologies for environmental diagnostics, threat detection, and biomedicine. Including Ottawa’s own wealth of talent such as Dr. Paul Corkum from the joint National Research Council of Canada (NRC) and University of Ottawa Attosecond Science Laboratory; Dr. Albert Stolow of NRC and University of Ottawa; colleagues Drs. Thomas Brabec, Ravi-Bhardwaj Vedula and Lora Ramunno in Physics; Dr. Tito Scaiano in Chemistry; and Drs. Henry Schriemer and Hanan Anis in the School of Information Technology and Engineering, the Institute was focused on cultivating a global and multidisciplinary approach to addressing these complex issues, providing both education and networking opportunities to academics, students and industry professionals alike. All lecturers and participants were invited to share their views during roundtable discussions and networking sessions. An additional open session (Advanced Applications and Technology Forum) of the ASI encouraged networking among specialists in related fields, with the intent to seed possible collaborations and follow-ups in the domains of material science, optics and polymer science. The Program was designed to perform coverage of the above topics, to offer intensive interactions of presenters and participants, to facilitate extensive discussions, networking and establishing novel cooperative and linkage initiatives, to foster transfer of advanced knowledge from academia to industry, to adjust basic research for urgent societal needs, to encourage integration of Partner Countries researchers into NATO countries research and development projects. It included regular tutorial lectures by invited speakers, selected oral contributions from participants, two poster sessions for participants, and the above mentioned open session called Advanced Technology and Applications Forum, where international cooperation, academia – industry interactions, and general issues on comercialization of technologies were discussed. Roundtable discussions included as well NATO Science for Peace and Security Programme overviews (Department of Foreign Affairs and International Trade Canada) and International Science and Technology Centres activities in former USSR countries as efficient managerial and financial tools towards integration of academia and industry, primary means for commercialization of research breakthrough and cooperation. In addition, tours at NRC Canada and University of Ottawa laboratories provided direct access to cutting-edge laser research aimed at ultrafast lasers applications for ultrasensitive probing of electron properties in matter. Meeting and discussions during this kind of tours enables nucleation of emerging cooperation of ASI attendees with Canadian centres of excellence. Networking stimulated new linkages between participants, exchange of knowledge and ideas gained in the field of laser

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applications as well as nucleation of novel bi-national and international targetoriented research with potential applications for new materials, biomedicine, environmental monitoring, defense and security. The Crowne Plaza Hotel located in downtown Ottawa, close to the University of Ottawa and other institutions such as the National Research Council laboratories, was found to be a perfect choice as a location for this meeting. The beautiful location with a panoramic view of Ottawa and the Ottawa Valley allowed the participants to work in a relaxed and motivating atmosphere and establish many fruitful personal research contacts and discuss in more details the topics of the lectures. The overseas participants had the opportunity to be exposed to the cultural diversity of the host country. During the Institute, it was demonstrated that lasers offer unique options of study of matter properties by means of precise resonant excitation of atoms, molecules, and solids; probing and controlling their excited states; extreme spatial confinement of energy; high localization of electromagnetic field; ultrashort time scale of light–matter interactions and control; and distant access to the species under investigation and control. Lasers were shown to find extensive applications in control and monitoring in material science, biomedicine, environmental research, security and defense. All of the testimonies received from participants were very positive, not only about the formal learning experience but also the opportunity for interactions with scientific leaders in the field and with fellow participants. In the first half of this volume the reader will find chapters that explain stateof-the-art methods of ultrashort laser pulse generation and characterization; and how such pulse generation enables the observation and control of excited state dynamics in atoms, molecules and biomolecules. New microscopies and microanalytical techniques that exploit quantum control spectroscopy and nonlinear optical effects are covered in detail. Applications ranging from anti-viral release from photocages to laser materials processing are touched upon. The second half of the volume changes tack and contains chapters describing the state-of-the-art in the spatial manipulation of light within nanostructured photonic materials; concluding with a chapter on silicon photonics: an integration platform that is the closest to engineering feasibility and is capable of subwavelength confinement of light and hence extreme miniaturization. I would like to take this opportunity to thank all the presenters and participants for their contribution to the success of the Institute. In particular, special appreciation must go to those presenters, all busy people, who nevertheless found time to contribute to this book. I am very grateful to my Co-Director, Dr. Sergey Gaponenko, for his contribution to the organization of the meeting and for graciously helping out with the Chairing. Thanks also to Dr. Paul Corkum and Dr. Stolow for their work on the Scientific Programme committee. The event was a major financial undertaking and it would not have been possible to hold without the considerable financial support provided by the North Atlantic Treaty Organisation (NATO), the Department for Foreign Affairs and International Trade Canada; the International Science & Technology Centre in

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Russia (ISTC); the Science & Technology Centre in Ukraine (STCU); the Canadian Institutes of Health Research (CIHR-IRSC); the Photonics Centre of the Ontario Centres of Excellence (OCE); the Canadian Institute for Photonic Innovations (ICIP-CIPI); the University of Ottawa (uOttawa); and several industrial sponsors (Delta Photonics, Enablence Technologies Inc., Ontario Photonics Industry Network, Ottawa Centre for Research and Innovation, OZ Optics Limited, Photonics Media, Pikaia Systems Inc., Technix by CBS, SPIE). I also extend thanks to the Natural Sciences and Engineering Research Council (NSERC-CRSNG) of Canada who supported a Strategic Workshop as part of the Applications Forum which was co-located with the NATO ASI. I am also indebted to my staff and student volunteers that helped out with the organization of the meeting. Special thanks go to Olga Arnaudova who was responsible for logistics; my assistant Erin Knight who recruited and organised the student volunteers and who was willing to work all hours to ensure that the administration of the event ran smoothly; Robert Radziwilowicz for supporting the events with the audio-visual and IT equipment requirements; Dr. Sofia A. Paredes for creative design, website editing, programme booklet and lecture note production; and Dr. Sawsan Abdul-Majid who with Olga, Erin, Sofia and many volunteers helped meet the strain of staffing the desk over the 10 working days of the Institute. A very special thank you also goes to Sofia Paredes as my editorial assistant. Without her sterling effort this book would never have come to fruition. Prof. Dr. Trevor J. Hall NATO ASI Co-director Centre for Research in Photonics – University of Ottawa (CRPuO) Ottawa, Canada, July 2009.

List of Abbreviations ACV AWG CARS CCD CEO CPA CSRS DFG DFWM DOS EIT EOS ESA ESI-MS FDTD FEM FFT FI FM FROG FTIR FWHM FWM HHG HSV IR ISC ISRS LDOS LED LFP LHM LIFT LO NA nc-OPA

Acyclovir Arrayed Waveguide Grating Coherent Anti-Stokes Raman Scattering Charge-Coupled Device Carrier Envelope Offset Chirped Pulse Amplification Coherent Stokes Raman Scattering Difference Frequency Generation Degenerate Four Wave Mixing Density of States Electromagnetically Induced Transparency Equations of State Excited State Absorption Electrospray-Ionisation Mass-Spectrometry Finite-Difference Time-Domain Finite-Element-Method Fast Fourier Transformation Faraday Isolator Frequency Modulation Frequency Resolved Optical Gating Fourier-Transform IR Spectrometer Full Width at Half Maximum Four Wave Mixing High (Order) Harmonic Generation Herpes Simplex Virus Infrared Intersystem Crossing Impulsive Stimulated Raman Scattering Local Density of States Light Emitter Diode Laser Flash Photolysis Left-Handed Material Laser Induced Forward Transfer Local Oscillator Numerical Aperture Noncollinear Optical Parametric Amplifier xvii

LIST OF ABBREVIATIONS

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NIR NLSE NMR NR NSAIDs OLCR OPA OPCPA PBG PC PCF PDT PL PML PPG PPLN QCS QEOS QM SAC-SPIDER SCG SERS SHG SHG-FROG SLM SNOM SOC SOI SPIDER SPM SWG TA TE TFRM TL TM TPF

Near-IR Nonlinear Schrödinger Equation Nuclear Magnetic Resonance Nonresonant Nonsteroidal Anti-inflammatory Drugs Optical Low-Coherence Reflectometry Optical Parametric Amplifier Optical Parametric Chirped Pulse Amplification Photonic Band Gap Photonic Crystal Photonic Crystal Fiber Photodynamic Therapy Photoluminescence Polarization Mode-Locked Photolabile Protecting Groups Periodically Poled Lithium Niobate crystal Quantum Control Spectroscopy Quotidian Equations of State Quantum Mechanical Shaper-Assisted Collinear SPIDER Supercontinuum Generation Surface Enhanced Raman Scattering Second Harmonic Generation Second Harmonic Generation- Frequency Resolved Optical Gating Spatial Light Modulator Scanning Near-Field Optical Microscope Spin-Orbit Coupling Silicon-on-Insulator Spectral Phase Interferometry for Direct ElectricField Reconstruction Self-Phase-Modulation Subwavelength Grating Transient Absorption Transverse Electric Time-Frequency Reflectance Map Transform Limited Transverse Magnetic Two-Photon Fluorescence

LIST OF ABBREVIATIONS

TRPES UV XC XFROG XUV ZDW

Time-Resolved Photoelectron Spectroscopy Ultraviolet Cross Correlation Cross-Correlation Frequency Resolved Optical Gating Extreme Ultraviolet Region Zero-Dispersion Wavelength

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Ultrashort Laser Pulse Generation Olivier Chalus, Jens Biegert ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels, Barcelona, Spain [email protected], [email protected]

Abstract Ultrashort laser pulse generation is a pre-requisite for many physics applications ranging from high field physics to extreme nonlinear optics. Meanwhile the creation of few cycle pulses is still not trivial and should be handled with a lot of care. In this chapter, we will highlight two methods to generate few cycle pulses. The first one consists, in the near-IR, in the use of filamentation to broaden the spectrum of the initial pulse to allow recompression to shorter duration, but also technique of direct recompression from the filamentation. The second one involves a complete new approach to generate few-cycle pulses in the mid-IR using optical parametric chirped pulse amplification. We furthermore show recent attempts to adequately characterize such few-cycle pulses and limitations of the used methods.

Keywords: Ultrashort pulse, filamentation, mid-IR, optical parametric chirped pulse amplification, few cycle, compression

1

Filamentation

1.1 Introduction During the last decade, techniques for the generation of intense few-cycle laser pulses have emerged that led to dramatic improvements in many fields of research, and that have revolutionized high field physics and extreme nonlinear optics. High order harmonic generation (HHG) (Macklin et al. 1993; Antoine et al. 1996) is one of the applications for such short pulses, which has culminated in the generation of single attosecond pulses in the extreme ultraviolet region (XUV) (Drescher et al. 2001). HHG is the coherent up-conversion of optical frequencies into the XUV spectral region, and is typically achieved by strongly focusing an intense nearinfrared driving pulse into a noble gas. Many-cycle driving pulses (>30 fs) yielded attosecond pulse trains (Paul et al. 2001), whereas few-cycle pulses, paired with carrier envelope phase (CEP) stability (Telle et al. 1999; Jones et al. 2000), led to the generation of isolated attosecond pulses (Drescher et al. 2001). T.J. Hall and S.V. Gaponenko (eds.), Extreme Photonics & Applications, © Springer Science + Business Media B.V. 2010

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Intense few-cycle pulses are mostly generated using gas-filled hollow fibers (Nisoli et al. 1997). Recently, an alternative technique based on filamentation in a noble gas was demonstrated, which is simpler to implement and does not suffer from the drawbacks of coupling an intense pulse into a narrow and elongated waveguide (Hauri et al. 2004a). The experimental arrangement and laser requirements for filamentation are similar to the previous technique, just with the hollow fiber being removed. As a result, implementation is conceptually extremely simple. Filament compression in the few-cycle regime is surprising because the high plasma density generated has such strong dispersion that a few-cycle laser pulse would be expected to broaden very fast. In addition, the CEP phase of the compressed pulse is maintained or even further stabilized, a fact one would not have expected. Meanwhile, simulations reproduce the experimental results well (Hauri et al. 2004b; Couairon et al. 2006; Apolonski et al. 2000). Filamentation is understood as a competition between several physical effects: Localization in the transverse diffraction plane results from the dynamics of the Kerr effect, that leads to beam focusing, and Multi-photon absorption that leads to ionization and plasma induced beam defocusing (Braun et al. 1995). Both effects are schematically illustrated in Fig. 1. Under appropriate laser parameters and ambient medium conditions, this competition can induce a self-guiding effect over several Rayleigh lengths, i.e. the beam takes the form of an intense narrow core during the nonlinear interaction with the medium, surrounded by a weak part

Fig. 1 Mechanism for channel formation: For a typical beam intensity profile with a maximum on axis, the intensity dependent refractive index (n = n0 + ΔnKerr(I) = n0 + n2I) acts like a succession of increasingly converging lenses. In the absence of a limiting effect, this would lead to a beam collapse on axis if the pulse power exceeds a critical value PCR = λ2/2πn0n2, which is several Gigawatt in air at 800 nm. On-axis beam collapse is arrested by multi-photon ionization. This occurs typically at intensities around 1013 W/cm² in gases, giving rise to a weakly ionized plasma with an electron density typically around 1016 cm−3, and a corresponding reduction of the local refractive index Δn = −N(I)/2NCR~10−4. Here, N(I) denotes the intensity dependent free electron density and NCR, the density above which the plasma becomes opaque. Thus, multiphoton ionization acts as a strong regulating mechanism, limiting the beam intensity on axis and the dissipation of laser energy during propagation.

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propagating linearly. Similar to the hollow-fiber technique, self-phase modulation during filamentation broadens the incident pulse spectrum to more than an optical octave, while self-guiding acts as a spatial filter leading to an improvement in spatial mode quality (Prade et al. 2006). Furthermore, since the pulse intensity is clamped inside the filament within a very narrow range (Lange et al. 1998), the intensity stability is improved by the process. The energy throughput tends to be slightly higher compared to the hollow fiber technique. The complex processes taking place during filamentation can lead to a variety of different effects, which more and more groups have begun to investigate. Of particular interest is the effect of self-compression to the few-cycle regime within the filament, which was predicted by numerical modeling (Hauri et al. 2004a). Theoretical analysis as well as numerical simulations suggest that the use of a longitudinally non-uniform gas distribution could lead to efficient selfcompression down to the single-cycle limit, thereby eliminating the need for any external chirped mirror compression (Couairon et al. 2005). Power scaling to above 1 mJ has been demonstrated as well (Stibenz et al. 2006). Another attractive aspect of filaments is the possibility to generate tunable ultrashort pulses by filamentation employing two-color driving fields and four-wave mixing (Theberge et al. 2006). One additional aspect of femtosecond filamentation, which has no counterpart in external guiding (hollow fiber), is pulse self-compression (Mlejnek et al. 1999; Mikalauska et al. 2002). The different nonlinear effects occurring in filamentation lead to an important restructuring of the pulse time profile. Some of these effects, such as self-focusing, pulse self-steepening, and self-phase modulation, act instantaneously. These effects tend to accumulate laser energy to the ascending part of the pulse, whereas the time-delayed effects, such as photo-ionization and the Raman effect, tend to cut off its trailing part. Their combination eventually leads to the formation of pulses that are significantly shorter than the incident pulse. One important facet, previously overlooked, is the fact that pulse reshaping by filamentation can be effective down to the fundamental limit of nearly one optical cycle.

1.2 Filamentation experiment We found that optimum pulse shortening from 43 to 5.7 fs could be achieved by using two successive gas cells at pressures of 840 and 700 mbar of argon, as shown in Fig. 2. The CEP-phase-locked input pulse with an energy of 0.84 mJ was loosely focused into the first cell where it generated a 10–15 cm long filament roughly in the center of the cell, as estimated from the length of the scattered broadband continuum. The emerging spectrum was recompressed with chirped

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mirrors, resulting in a shortening by a factor four while retaining 94% of the input energy. Sending the 10.5-fs pulse into the second gas cell, another 15–20 cm long filament was formed, leading after chirped-mirror recompression, to a 5.7-fs pulse with 27% of the initial pulse energy in an excellent spatial profile. Figure 3a and b show a spectral phase interferometry for direct electric-field reconstruction (SPIDER) (Kornelis et al. 2003) measurement of the pulse temporal profile, spectral phase and spectrum. The excellent spatial quality of this pulse is revealed in the inset of Fig. 3a. CEP phase stability is confirmed through f-2f spectral interferometry (Telle et al. 1999; Mehendale et al. 2000) with the persistence of fringes confirming the phase-preserving nature of the filaments. Figure 3c shows a time series of such a measurement where the CEP-lock, with one feedback for the oscillator only, has been switched on after roughly 1.2 s. The measurement clearly confirms the formation and steadiness of fringes, hence corroborating CEP-phase conservation. Figure 3e shows the spectrum measured after the second gas-cell (before the chirped mirror compressor), optimized for maximal spectral width at a pressure of 1,060 mbar. It supports a transform-limited pulse duration of 1.75 fs shown in Fig. 3d, assuming a flat spectral phase. The spectrum after the first cell, at a pressure of 840 mbar, is shown in Fig. 3f.

Fig. 2 Experimental setup: Two cells are filled with argon at 840 and 700 mbar respectively. CEP-phase-locked 43-fs near-IR pulses with 0.84 mJ energy form a 10–15-cm long filament in the middle of the first cell and are compressed with chirped mirrors to 10.5 fs at 790 µJ. Sending those pulses into the second cell results in a filament also 15–20 cm long that, after final recompression, leads to a 5.7-fs pulse with 45% of the initial pulse energy in an excellent spatial profile.


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We attribute the observed differences between results shown in Fig. 3a and d, respectively Fig. 3b and e to a large extent as follows: First, the difference in the spectra is obviously due to the chirped mirrors. The phase characteristics of the chirped mirrors used in the experiment are insufficient to produce a pulse much shorter than 6 fs. This limitation not only modifies the spectrum but, more

Fig. 3 (a) Temporal intensity profile of compressed pulses as obtained with SPIDER for pressure of 840 and 700 mbar of Argon. The full width at half maximum (FWHM) pulse duration of 5.7 fs corresponds to 2.1 cycles. The spatial profile, measured with a high resolution charge-coupled device, CCD (WinCam, DataRay), is excellent as can be seen in the inset. The associated spectrum and spectral phase are given in (b). (c) Time series from f-2f spectral interferometry measurement. The CEP-phase lock is switched on after roughly 1.2 s with the emergence of fringes confirming the CEP phase lock. (d) Transform-limited pulse duration supported by the normalized spectrum after the second gas-cell, shown in (e), before the chirped mirror compressor, assuming a flat phase, perfect bandpass and phase characteristics of the chirped mirrors. (f) Normalized spectrum after the first cell. The pressures in the first and second cell were optimized for maximal spectral broadening and were measured to be 840 and 1,060 mbar respectively.

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importantly, limits the compression. The dispersion introduced by the mirrors being known, it will be possible to design and fabricate appropriate chirped mirror (Matuschek et al. 2000; Sansone et al. 2004) structures to obtain better compression. Secondly, the dispersion in the exit window, consisting of a 0.7 mm thick fused silica plate at Brewster angle, has a significant effect on such ultra broadband pulses. For instance, it increases the duration of a transform-limited single-cycle pulse sevenfold. The robustness of the present experimental technique to obtain shorter pulses with respect to variations in input pulse and operating parameters is essential. We have therefore carefully checked the influence of these changes on the final pulse energy, duration and spatial profile and have found that this method is surprisingly insensitive to such changes. Therefore self-compression through filamentation is a very robust and reliable method to generate intense few-cycle pulses, e.g. a change in pressure of 100 mbar influenced the pulse duration by 0.2 fs only.

1.3 Numerical simulation of filamentation The measured results can be well reproduced by a code calculating the propagation of intense short laser pulses in a transparent medium (Baltuska et al. 2002; Mechain et al. 2004). Our code solves the 3D nonlinear envelope equation describing the evolution of the field envelope. This approach has been demonstrated to be valid down to the single-cycle limit (Brabec and Krausz 1997). It accurately reproduces experimental results on filamentation in gaseous and solid media. The model includes the physical effects of diffraction, group velocity dispersion, self-focusing, self-steepening, space-time focusing, Raman scattering, ionization, plasma defocusing as well as photo-absorption and plasma recombination. We have calculated the propagation inside the second gas cell, by using various input conditions close to the measured input pulse parameters, such as the pulse amplitude and phase (obtained from a SPIDER measurement). Figure 4a gives the calculated spatial distribution of laser intensity, which confirms the formation of a filament. Figure 4b and c show the spatio-temporal intensity distribution at two different propagation distances. The temporal evolution is illustrated by the transition between Fig. 4b and c. We find specific locations where self-shortening of the laser pulse takes place over several centimeters with a minimum pulse duration nearing 1.3 optical cycles. Interestingly, the calculated phase front of an isolated self-compressed pulse such as in Fig. 4c reveals a near flat surface over several centimeters, evolving into a parabolic shape (diverging beam). In fact, the comparison between numerical and experimental results shows that the pulse of Fig. 3a with a diverging beam of excellent quality was collected at a location close to optimal. It suggests that the final external compression stage is required by the presence of the exit window.


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Fig. 4 Modeled pulse propagation for second gas cell: (a) Intensity profile confirming the formation of filament. (b, c) Space time profile of the pulse at two locations inside the filament. (b) is calculated 92 cm from the focusing mirror for the second cell. Further propagation (25 cm) leads to a tightly spatially and temporally confined light bullet.

1.4 Measurement of ultrashort pulse From the previous discussion and the simple picture of a filament being a dynamic equilibrium between various effects, it is safe to assume that its spatio-temporal properties might vary significantly. We found it important to corroborate our theoretical findings through a detailed experimental investigation by performing a transverse scan by a 1-mm aperture in 100-µm steps through the 4 mm diameter output beam. The selected output beam is sent to the SPIDER in order to measure the transmitted temporal profile. Figure 5 shows the evolution of the temporal profile as a function of the transverse iris position in the output beam. A single 4.9-fs pulse can be measured only by selecting the center part of the beam. The temporal structure appearing on either sides of this short pulse remains below 10% of the peak intensity. Towards the outer parts of the beam, the temporal profile changes until a double pulse is formed. It is clear that without any spatial selection of the short pulse component, throughput is maximized at the expense of pulse contrast. Depending on the requirements for a particular application, one has to trade-off pulse duration versus contrast and energy throughput. This method, even though being conceptually simple still requires external recompression due to the dispersion of the cell windows. We have therefore numerically studied a simpler setup that allows for filamentation and propagation in vacuum directly to the experiment; i.e. HHG or some strong field physics experiment inside another vacuum chamber.

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Fig. 5 Spatial dependence of the temporal profile measured at the output of the two-stage filamentation system. When the central part of the filament beam is selected, the profile corresponds to a single 4.9-fs pulse (at x = 0 mm transverse position). Towards the outer part of the beam (x > 0 mm transverse position), the energy is distributed on satellites structures leading to a double-pulse structure in the temporal profile.

1.5 Self-compression by filamentation Instead of producing a filament in a cell filled with a gas of uniform density, we investigate the case of a gas density gradient along the propagation axis. We show that this design gives even better pulse compression, avoids cyclic compression stages, and therefore limits the energy losses. The scheme permits easy extraction of the isolated single optical cycle pulse for delivery to an interaction chamber. To evaluate the pulse’s temporal characteristics, we perform a numerical simulation with a three-dimensional nonlinear envelope propagation code (Mechain et al. 2004; Couairon et al. 2002, 2003). We have concentrated on the nonlinear propagation of a femtosecond laser pulse at 800 nm for a large number of configurations of gas density gradients, with a practical design as guideline. The parameter correspond to those of argon gas with potential Ui = 15.76 eV, nonlinear index n2 = 4.9 × 10−19 p (cm2/W), where p denotes the pressure in bars, and multiphoton ionization rate σ11, in agreement with Keldysh’s formulation. For the sake of simplicity we always chose the same incident laser characteristics, which correspond to presently commercially available kilohertz laser systems and adopt a


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converging laser geometry. The incident laser pulse had a duration of 30 fs and an energy of 1 mJ. The beam of waist w0 = 2 mm is focused with a lens f = 80 cm. The input pulse is assumed to have a Fourier limited bandwidth. Several particularly favorable cases emerge from computations performed with a large number of probed gradient configurations. To illustrate these cases, we consider the argon gas density shown by dotted lines in Fig. 6. It has a steep front edge followed by a smoother decreasing density gradient. The geometric focus is at z = 80 cm, slightly before the maximum gas density at z = 85 cm. The computed width (FWHM) of the propagating beam is shown by full lines in the same figure. The high intensity at the front edge leads to ionization and beam defocusing, immediately followed by a refocusing stage. Figure 7 shows the peak intensity and electron density produced in the argon gas as a function of the propagation distance. Ionization, through plasma defocusing that saturates Kerr self-focusing, fixes the upper limit of the pulse intensity to a value I = 6 × 1013 W/cm2. The electron density reaches 4 × 1017 cm−3 at the focus, which produces a defocusing effect sufficient to slow the refocusing dynamics occurring at constant pressure filamentation (Mlejnek et al. 1998). A plateau at about 1017 cm−3 is reached and sustained over the whole extent of the pressure gradient, thus maintaining the conditions for an efficient pulse compression.

Fig. 6 Pressure distribution (dotted curve, scale on the right-hand axis) and computed beam width (continuous curve, scale at the left) along the propagation axis. The linear focus is at z-80 cm.

Fig. 7 Peak intensity (continuous curve, left-hand axis) and electron density (dashed curve, scale at the right) as functions of the propagation distance.

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The efficiency of the shortening can be seen on Fig. 8 that shows the computed pulse duration as a function of the propagation distance when the beam intensity is integrated over 100 μm. The shortening process is clearly visible for 80 ≤ z ≤ 120 cm. The core of the beam (i.e. the partial power contained in the cylinder of radius r < 100 µm) reaches a single cycle pulse duration during the descending part of the pressure gradient. Figure 9 shows the temporal profile of the pulse intensity integrated over a radius of 100 µm, obtained at z = 115 cm. For comparison, the input pulse is shown in dashed line. The compression ratio in this case is about 12. The short pulse generated in the pressure gradient persists upon further propagation into vacuum, where its peak intensity starts decreasing slightly because of diffraction (z > 130 cm). This single cycle pulse can be easily used since it is long lived over a large distance, therefore avoiding any critical positioning of an extraction setup. Since the pressure gradient along z ensures a smooth transition between gas and vacuum, an “interaction chamber” can simply be achieved by adding a transverse gas jet at any location beyond the decreasing part of the gradient. We have described an efficient self-compression scheme by filamentation based on the presence of suitable pressure gradients. The process yields single cycle optical pulses.

Fig. 8 Evolution of the pulse duration (FWHM) as a function of the propagation distance. The duration is computed after radial integration of the beam intensity over a beam radius of 100 μm.

Fig. 9 Normalized intensity of the single-cycle pulse at z-115 cm. The dashed curve indicates the input pulse duration. The pulse has been averaged over the radius of 100 μm.


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Ultrashort pulse source in the mid-IR

Laser technology has advanced in recent years, enabling the production of near-IR pulse durations corresponding to only a few cycles of the laser field, at moderately high repetition rates and with stabilised CEP. These sources are nearly exclusively based on chirped pulse amplification (CPA) in Ti:Sapphire with subsequent broadening via gas-filled hollow fibres (Nisoli et al. 1998) or filamentation (Hauri et al. 2004a) and compression with chirped mirrors. State of the art pulse durations at centre wavelengths in the visible to near-IR currently lie in the few-cycle range at repetition rates up to a few kHz (Kling and Krausz 2008), but there exist significant motivations for the development of much less complex to operate new sources, particularly emitting at different wavelengths such as in the mid-IR. A few-cycle Mid-IR source has recently been demonstrated via four wave mixing in a filament (Fuji and Suzuki 2007), but this is based on a Ti:Sapphire CPA system, inheriting many of the associated disadvantages, and may not be scalable to high energies. Mid-IR wavelengths are interesting, e.g. from an atomic physics application point of view, since they allow for a much clearer investigation of tunneling processes, whereas near-IR pulses operate in a mix of multi-photon and tunneling regimes. Attosecond pulses with a carrier frequency corresponding to extreme ultraviolet wavelength can be produced from short-pulse laser systems, using high order harmonic generation (HHG) as coherent up-shifting mechanism from the near-IR drive laser (McPherson et al. 1987; Ferray et al. 1988). Here, changing drive laser wavelengths to the mid-IR is expected to yield shorter attosecond pulses due to a square of wavelength dependence of the shortest wavelength reachable via HHG (Sheehy et al. 1999; Gordon and Kartner 2005). Recent experiments have confirmed this scaling of the harmonic cutoff with drive wavelength, while showing that predicted losses in harmonic yield (Tate et al. 2007) can be compensated by taking advantage of more favourable HHG phasematching at longer wavelengths (Popmintchev et al. 2008). Based on their results we expect a 3 µm source to generate harmonic spectra extending to a photon energy well above 1 keV. Higher repetition rates help to improve signal to noise ratio for most experiments, but they are also essential for some in strong field physics; for instance, particle coincidence experiments with reaction microscopes (COLTRIMS) (Moshammer et al. 1996) permit the investigation of atomic and molecular processes with unprecedented scrutiny, but are limited mainly by the stability of current lasers due to the low cross sections of the processes under investigation; the measurement time is, in practice, nearly always longer than the time over which the best lasers can deliver constant performance. Using a 100 kHz repetition rate, experiments taking 6 days with a 1 kHz system can be completed in 90 min, greatly reducing the demands on the laser system stability.

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Maintaining CEP stability is a key difficulty for most few-cycle laser sources, with current state of the art feedback stabilised systems capable of CEP locked operation for several hours, though locked durations of tens of minutes to a few hours are more common. Total CEP stability of the laser source is essential for many experiments, e.g. the measurement of double ionization (Weber et al. 2000), which demands a few-cycle laser source to be phase-stable with negligible amplitude instabilities and drift over about 12 h with a 1 kHz repetition rate; an unrealistic requirement from current electronically CEP stablilised systems. To date, various techniques allow the conversion of few-cycle near-IR sources to longer wavelength, however these sources are based on Ti:Sa CPA sources, and inherit the associated problems of long term stability and operational complexity. Difference frequency generation between different components of the same pulse followed by parametric amplification can create phase stable pulses with millijoule energies and few-cycle durations (Vozzi et al. 2006), but this is limited to near-IR and visible wavelengths. Parametric amplification of a white-light continuum can also provide short pulses in the visible-2 µm range (Cirmi et al. 2008), but as of yet cannot provide pulses at a longer wavelength. With the above-mentioned limitations in mind, we have designed a completely different mid-IR (3.2 µm) source with the goal for long-term stability and compactness based on optical parametric chirped pulse amplification (OPCPA) (Dubietis et al. 1992; Ross et al. 1997). The use of OPCPA as the amplification technique allows broadband amplification at arbitrary wavelengths (Fuji et al. 2006), and does not use any gain storage as in a traditional laser system, thereby freeing the repetition rate of the laser from problems associated with thermal deposition in the amplifiers. Figure 10 shows the layout of this source, which incorporates the stability and hands-off operability of a fibre front-end, stability of a diode-based pump laser, and the use an intrinsically optically self-CEP-stable seed (Baltuska et al. 2002). The source generates 96 fs pulses at 3.2 µm (9.0 cycles) with an amplified spectrum supporting a Fourier transform limited pulse duration of 45 fs, or 4.2 cycles. Currently we measure compressed pulse energies

Fig. 10 Mid-IR OPCPA source layout. Two arms of a fibre laser are combined through DFG to generate 3.2 µm radiation. This is then amplified by a double stage OPCPA pumped by a Nd:YVO4 laser and finally compressed by a Martinez-type compressor.


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of 1.2 µJ, at a repetition rate of 100 kHz, while our use of OPCPA ensures this is inherently scalable to higher energies (Ross et al. 2002), with multi-joule near-IR OPCPA sources already demonstrated (Chekhlov et al. 2006). The mid-IR seed beam is generated by single path difference frequency generation (DFG) between a broadband beam centred at 1.07 µm and a beam at 1.58 µm, both of which are produced by a commercially available fibre laser (Toptica FFS) (Fig. 10). The fibre laser operates at 100 MHz rep rate and has two outputs, one providing 75 fs pulses at 1.58 µm with 160 mW average power while the second arm propagates identical pulses through a highly nonlinear fibre to generate a supercontinuum, from which a 1.0–1.15 µm spectral band is selected and compressed, yielding 65 fs pulses with an average power of 12.6 mW. To create the difference frequency, the 1.07 and 1.58 µm pulses from the fibre laser were combined in a 2 mm long, periodically poled, MgO-doped Lithium Niobate (PPLN) crystal (Chalus et al. 2008). The two beams were collinearly focused on to the crystal, achieving respectively an intensity of 0.3 and 3 GW/cm2. The resulting 3.2 µm DFG pulses are slightly down-chirped and have an average power of 1.6 mW at 100 MHz, corresponding to a quantum efficiency of more than 40% for the conversion process. The spectrum of the DFG pulse was measured using a Fourier-transform IR spectrometer (FTIR) and is shown in Fig. 11. The spectrum spans from 2.9–3.8 µm at the 1/e2 level, and corresponds to a transform-limited pulse duration of 33 fs or three cycles. The down-chirped DFG output is then further negatively stretched via propagation through an uncoated 10 cm long Sapphire rod, producing a 6 ps stretched pulse with which we seed the OPCPA amplifier chain.

Fig. 11 Spectra at each nonlinear stage. The mid-IR spectra after the difference frequency generation (red), first (orange) and second (blue) OPCPA stages.

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We amplify the stretched mid-IR seed beam, which is the idler wave, in two collinear geometry OPCPA amplifiers. The stability and beam quality of the pump laser in any OPCPA system is of crucial importance, and here we use acommercially available Nd:YVO4 mode-locked laser (Lumera), delivering 10 W at 100 kHz with a pulse duration of 8.7 ps, and an extremely good spatial profile with an M2 of 1.1. The pump laser is electronically synchronised to the fibre laser, using the cavity of the pump as reference on which the oscillator of the fibre laser is slaved. The RMS timing jitter between the two laser pulse trains is typically sub-350 fs, less than 5% of the pump pulse duration. In the first parametric amplification stage the mid-IR seed is focused into a PPLN crystal identical to the DFG crystal. The pump beam power was set to 2.6 W via a series of waveplates and polarisers, generating a focussed intensity of 60 GW/cm2 at the crystal, producing a gain of 4 × 103. The amplified mid-IR beam is recollimated and dielectric filters are used to reject the pump beam and any unwanted frequencies before propagation to the second OPCPA stage. The second OPCPA stage again uses a 2 mm long PPLN crystal at the focus of the pump and seed beams, with identical characteristics to the DFG stage. The seed and pump beams are focussed to intensities of 83 MW/cm2 and 61 GW/cm2 respectively. As in the first stage the beam is recollimated and passed through dielectric filters. The measured pulse energy after the two amplifier stages is 2.5 µJ at a repetition rate of 100 kHz. The amplified spectrum is shown in Fig. 11, and it spans from 2.9–3.3 µm, at the 1/e2 level, corresponding to a transform-limited pulse duration of 45 fs, or 4.2 cycles at 3.2 µm (Chalus et al. 2009). The loss of bandwidth on the long wavelength side of the spectrum is due to the phase-matching in the amplification process and affects only slightly the transform limited pulse duration. The modulation seen on the spectrum has been identified as originating in the fibre laser, and is not an artifact of the OPCPA process – we hope to resolve this issue in the near future. To compress the down-chirped amplified pulse we used a Martinez type 4-f grating sequence with two 200 lines/mm gold coated gratings designed for 3 µm. The total efficiency of the compressor is 48%, giving a compressed energy of 1.2 µJ, with most losses due to the uncoated lenses in the setup. The pulse duration was measured using SHG-FROG (Fig. 12), and with our simple compressor, we have measured pulses as short as 96 fs, corresponding to 9.0 cycles at 3.2 µm, and within a factor of two of the 45 fs, 4.2 cycle transform limited pulse duration (Fig. 13). To remove the time-reversal ambiguity in the FROG we directed the optimally compressed pulse through an 8 mm thick Silicon plate. Comparing the change in the FROG retrieved phase with the known phase introduced by the silicon allowed us to determine the correct direction for the time axis. We also measured a knife-edge spatial profile of the compressed pulse 1.5 m after exiting the compressor (Fig. 13 inset), showing an excellent beam quality, with close to a Gaussian profile.


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(b)

(a)

Fig. 12 Measured and retrieved FROG traces. (a) Measured FROG trace of the mid-IR pulse – the colour scale represents the square root of the normalized intensity, chosen to enhance the small features (b) Retrieved FROG trace of the id-IR pulse using the same colour scale.

(a)

(b)

Fig. 13 Reconstructed pulse from the FROG measurement. (a) Temporal profile and instantaneous frequency retrieved by the FROG: 96 fs FWHM = 9.0 cycles (at 3.2 μm 1 cycle = 10.7 fs) The spatial profile is shown in inset, width of image corresponds to 5 mm. (b) Spectrum and phase retrieved by the FROG.

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The energy achieved is sufficient for spectroscopy (Thorpe and Ye 2008) as well as for some strong field physics experiments, since focusing to a few times the diffraction limit already yields intensities in excess of 1013 W/cm2, and the total energy by no means represents the achievable limit from our system. As shot noise in experiments typically scales inversely with repetition rate, our 100 kHz system can produce data with higher signal to noise ratio than would be expected from comparable energies at kHz repetition rates.

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Nisoli, M., de Silvestri, S., Svelto, O., Szipocs, R., Ferencz, K., Spielmann, Ch., Sartania, S., Krausz, F.: Compression of high energy laser pulses below 5 fs. Opt. Lett. 22(8), 522–524 (1997) Nisoli, M., Stagira, S., De Silvestry, S., Svelto, O., Sartania, S., Cheng, Z., Tempea, G., Spielmann, Ch., Krausz, F.: Toward a Terawatt-scale sub-10 fs laser technology. IEEE J. Sel. Top. Quant. Electron. 4, 414–420 (1998) Paul, P.M., Toma, E.S., Breger, P., Mullot, G., Auge, F., Balcou, Ph., Muller, H., Agostini, P.: Observation of a train of attosecond pulses from high harmonic generation. Science 292(5522), 1689–1692 (2001) Popmintchev, T., Chen, M., Cohen, O., Grisham, M., Rocca, J., Murnane, M., Kapteyn, C.: Extended phase matching of high harmonics driven by mid-infrared light. Opt. Lett. 33(18), 2128–2130 (2008) Prade, B., Franco, M., Mysysrovicz, A., Couairon, A., Buersing, H., Eberle, E., Krenz, M., Seiffeur, D., Vasseur, O.: Spatial mode cleaning by femtosecond filamentation in air. Opt. Lett. 31(17), 2601–2603 (2006) Ross, I., Matousek, P., Towrie, M., Langley, A., Collier, J.: The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers. Opt. Commun. 144(1–3), 125–133 (1997) Ross, I., Matousek, P., New, G., Osvay, K.: Analysis of optimization of optical parametric chirped pulse amplification. J. Opt. Soc. Am. B 19(12), 2945–2956 (2002) Sansone, G., Benedetti, E., Calegari, F., Vozzi, C., Avaldi, L., Flammini, R., Poletto, L., Villoresi, P., Altucci, C., Velotta, R., Stagira, S., De Silvestri, S., Nisoli, M.: Isolated singlecycle attosecond pulses. Science 314(5798), 443–446 (2006) Sheehy, B., Martin, J., DiMauro, L., Agostini, P., Schafer, K., Gaarde, M., Kulander, K.: High harmonic generation at long wavelengths. Phys. Rev. Lett. 83(25), 5270–5273 (1999) Stibenz, G., Zhavoronkov, N., Steinmeyer, G.: Self-compression of millijoule pulses to 7.8 fs duration in a white-light filament. Opt. Lett. 31(2), 274–276 (2006) Tate, J., August, T., Muller, H., Salieres, P., Agostini, P., DiMauro, L.: Scaling of wave-packet dynamics in an intense midinfrared field. Phys. Rev. Lett. 98, 013901 (2007) Telle, H.R., Steinmeyer, G., Dunlop, A.E., Stenger, J., Sutter, D., Keller, U.: Carrier-envelope offset phase control: a novel concept for absolute optical frequency measurement and ultrashort pulse generation. Appl. Phys. B 69(4), 327–332 (1999) Theberge, F., Akozbek, N., Liu, W. Becker, A., Chin, S.: Tunable ultrashort pulses generated through filamentation in gases. Phys. Rev. Lett. 97, 023904 (2006) Thorpe, M.J., Ye, J.: Cavity enhanced direct frequency comb spectroscopy. J. Opt. Soc. Am. B 91(3–4), 397–414 (2008) Vozzi, C., Cirmi, G., Manzoni, C., Benedetti, E., Calegari, F., Sansone, G., Stagira, S., Svelto, O., De Silvestri, S., Nisoli, M., Cerullo, G.: High-energy, few-optical-cycle puses at 1.5 μm with passive carrier envelope-phase stabilization. Opt. Express 14(21), 10109–10116 (2006) Weber, T., Khayyat, K., Dorner, R., Mergel, V., Jagutzki, O., Schmidt, L., Afaneh, F., Gonzalez, A., Cocke, C., Landers, A., Schmidt-Bocking, H.: Kinematically complete investigation of momentum transfer for single ionization in fast proton–helium collisions. J. Phys. B 33(17), 3331–3344 (2000)

Femtosecond Dynamics and Control: From Rydberg Molecules to Photochemistry and Photobiology Helen H. Fielding Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, United Kingdom [email protected]

Abstract Significant advances in laser technology have led to an increasing interest in the observation and control of excited state dynamics in atoms, molecules and biomolecules. We begin by describing a simple intuitive control scheme based on an analogue of Young’s double-slit experiment in a Rydberg atom. We then describe how sequences of optical pulses can be employed to control the angular momentum of Rydberg electron wave packets, and how phase-shaped optical pulses can be exploited to localise a Rydberg wave packet in time and space. Finally, we describe recent experiments unravelling the femtosecond dynamics of benzene in its first electronically excited state, and our progress in the development of a new experiment for observing the femtosecond dynamics of biological chromophores in their native protein environments.

Keywords: Rydberg, photochemistry, ultrafast molecular dynamics, time-resolved photoelectron spectroscopy

1

Introduction

The typical vibration period of a covalent bond in a molecule is of the order of tens of femtoseconds. The application of femtosecond lasers to probe molecular dynamics on this timescale was pioneered by Ahmed Zewail in the 1990s and its significance was recognised by the award of the 1999 Chemistry Nobel Prize (Zewail 2000). A key challenge for the 21st century is to develop a more detailed understanding of molecular dynamics at the level of quantum mechanics and to control molecular processes.

T.J. Hall and S.V. Gaponenko (eds.), Extreme Photonics & Applications, © Springer Science + Business Media B.V. 2010

19

20

H.H. FIELDING

1.1 Coherent control The term coherent control describes optical methods in which quantum interference effects are used to control atomic and molecular processes. The motivation behind the concept of coherent control is to achieve a superior fundamental understanding of atoms and molecules and their interactions with light and, potentially, to control the products of photochemical reactions. An early example of an optical control experiment is Young’s double slit experiment (Young 1804). In this experiment, a monochromatic plane wave is incident on two closely spaced slits, from which two new spherical waves emerge and travel to an observation screen. The distance from the slits to the screen is large compared to the separation between the slits. Each spherical wave has a fixed phase relation to the incident light wave and, therefore, to a wave originating from the other slit. At the observation screen, a pattern of equally spaced bright and dark fringes is observed. Whether a fringe is bright or dark is dependent upon whether the two waves are in or out of phase with one another at that point on the screen. Mathematically, the intensity of light at any point of the screen is proportional to the square of the sum of the two electric fields,

ES

2

= E1 + E 2

2

2

2

= E1 + E 2 + 2 E1 E 2 cos φ , where φ is the phase difference

that controls the intensity. One can also set up a matter equivalent of Young’s double slit experiment, e.g. by exciting two time-dependent wave functions, Ψ1 and Ψ2 , in an atom or molecule and allowing them to evolve independently until they overlap spatially and interfere with one another. The resulting superposition wave function is the square of the coherent sum of the individual wave functions, ΨS

2

2

= Ψ1 + Ψ2 , and it

follows that by manipulating the relative phases of the individual wave functions, it is possible to control the interference pattern and hence the resulting wave function.

2

Young’s double slit experiment in an atom

In this section, we explore the dynamics of Rydberg electron wave packets, focusing on experiments that exploit sequences of phase-locked optical pulses to control wave packet composition and dynamics – analogues of Young’s double slit experiment. Rydberg wave packets are ideal laboratories for developing and testing intuitive control schemes since one can determine the quantum interfering paths and wave packet dynamics accurately in these quasi one-electron systems (Wals et al. 1995; Wolde et al. 1988; Averbukh and Perelman 1989; Verlet and Fielding 2001; Jones and Noordam 1998). Briefly, an electron wave packet is created from a

FEMTOSECOND DYNAMICS AND CONTROL

21

superposition of Rydberg states and at short times it behaves very much like a classical electron: it remains fairly well localised and oscillates between the inner and outer turning points of the Coulomb potential well. The period of motion is inversely proportional to the energy spacing of the states in the superposition, tcl = 2π / ΔEn = 2πn3 , where n is the average principal quantum number. As a consequence of the anharmonicity of the Coulomb potential, the wave packet disperses and effectively becomes delocalised. A series of interference patterns with localised islands of probability distribution are generated at well-determined times. For a wave packet composed of p Rydberg states, p − q ( q = 1,2, K , p − 1 ) miniature wave packets may be observed at times Tq = TR / q (a qth order partial revival), where TR = ntcl / 3 is the full revival time. Note that this definition of TR is half the revival time defined by Averbukh (Averbukh and Perelman 1989). Dispersion and the formation of a second order partial revival is illustrated in Fig. 1, which is a plot of the probability density of a radial wave packet as a function of time t, and radial coordinate r, Ψ (r , t ) = r 2 2

∑aR n

n

nl

2

(r )e −iω t . a n and ωn are n

the amplitudes and angular frequencies of each Rydberg state in the superposition. In this example, the laser pulse is centered at n = 40, and its frequency profile is that of a bandwidth-limited 1 ps Gaussian pulse.

Fig. 1 Time-dependent radial distribution function of a radial wave packet excited with a bandwidth limited 1 ps Gaussian laser pulse centered around n = 40 in the hydrogen atom. Dark indicates high intensity. The classical period of motion is tcl ≈ 10 ps, and the second order partial revival is observed around TR/2 ≈ 65 ps.

22

H.H. FIELDING

The dynamics of the wave packet are governed by the states contributing to the superposition; therefore, by controlling the states in the superposition, the dynamics can be controlled. The optical Ramsey detection method is an example of control using pairs of optical pulses (Noordam et al. 1992). In the weak field limit, the population of the initial state remains virtually unchanged following excitation to a superposition of Rydberg states, and the excitation may be treated perturbatively. The initial state couples to the Rydberg states through Rabi oscillations at the frequency of the laser field. As the initial state is not depleted, a second identical wave packet may be launched using a second, delayed laser field, and there will be a similar coupling between the initial state and the Rydberg states. The phase difference, φ , between the paths coupling the individual Rydberg wave packets to the initial state is determined by the delay between the two laser pulses, tφ = φ ω , where ω is the optical frequency. If the first wave packet is still in the core region when the second wave packet is created, as tφ is varied, the total Rydberg population will be the coherent sum of the two individual wave packets. When the coupling pathways are in phase, constructive interference occurs and the total Rydberg population is enhanced. When the coupling pathways are out of phase, destructive interference occurs and the Rydberg population is effectively dumped back down to the initial state. The net effect is that interference fringes are observed in the total Rydberg population, analogous to Ramsey fringes. If the first wave packet is away from the core region when the second wave packet is created, the two wave packets will not be overlapped spatially and the total Rydberg population will simply be the incoherent sum of the two individual wave packets. The total Rydberg state population can be measured using a delayed electric field to field-ionise the Rydberg states with almost unit efficiency. The use of a pulse pair as a detection method may be extended to control the composition of a wave packet, and therefore to control the wave packet dynamics. The generation of bespoke wave packets using phase-locked pairs of electron wave packets was first reported by Noel and Stroud (Noel and Stroud 1995, 1996) and has been the subject of a number of theoretical investigations (Noordam et al. 1992; Chen and Yeazell 1998, 1999). We have carried out an analogue of Young’s slit experiment in Xe (Verlet et al. 2003). A pair of phase-locked radial wave packets, separated by half a classical orbit period, was excited in a quasi one-electron atomic system. The wave packets were allowed to evolve to the second order partial revival where they overlapped with one another spatially and could interfere. The resultant wave packet can be written, ΨS (t ) = Ψ1 (t + tcl / 2 + tφ ) + Ψ2 (t ) , where ΨS is the coherent sum of the two individual wave packets Ψ1 and Ψ2 , and tφ is the delay introduced between the two optical pulses, corresponding to an optical phase φ . We monitored the resulting wave packet dynamics in Xe around the second order partial revival time, TR / 2 = nt cl / 6 , measuring ΨS (t ≈ TR / 2) Ψ (0) using the optical Ramsey method. The two cases where k is either even or odd are shown in Fig. 2a and b


23

respectively, for wave packets excited at an average energy corresponding to n = 34.9, for which tcl = 6.5 ps and TR / 2 = 75.2 ps. At the second order partial revival a single wave packet splits into two smaller partial wave packets. Electron distribution therefore passes through the core region twice per orbit period and the whole system may be written as (Averbukh and Perelman 1989)

[

]

ΨTR / 2 (r , t ) = 2−1 / 2 Ψcl (r , t )e− iπ / 4 + Ψcl (r , t + tcl / 2)e + iπ / 4 ,

(1)

where Ψcl (r , t ) represents the original wave packet. The two scaled components are separated by half the classical orbit period and, most importantly, have a π / 2 phase difference between them. Now, if instead of a single wave packet, two wave packets are launched in the system such that ΨS (t ) = Ψ1 (t ) + Ψ2 (t + tcl / 2 + tφ ) , they will interfere so that the spectrum at the time of the second order partial revival resembles a full revival, as illustrated in Fig. 2a and b. This may be explained in terms of the cartoons presented on the right hand side of Fig. 2, and by considering Equation (1). Briefly, for a second wave packet excited at a time Δt = tcl / 2 + tφ after the first, such that Ψ2 (t + (Δt + tφ )) , the phases of the scaled wave packets in Equation (1) will be reversed around TR / 2 with respect to Ψ1 (t ) . The dynamics of the resultant wave packet, ΨS (t ) , are then monitored around this time (with a third pulse in an opti-

cal Ramsey fashion). When φ = (2k − 1)π / 2 and k is even, the partial wave

packets at the inner turning point of both Ψ1 (t ) and Ψ2 (t ) are in phase and interfere constructively, whilst those at the outer turning point are out of phase and cancel each other (Fig. 2a). Alternatively, if k is odd, the opposite occurs and the partial wave packets at the inner turning points are out of phase and those at the outer turning point are in phase, as shown in Fig. 2b. We have also demonstrated how to control the orbital angular momentum composition of a wave packet using phase-locked pairs of electron wave packets (Verlet et al. 2002; Carley et al. 2005). In one experiment, a pair of wave packets composed of superpositions of two non-interacting Rydberg series, with different orbital angular momenta l and quantum defects μ l , were excited in Xe. The quantum defect difference between the two orbital angular momentum series was Δμl ≈ 0.25 , resulting in an angular beat with period t μ = tcl / Δμl ≈ 4tcl . Therefore, after two classical orbit periods the pathways connecting the two angular momentum components are out of phase with one another. If a second wave packet is launched at this time, it can be created so that it interferes constructively with one orbital angular momentum component of the evolved wave packet and destructively with the other, controlling the orbital angular momentum character of the electron wave packet, ΨS (t ) = Ψ1 (t + 2tcl + tφ ) + Ψ2 (t ) .

24

H.H. FIELDING

Fig. 2 Observed dynamics of Rydberg electron wave packets in Xe excited at an average energy corresponding to n = 34.9, following two pulses separated by tcl/2 + φ, and monitored by a third pulse around the time of their second order partial revivals, where φ = π/2 (a), φ = 3π/2 (b), and φ = π or 0 (c). In all three cases, the top trace has been calculated whilst the bottom trace has been observed experimentally. The time axis has been labeled in units of the classical period to highlight the different dynamics. The additional plots to the left of (c) indicate the two possible calculated population distributions for the nd [3 / 2]1 states, when the phase φ = π or 0. Cartoon plots on the right hand side of each figure are included to highlight the influence of phase on the wave packet interference patterns. We show the resultant dynamics of the wave packet (black) if the additional phase introduced φ = (2k − 1)π/2, where k is even (right of (a)) and odd (right of (b)). The two interfering wave packets at their respective second order partial revivals (white Ψ1 and grey Ψ2) have their relative phases indicated. (c) shows the case when the phase introduced φ = kπ, where k is zero. Again the resultant dynamics are indicated in black.

3

Controlling Rydberg electron wave packet localization in H2

Molecular Rydberg wave packets are more complex than atomic Rydberg wave packets because of the additional degrees of freedom introduced by the molecular core. Molecular Rydberg wave packets exhibit many of the dynamical complications


25

associated with larger molecules of more interest to chemists and biologists, such as the high density of rovibronic states and non-adiabatic coupling, yet they remain highly tractable (Fielding 2005). In our group, we have investigated the influence of the molecular core on the radial dynamics of Rydberg wave packets in molecules (Stavros et al. 2000; Smith et al. 2003) and have used sequences of pulses to control the rotational quantum state composition (Minns et al. 2003a) and the ionization/predissociation ratio in NO (Minns et al. 2003b). More recently, we have shown how the slew rate of an external electric field can be employed as a complementary tool to control the rotational quantum state during field-ionisation (Patel et al. 2007). In this section, we describe our investigations of molecular Rydberg wave packet dynamics in the more general case of coupled oscillators, specifically the coupled Rydberg channels corresponding to different rotational states of the H +2 ion core of the highly excited H2 molecule (Kirrander et al. 2007, 2008). By shaping the phase profile of the excitation pulse we can shift particular features in the wave packet evolution to any chosen time. One context where such control could be particularly important is in time-domain (pump-dump) coherent control experiments in polyatomic molecules (Tannor and Rice 1985) where, for instance, the localization of a wave packet in a particular Franck–Condon region of a potential energy surface could be desirable. The Rydberg wave packet is written ψ (r , t ) = ∑ c n (t )exp(− iE n t )ψ n (r ) , n

where each state n is characterized by an energy E n and a wavefunction ψ n (r ) , and r corresponds to the radial coordinate of the Rydberg electron. The dimensionless expansion coefficients cn (t ) change during the excitation process, but remain constant afterwards. In the weak field regime, where first order perturbation theory can be used, the coefficients are given by, c n (t ) = iDns cef (E n , t ) .

Dns are the dipole transition moments connecting the state n and the initial state s,

and cef (E n , t ) is the complex excitation function (Shapiro 1990; Shapiro and Brumer 2003; Taylor and Brumer 1983), defined by

cef (En , t ) =

∫

∞

−∞

dEε (E )

∫

t

−∞

dt ' exp(i (En − E )t ′) ,

(2)

where ε (E ) is the Fourier transform of the optical excitation pulse ε (t ) , using natural units (Kirrander et al. 2008). The wave packet evolution following excitation by an unshaped laser pulse is presented in Fig. 3. The almost instant excitation with the reference pulse is followed by free evolution of the wave packet. The overall dynamics follows the typical Rydberg pattern of oscillations between the core, r ≈ 0 , and the outer turning point. Note that the wave packet in channel j = 1, corresponding to the molecular ion core in a rotational quantum state N + = 0 , has its outer turning

26

H.H. FIELDING

point at larger distances. The outer turning point given by rcl = 2n 2 , where n is the mean principal quantum number: n = 24.5 in channel 1, corresponding to rcl = 1,200 au, and n = 17.5 in channel 2, corresponding to rcl = 615 au.

Fig. 3 The contour plot shows the wave packet probability density as a function of radial distance r (au) and time t (ps). Negative radial distances r correspond to the dynamics in channel j = 2, while positive radial distances correspond to channel j = 1 (see text). In both channels, the wave packet oscillates between the core and the outer turning point. The excitation is almost instant and is quickly followed by free evolution of the wave packet until the wave packet in channel 2 returns to the core and can scatter into channel 1.

The effect of the coupling between the channels can be seen clearly in the oscillation period, t cl . This is, strictly speaking, only defined for a single (or uncoupled) channel. In a system of two uncoupled channels, the different mean principal quantum number would give two separate time scales: t cl = 2.2 ps in channel 1 and t cl = 0.8 ps in channel 2. In this example, due to coupling between the channels, both time scales are present in both channels. The shorter time scale in channel 2 is immediately visible, while the longer period motion in channel 1 has an additional modulation of the probability density due to scattering from channel 2 into channel 1, discernible in the free evolution of the wave packet for t > 0.8 ps. The phase profile Φ (E ) of the excitation pulse in equation (4),

ε (E ) = ε (E ) exp(iΦ (E )) , can be optimized to localize the wave packet in various

target sets of coordinates at specific target time. The phase profile for a given target can be determined by allowing the wave packet to explore all the phase space available to it and looking for the time, t s , at which the specific target feature occurs. The time t s , can then be shifted to the target time, t , using Φ (E n ) = mod(E n (t − t s ), 2π ) .


27

Fig. 4 The wave packet probability density in the two channels j = 1 and j = 2 around the target time t = 10.0 ps after excitation. The contour plots show the wave packet dynamics as a function of radial distance r (au) and time t (ps). Negative radial distances r correspond to the dynamics in channel j = 2, while positive radial distances correspond to channel j = 1. Pulse A gives the unshaped wave packet. Pulse B generates a wave packet localized close to the outer turning point in both channels. Pulse C generates a wave packet with little probability density in channel j = 1 and most of the probability density localized at the outer turning point in channel j = 2. This corresponds to the molecule predominantly localized in the N + = 2 rotational state. Pulse D generates a wave packet with most probability density in channel j = 1, which corresponds to the molecule predominantly localized in the N + = 0 rotational state.

Our specific targets are: Localisation of the wave packet at the outer turning point in both channels; Localisation of the wave packet at the outer point of just one channel, so that the probability of detecting the wave packet in that particular channel is maximal. The target time t = 10.0 ps was selected to be sufficiently short to minimize dispersion of the wave packet, but long enough so that the amplitude of the phase-shaped excitation pulses could return to zero. The results of the optimizations are shown in Fig. 4. In A the pulse is unshaped. In B the pulse is shaped to localize the wave packet close to the outer turning points in both channels with a (0.60, 0.40) distribution between the channels ( j = 1, j = 2). In C the wave packet localized at the outer turning point in channel j = 2 with a (0.26, 0.74) distribution. This corresponds to the molecule predominantly in the N + = 2 rotational state. Finally, in D the wave packet is localized in channel j = 1 with a (0.82, 0.18) distribution between the two channels, which corresponds to the molecule being localized predominantly in the N + = 0 state.

28

H.H. FIELDING

4

Femtosecond dynamics of the S1 excited state of benzene

Armed with the ability to design pulse sequences or phase profiles to control wave packet composition and localization in small, theoretically tractable systems, the next question is: Can we understand the dynamics in polyatomic molecules in sufficient detail that we can begin to devise intuitive schemes to control their photophysics or photochemistry? Although it is not yet possible to calculate the potential energy surfaces of polyatomic molecules in sufficient detail to be able to design accurate pulse sequences or phase profiles to create a specific photoproduct, it should be possible to use a combination of theory and chemical intuition to determine which vibrational modes will be important for steering a molecular wave packet towards a particular feature on the excited state potential energy surface. It should then be possible to optimize the pulse sequence or phase profile experimentally. With this in mind, we set out to explore the dynamics of benzene in its first electronically excited state, with the ultimate aim of controlling the yield of the fulvene isomer. fulvene

benzene ε1a ε1b

ε2

ε3

D'0 D0 254 nm

235 nm

Δt S1

T2 T1

243 nm

S0 Fig. 5 Schematic energy level diagram of our pump-probe TRPES study of intramolecular dynamics of benzene. The optically bright S1 state is accessed from the ground S0 state. The evolution of the system on the excited state is monitored by applying a delayed femtosecond probe pulse at 254 or 235 nm. ε1a and ε1b represent photoelectron energies associated with ionisation from S1, ε2 represents the photoelectron energies associated with T2 and ε3 represents photoelectron energies associated with ionisation from T1. The ionisation limit of the fulvene cation is shown on the left hand side of the figure. The benzene and fulvene cations are represented by HOMOs from which an electron has been removed. The S1 and T1 states of benzene are represented by the LUMO into which excitation takes place from the HOMO.


29

In this section, we describe our detailed investigations of the femtosecond dynamics of benzene in the S1 excited state (Parker et al. 2009a), using timeresolved photoelectron spectroscopy (TRPES) and speculate on potential control scenarios. The energy level scheme relevant to our experiment is shown in Fig. 5. We employ femtosecond TRPES (Gessner et al. 2006; Stolow et al. 1999, 2004; Stolow and Jonas 2004) to investigate the dynamics at the onset of the “channel 3” region (Clara et al. 2000; Moss and Parmenter 1986; Longfellow et al. 1988; Callomon et al. 1966, 1972; Riedle et al. 1982). A femtosecond pump pulse at 243 nm prepares benzene in the excited S1 state, with 3,070 cm−1 of excess vibrational energy. The subsequent evolution of the resulting wave packet is then monitored by projecting its complete wave function onto all accessible cation states using a delayed femtosecond probe pulse of wavelength 254 or 235 nm. The total integrated photoelectron signal provides a measure of the ionization probability as a function of time, and the photoelectron spectrum provides a map of timedependent overlap integrals between the excited state and the accessible cation states, revealing detailed information about the evolution of the electronic and vibrational character of the wave packet on the excited state. The total integrated photoelectron signal is plotted in Fig. 6a as a function of the delay between the pump and probe pulses for probe wavelengths 254 and 235 nm. Both decay curves can be fitted to biexponential functions, 0 I = I 0fast exp − t / τ fast + I slow exp − t / τ fast , as observed in previous femtosecond

(

)

(

)

pump-probe ionisation measurements (Clara et al. 2000). The amplitude of the 0 slow component, I slow = 1 − I 0fast , as a function of probe wavelength, is plotted in Fig. 6b: it is a step function centered at approximately 250 nm (4.96 eV), implying that when the probe energy is greater than 4.96 eV it is possible to ionise more of the excited potential energy surface. Such a distinct threshold is a signature of a new ionisation pathway opening up, suggesting that there is a change in the electronic character along the adiabatic potential surface of the excited state. The excited state dynamics can be unravelled by analysing the timedependence of different components of the photoelectron spectra. Plots of the difference spectra (recorded using the higher energy 235 nm probe) as a function of time for three different ranges of photoelectron energy, ε1a = 1.25 − 1.35 eV plus

ε1b = 1.13 − 1.23 eV, ε 2 = 0.75 − 0.95 eV and ε 3 = 0.35 − 0.45 eV, are presented in Fig. 6c. Oscillations are observed for photoelectron kinetic energies ε1a , ε1b and ε 2 . The oscillations have period τ osc = 1.2 ± 0.1 ps, which is much longer than any fundamental vibration period of S1 benzene. Photoelectron energies ε1a and ε1b are separated by 0.1 eV, which corresponds to the v1(a1g) breathing mode of the benzene cation, which is a signature that the wave packet contains a component associated with the v1 mode of benzene which is known to show a long vibrational progression upon ionization (Long et al. 1983). The oscillations at ε1a and ε1b are in phase with one another, but are out of phase with the oscillation at

30

H.H. FIELDING

ε 2 , and the sum of the amplitudes of the oscillations at ε1a and ε1b is equal to the amplitude of the out of phase oscillation at ε 2 , demonstrating that we are ionising a bound wave packet at two extremes of an oscillatory motion on the adiabatic potential energy surface. For photoelectron energies ε 3 , the difference spectrum shows an increase to a maximum at 600 fs (the time it takes for the bound wave packet to reach its outer turning point), suggesting that a portion of the excited state population is being transferred from the Franck-Condon region (monitored at ε1a and ε1b ) to a dark state (monitored at ε 3 ) via a doorway state (monitored at

ε 2 ). The fact that the population of ε 2 does not decrease as ε 3 rises, indicates that there is a strong coupling between ε 2 and ε 3 that transfers population to ε 3 immediately and that it is never detected in ε 2 .

Fig. 6 (a) Experimental integrated photoelectron yield as a function of delay for probe wavelengths of 254 nm (lower curve/blue points) and 235 nm probe (upper curve/red points). The solid lines represent least squares fits to the observed data with a biexponential function. The absolute value of the exponential tells us very little about the dynamics, only how quickly population leaves the observation window. The amplitude of the slow component is plotted as a function of probe wavelength in (b). The step at 250 nm is the signature of a new ionisation pathway opening up. (c) Integrated intensities of photoelectrons as a function of pump-probe delay with the exponential decay removed. The open circles correspond to integration over the range ε1a = 1.25–1.35 eV and ε1b = 1.13–1.23 eV, and the closed circles correspond to integration over the range ε2 = 0.75–0.95 eV. The filled triangles correspond to integration over the range ε3 = 0.35– 0.45 eV. The solid lines in the lower half of the graph are a least squares sinusoidal fit to the data and have periods 1.2 ± 0.1 ps.


31

Fig. 7 Pairs of pulses with different central wavelengths generated from a triangular phase mask. (a) Triangular phase mask applied by the dual-array 640 pixel SLM in the visible with a central wavelength of 508 nm. (b) Spectral intensity (black) and phase (red) showing the central wavelengths of the UV pulses: 253.6 and 254.6 nm, i.e. separated by 1 nm. The retrieved spectral phase shows a clear change in slope at the central frequency of the spectrum, 254 nm, as programmed on the pulse shaper at 508 nm. (c) Time domain electric field intensity (black) and phase (red) of the UV pulse, showing two distinct subpulses separated by 650 ps. (d) Raw XFROG trace.

These observations can be explained in terms of intersystem crossing (ISC) involving the S1, T1 and T2 states of benzene – the energies of the measured photoelectrons correspond extremely well with those expected for ionisation of the S1, T1 and T2 states based on simple Frank-Condon arguments and taking into account the excess vibrational energy in all of the states involved. Once population resides in T2, strong vibronic coupling between T2 and T1 facilitates rapid population transfer to the dense bath of states which are energetically accessible in T1, resulting in the step in the photoelectron counts observed at low photoelectron energies (Fig. 6b). The small population of states in T2 which do not couple well

32

H.H. FIELDING

to T1, are then free to oscillate back to the S1 state. Such rapid intersystem crossing challenges currently accepted models for ultrafast non-adiabatic processes in which ISC is assumed to be too slow to be important, unless certain conditions are fulfilled to increase the spin-orbit coupling (SOC) such as the presence of a heavy atom or nitro group (Zugazagoitia et al. 2008; Aloise et al. 2008; Hare et al. 2007; Mohammed and Vauthey 2008; Serrano-Perez et al. 2007). The next challenge is to control the dynamics of benzene. Amplitude-shaped optical pulses offer the possibility of cutting out particular vibrational modes from the wave packet created on the S1 potential energy surface, e.g. those that are responsible for the 1.2 ps beat or the initial rapid decay to S0 that is responsible for the 25% loss of population observed with the higher energy probe pulse. Phaseshaped optical pulses provide the opportunity to localize the wave packet at a particular point on the potential energy landscape, e.g. at the prefulvene geometry. Although pulse shapers operating in the ultraviolet (UV) are not available commercially, we have developed a robust method for producing tuneable, shaped, UV femtosecond laser pulses for controlling the excited state dynamics of organic molecules (Parker et al. 2009b). A reflective mode, folded, pulse shaping assembly employing a spatial light modulator (SLM) shapes femtosecond pulses in the visible region of the spectrum. The shaped visible light pulses are frequency doubled to generate phase- and amplitude-shaped, ultrashort light pulses in the deep UV. This approach benefits from a simple experimental setup and the potential for tuning the central frequency of the shaped UV waveform. A number of pulse shapes have been synthesised and characterised using cross-correlation frequency resolved optical gating (XFROG), including pairs of pulses with different central wavelengths, as illustrated in Fig. 7.

5

A new experiment for investigating the photochemistry of biological chromophores in the gas-phase

Many of the challenges in modern science lie in unravelling the dynamics of complex molecular systems to determine the relationship between structure and function. However, our understanding of the interaction between a molecule and its environment, e.g. a biological chromophore embedded in a protein, is still in a primitive state. A protein environment provides both a static and dynamical constraint on the motions of the atoms within a chromophore, and effectively catalyses a photoreaction path that may not be observed in the isolated chromophore or in solution, but which may be crucial for a specific biological function. We have recently built a unique instrument combining electrospray-ionisation mass-spectrometry (ESI-MS) to generate a 1 kHz source of mass and charge selected ‘native’ protein ions in the gas phase for photoelectron imaging of biological chromophores. m/z selected ion packets from a modified commercial ESImass-spectrometer are directed into the interaction region of a custom-designed


33

photoelectron imaging spectrometer where they interact with nanosecond or femtosecond laser pulses to resonantly excite the biological chromophore and ionise it. This new tool promises to provide valuable new insight into the ultrafast dynamics of biological chromophores and to have exciting applications in the life sciences.

6

Summary

We have described a series of investigations of ultrafast dynamics and control in electronically excited atoms and molecules. We began with a simple intuitive control scheme based on an analogue of Young’s double-slit experiment in a Rydberg atom. We then described how sequences of optical pulses can be employed to control the composition of Rydberg wave packets in atoms and molecules, and how sculpting the phase profile of an optical pulse can be exploited to control the trajectory of a Rydberg wave packet through a given phase space and hence localise the wave packet in time and space. We then moved on to more complex polyatomic systems with many degrees of freedom and described recent experiments investigating the femtosecond dynamics of benzene in its electronically excited state and proposals for controlling the photochemistry and photophysics. Finally, we summarised steps towards the development of a new tool for investigating the femtosecond dynamics of biological chromophores in their native protein environments. Acknowledgments The author would like to thank many coworkers for their contributions to the work described in these proceedings: Elizabeth Boléat, Elizabeth Gill, Nicholas Jones, Christian Jungen, Adam Kirrander, Adam McKay, Russell Minns, Abigail Nunn, Dorian Parker, Rakhee Patel, Tom Penfold, Maria Sanz, Robert Smith, Vasilios Stavros, Jan Verlet, Graham Worth. The author also acknowledges financial support from the EPSRC, The Leverhulme Trust and The Royal Society.

References Aloise, S., Ruckebusch, C., Blanchet, L., Rehault, J., Buntinx, G., Huvenne, J.P.: The benzophenone S1(n,π*)->T1(n,π*) states intersystem crossing reinvestigated by ultrafast absorption spectroscopy and multivariate curve resolution. J. Phys. Chem. A 112(2), 224–231 (2008) Averbukh, I.S., Perelman, N.F.: Fractional revivals – universality in the long-term evolution of quantum wave-packets beyond the correspondence principle dynamics. Phys. Lett. A 139(9), 449–453 (1989) Callomon, J.H., Dunn, T.M., Mills, I.M.: Rotational analysis of 2600 Å absorption system of benzene. Phil. Trans. Royal Soc. Lond. A 259(1104), 499 (1966) Callomon, J.H., Parkin, J.E., Lopez-Delgado, R.: Nonradiative relaxation of excited A1B2u state of benzene. Chem. Phys. Lett. 13(2), 125 (1972)

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Carley, R.E., Boléat, E.D., Patel, R., Minns, R.S., Fielding, H.H.: Interfering Rydberg wave packets in Na. J. Phys. B 38(12), 1907–1922 (2005) Chen, X., Yeazell, J.A.: Analytical wave-packet design scheme: control of dynamics and creation of exotic wave packets. Phys. Rev. A 57, R2274 (1998) Chen, X., Yeazell, J.A.: Phase-conjugate picture of a wave-packet interference design for arbitrary target states. Phys. Rev. A 59(5), 3782–3787 (1999) Clara, M., Hellerer, Th., Neusser, H.J.: Fast decay of high vibronic S-1 states in gas-phase benzene. Appl. Phys. B 71(3), 431–437 (2000) Fielding, H.H.: Rydberg wave packets in molecules: from observation to control. Ann. Rev. Phys. Chem. 56, 91 (2005) Gessner, O., Lee, A.M.D., Shaffer, J.P., Reisler, H., Levchenko, S.V., Krylov, A.I., Underwood, J.G., Shi, H., East, A.L.L., Wardlaw, D.M., Chrysostom, E.T., Hayden, C.C., Stolow, A.: Femtosecond multidimensional imaging of a molecular dissociation. Science 311(5758), 219–222 (2006) Hare, P.M., Crespo-Hernandez, C.E., Kohler, B.: Internal conversion to the electronic ground state occurs via two distinct pathways for pyrimidine bases in aqueous solution. Proc. Nat. Acad. Sci. U.S.A. 104(2), 435–440 (2007) Jones, R.R., Noordam, L.D.: Electronic wave packets. Adv. At. Mol. Opt. Phys. 38, 1 (1998) Kirrander, A., Fielding, H.H., Jungen, C.: Excitation, dynamics and control of rotationally autoionizing Rydberg states of H-2. J. Chem. Phys. 127(16), 164301 (2007) Kirrander, A., Jungen, C., Fielding, H.H.: Localization of electronic wave packets in H-2. J. Phys. B 41(7), 074022 (2008) Long, S.R., Meek, J.T., Reilly, J.P.: The laser photoelectron spectrum of gas-phase benzene. J. Chem. Phys. 79(7), 3206–3219 (1983) Longfellow, R.J., Moss, D.B., Parmenter, C.S.: Rovibrational level mixing below and within the channel 3 region of S-1 benzene. J. Phys. Chem. 92(19), 5438–5449 (1988) Minns, R.S., Patel, R., Verlet, J.R.R., Fielding, H.H.: Optical control of the rotational angular momentum of a molecular Rydberg wave packet. Phys. Rev. Lett. 91(24), 243601 (2003a) Minns, R.S., Verlet, J.R.R., Watkins, L.J., Fielding, H.H.: Observation and control of dissociating and autoionizing Rydberg electron wave packets in NO. J. Chem. Phys., 119(12), 5842– 5847 (2003b) Mohammed, O.F., Vauthey, E.: Excited-state dynamics of nitroperylene in solution – solvent and excitation wavelength dependence. J. Phys. Chem. A 112(17), 3823–3830 (2008) Moss, D.B., Parmenter, C.S.: A time-resolved fluorescence observation of intramolecular vibrationally redistribution within the channel-3 region of S-1 benzene. J. Phys. Chem. 90(6), 1011–1014 (1986) Noel, M.W., Stroud, C.R.: Young’s double-slit interferometry within an atom. Phys. Rev. Lett. 75(7), 1252–1255 (1995) Noel, M.W., Stroud, C.R.: Excitation of an atomic electron to a coherent superposition of macroscopically distinct states. Phys. Rev. Lett. 77(10), 1913–1926 (1996) Noordam, L.D., Duncan, D.I., Gallagher, T.F.: Ramsey fringes in atomic Rydberg wave packets. Phys. Rev. A 45(7), 4734–4737 (1992) Parker, D.S.N., Minns, R.S., Penfold, T.J., Worth, G.A., Fielding, H.H.: Ultrafast dynamics of the S1 excited state of benzene. Chem. Phys. Lett. 469(1–3), 43–47 (2009a) Parker, D.S.N., Nunn, A.D.G., Minns, R.S., Fielding, H.H.: Frequency doubling and Fourier domain shaping the output of a femtosecond optical parametric amplifier: easy access to tuneable femtosecond pulse shapes in the deep ultraviolet. Appl. Phys. B 94(2), 181–186 (2009b) Patel, R., Jones, N.J.A., Fielding, H.H.: Rotational-state-selective field-ionization of molecular Rydberg states. Phys. Rev. A 76(4), 043313 (2007) Riedle, E., Neusser, H.J., Schlag, E.W.: Sub-doppler high-resolution spectra of C6H6 – anomalous results in the channel 3 region. J. Phys. Chem. 86(25), 4847–4850 (1982)


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Serrano-Perez, J.J., Merchan, M., Serrano-Andres, L.: Quantum chemical study on the population of the lowest triplet state of psoralen. Chem. Phys. Lett. 434(1–3), 107–110 (2007) Shapiro, M.: Half Collision Resonance Phenomena in Molecules. In Garcia-Sucre, M., Raseev, G., Ross, S. (eds.), AIP Conf. Proc. 225 (New York: AIP) p. 230 (1990) Shapiro, M., Brumer, P.: Principles of the Quantum Control of Molecular Processes (1st edn.). Wiley, New York (2003) Smith, R.A.L., Stavros, V.G., Verlet, J.R.R., Fielding, H.H., Townsend, D., Softley, T.P.: The role of phase in molecular Rydberg wave packet dynamics. J. Chem. Phys. 119(6), 3085– 3091 (2003) Stavros, V.G., Ramswell, J.A., Smith, R.A.L., Verlet, J.R.R., Lei, J., Fielding, H.H.: Vibrationally autoionizing Rydberg wave packets in NO (vol 83, 2552 1999). Phys. Rev. Lett. 84(8), 1847–1847 (2000) Stolow, A., Jonas, D.M.: Multidimensional snapshots of chemical dynamics. Science 305, 1575– 1577 (2004) Stolow, A., Blanchet, V., Zgierski, M.Z., Seideman, T.: Discerning vibronic molecular dynamics using time-resolved photoelectron spectroscopy. Nature 401(6748), 52–54 (1999) Stolow, A., Bragg, A.E., Neumark, D.M.: Femtosecond time-resolved photoelectron spectroscopy. Chem. Rev. 104(4), 1719–1757 (2004) Tannor, D.J., Rice, S.A.: Control of selectivity of chemical reaction via control of wave packet evolution. J. Chem. Phys. 83(10), 5013–5018 (1985) Taylor, R.D., Brumer, P.: Pulsed laser preparation and quantum superposition state evolution in regular and irregular systems. Faraday Discuss. Chem. Soc. 75(75), 117–130 (1983) Verlet, J.R.R., Fielding, H.H.: Manipulating electron wave packets. Int. Rev. Phys. Chem. 20(3), 283–312 (2001) Verlet, J.R.R., Stavros, V.G., Minns, R.S., Fielding, H.H.: Controlling the angular momentum composition of a Rydberg electron wave packet. Phys. Rev. Lett. 89(26), 263004 (2002) Verlet, J.R.R., Stavros, V.G., Minns, R.S., Fielding, H.H.: Controlling the radial dynamics of Rydberg wave packets in Xe using phase-locked optical pulse sequences. J. Phys. B 36, 3683 (2003) Wals, J., Fielding, H.H., van Linden van den Heuvell, H.B.: The role of the quantum-defect and of higher-order dispersion in Rydberg wave packets. Phys. Scr., T58, 62 (1995) Wolde, A. ten, Noordam, L.D., Lagendijk, A., van Linden van den Heuvell, H.B.: Observation of radially localised atomic electron wave packets. Phys. Rev. Lett. 61(18), 2099–2101 (1988) Young, T.: The Bakerian lecture: experiments and calculations relative to physical optics. Phil. Trans. Royal Soc. Lond. 94, 1 (1804) Zewail, A.H.: Femtochemistry: Atomic-scale dynamics of the chemical bond using ultrafast lasers. Angew. Chem. Int. Ed. Engl. 39(15), 2587–2631 (2000) Zugazagoitia, J.S., Almora-Diaz, C.X., Peon, J.: Ultrafast intersystem crossing in 1-nitronapthalene. An experimental and computational study. J. Phys. Chem. A 112(3), 358–365 (2008)

Coherent Control for Molecular Ultrafast Spectroscopy Tiago Buckup, Marcus Motzkus Physikalische Chemie, Philipps-Univertät Marburg, D-35043, Marburg, Germany [email protected]

Jürgen Hauer Physikalische Chemie, Philipps-Univertät Marburg, D-35043, Marburg, Germany Institute for Physical Chemistry, A-1090, Vienna, Austria

Abstract Quantum control spectroscopy (QCS) is a powerful modern experimental concept to disentangle complex dynamics in molecular quantum systems and combines coherent control methods with time-resolved optical spectroscopy. By manipulating the photo-induced reaction pathway with specifically tailored excitation light fields, it offers a new spectroscopic degree of freedom in addition to classic spectral and temporal coordinates. Particularly interesting in QCS is the excitation with well-defined pulse sequences which will be shown on two examples. In a first series of experiments the comparison of excitations with one or two pulses allows to unravel the complex energy flow network in all-trans-β-carotene and its homologs. In a second experiment, we go a step further and use multipulse excitation via phase shaping in a prototype dye molecule to analyze the role of electronic coherence time for population and vibrational coherence enhancement.

Keywords: Coherent control, open-loop control, pulse shaping, carotenoids

1

Introduction

The modification of molecular quantum pathways by coherent control offers the possibility of a deeper insight into the involved dynamics (Rabitz et al. 2000; Brumer and Shapiro 1986, 1989; Tannor and Rice 1998; Lozovoy et al. 1999). The underlying principles can either be understood in a time approach (Tannor


37

38

T. BUCKUP, M. MOTZKUS AND J. HAUER

and Rice 1998) or frequency domain scheme (Brumer and Shapiro 1989). Closed loop schemes based on evolutionary algorithms reveals the potential of coherent control experiments (Judson and Rabitz 1992; Bardeen et al. 1997; Assion et al. 1998), with successful optimization of dynamics even in complex biomolecules (Herek et al. 2002; Wohlleben et al. 2005; Prokhorenko et al. 2006, 2007). Differing from the often pursued goal of optimization, coherent control can also efficiently be used to trace the mechanisms of chemical reactions or photophysical processes. In contrast to mere optimization, such a quantum control spectroscopy (QCS) approach exploits the comparison between shaped and unshaped (Fourierlimited) pulses and focuses on specific shaped pulses typically in an open loop experiment to trigger theoretically well understood dynamics. The influence of multipulses on vibrational wave packet motion can be mentioned as an example of such a pulse shape (Weiner et al. 1990). Although open loop experiments are the more direct approach to QCS, the combination of mathematical analysis of generations (White et al. 2004) with the concept of phase and amplitude parameterization (Zeidler et al. 2001; Hornung et al. 2001) in a closed loop genetic algorithm can also lead to a better understanding of the involved mechanisms. In QCS the application of shaped pulses should be seen as a spectroscopic degree of freedom beside classic spectral and temporal coordinates. This approach offers therefore evident parallels to modern pulsed NMR experiments, where a multitude of tasks ranging from protein structure elucidation (Wüthrich 1986) to imaging applications (Ciobanu et al. 2003) can be accomplished without general changes in the instrumentation used, showing the full potential of applying shaped excitation pulse sequences as a spectroscopic concept. The parallels between NMR and nonlinear optical spectroscopy become especially obvious in case of 2Dshaping based multidimensional optical spectroscopy (Tian et al. 2003; Wagner et al. 2005; Gundogdu et al. 2007; Brixner et al. 2005). This article aims to discuss some generalizable examples of quantum control spectroscopy application in time-resolved experiments. In Section 2 the interaction of two excitation pulses in a transient absorption (TA) experiment is used to disentangle the complex ultrafast population dynamics of carotenoids. The idea behind this three pulse pump-probe (Fig. 1a), called pump-depletion-probe (Gai et al. 1997; Wohlleben et al. 2004; Buckup et al. 2004, 2005, 2006; Larsen et al. 2003), is to launch the system to an excited state and then interfere with its evolution at a point which is normally inaccessible with just one pump pulse. In Section 3, the interaction of a pulse train (Fig. 1b and c) is discussed. It will be shown that the precise control of the sub-pulse separation leads to tailoring of vibrational coherence and population transfer between the ground- and the excitedstate (Hauer et al. 2006a, 2007a, b, 2008; Buckup et al. 2008).

COHERENT CONTROL FOR MOLECULAR ULTRAFAST SPECTROSCOPY

39

Fig. 1 Pulse sequence used in this work and the respective delays in (a) pump-depletion-probe, (b) tailored four-wave-mixing and (c) arbitrarily shaped pump-probe.

2

Pump-depletion-probe spectroscopy on carotenoids

2.1 The complex deactivation network of carotenoids Carotenoids are important biological molecules performing a variety of critical functions in nature like acting as photoprotection chromophores, supporting structural elements and collecting energy for photosynthesis in the light harvesting (Polivka and Sundstrom 2004). Recently, much interest has been directed to the details of the deactivation pathways of carotenoids that follow excitation by

40


ultrashort laser pulses. More specifically, considerable interest in carotenoid reaction dynamics has been given to the presence and role of intermediate excited states, additional to the three commonly accepted energy levels of S0, S1 and S2 states (Fig. 2). The linear absorption spectrum of carotenoids contains only one bright state, conventionally labeled S2 or 1Bu+ in the idealized linear C2h symmetry (Christensen 1999). The traditionally accepted mechanism of deactivation from S2 involves internal conversion to the dark S1 (2Ag−) state, which is one photonforbidden from the ground-state S0 (1Ag−). The S1 decay to the ground state follows the energy-gap law (Englman and Jortner 1970) of weak vibronic interaction quite well (Andersson and Gillbro 1995; Frank et al. 2002). However, calculations with full configuration interaction indicate that below S2, there exists not one but, depending on the conjugation length, up to three dark states (Tavan and Schulten 1986). Particularly interesting is one spectral signature found in the time-resolved transient absorption signal of carotenoids, the so called S* band, which is characterized by a TA signal on the lower-energy side of the S0–S2 absorption. Several works claim that this contribution originates from one of these dark states. Depending on the carotenoid length, either one or two additional states would take part in the internal conversion, with additional transient absorption signal in the near IR (Ikuta et al. 2006; Fujii et al. 2003; Sashima et al. 1999). In another set of experiments, S* is explained as a vibrationally excited ground-state (hot-S0) in long carotenoids (Andersson and Gillbro 1995; Wohlleben et al. 2004; Buckup et al. 2006) as well for carotenoids in artificial light harvesting complexes (dyads) (Savolainen et al. 2008a, b). Recently, the presence of different conformers in solution was discussed as a possible explanation to these challenging S* signal (Frank et al. 2007; Niedzwiedzki et al. 2007). In order to help to disentangle the nature of these spectral signatures, we addressed the question whether these intermediary dark states are playing any role in the deactivation network of carotenoids in solution and what is the nature of S* by using a modified pump-probe technique, namely pump-depletion-probe. The second pump pulse interferes with the population on S2, thus depleting the state and modulating the usual decay pathway. The depletion pulse overlaps with the transient S2–Sn IR excited state absorption (ESA) bands observed for several

Fig. 2 (a) Energy level scheme for carotenoids with possible energy positions of the S* state. (b) Carotenoids investigated in this contribution. M11 is also usually called all-trans-β-carotene.


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carotenoids (Fujii et al. 2003; Cerullo et al. 2002; Polli et al. 2004). We show for a series of all-trans-carotenoids with N = 11, 13 and 15 conjugated bonds that there is no evidence for an additional electronic state. A deactivation mechanism that includes the hot-ground state supports the observed results nicely in the framework of a simple three state model (S2, S1 and S0).

2.2 Pump-depletion-probe experimental setup In our pump-probe spectroscopy setup, the pump laser pulse is provided by a home-built noncollinear optical parametric amplifier (nc-OPA), which was tuned to the S2 0–0 transition vibrational band for carotenoids N ≥ 11: 495, 525, and 555 nm for M11, M13, and M15, respectively. Pulse durations were below 35 fs. White-light continuum was used as probe pulse in all experiments which was generated in a sapphire crystal by a fraction of the energy of the fundamental laser 800 nm. In the pump-depletion-probe technique, an additional beam (depletion) is added as second pump- with variable delay T. The depletion, or control, pulse had a spectrum centered at 1,000 nm with a full width at half maximum FWHM around 50 nm and pulse duration below 40 fs. In order to find precisely the zero delay between the visible pump pulse and near-IR (NIR) depletion beam, a characterization setup was built to measure the cross correlation (XC) at the exact position of the sample. The zero delay (τ) between the white light and pump laser was determined using the nonresonant coherent artifact in pure benzene generated in the same cell. The zero-delay point could be determined with an error smaller than 10 fs. The energies used for the pump were less than 80 nJ and for the depletion about 1.4 μJ. After focusing with concave mirrors these values correspond to 4 × 1014 and 5 × 1015 photons/cm2, respectively. The all-trans-β-carotene (M11) was purchased from Aldrich and used as received. The all-trans-carotenes M13 and M15 were synthesized by a reduction dimerization (McMurry reaction) as described in (Yoshizawa et al. 2003). All samples were prepared in N2 atmosphere, in order to avoid degradation. They were dissolved in benzene without any further purification, resulting in a maximum static absorption of the S0–S2 transition between 0.25 and 0.4 optical density. A 0.2-mm-thick sealed rotating cell with windows of 1 mm thickness was used in all measurements.

2.3 Results and discussion The depletion experiment consists of keeping the delay τ between the pump and probe constant and then scanning the delay T between the additional depletion

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pulse and the fixed pump-probe pulses. The idea of this scheme is to change the population after the excitation by intercepting the normal relaxation by depletion and to identify specific states or nodes of the energy flow network which are dependent on this additional pulse. The results are summarized in Fig. 3 and lead to a consistent behavior independent of the molecule: The depleted band is centered at the S1 absorption maximum (565, 590, 630 nm for M11, M13 and M15, respectively), while the S* band at the blue shoulder of the ESA is not affected at all by the depletion pulse (Fig. 3a–c). Further information can be obtained by scanning the depletion pulse while keeping the pump and probe pulses fixed in time. These curves are called depletion action traces because they measure the amplitude of the depletion effect against the delay between the pump and the depletion pulses. The depletion action trace measured at the maximum of the S1 absorption band (Fig. 3d–f) was fitted with one single exponential for the rise and two exponentials for the decay. The rise of the depletion action trace was always faster than our experimental resolution yielding values below 25 fs. The recovery of the depletion action trace shows a more complex dynamics which depend also on the specific molecule. It consists of two components, a recovery part described by a single time constant and a plateau, which lives as long as the S1 state. For M11 (Fig. 3d), the depletion action recovery time showed a time constant of 145 ± 10 fs using 1,000 nm which matches those decay times found previously using 795 nm for depletion (Wohlleben et al. 2004). For the M13 the depletion recovers with a time constant of 90 ± 10 fs (Fig. 3e), while for the M15 the recovery was about 50 ± 10 fs (Fig. 3f).

Fig. 3 Depletion spectra (first row) probed at τ = 2 ps and depletion transients (second row) for M11 (a and d), M13 (b and e) and M15 (c and f). The difference between the ESA measured without (continuous line) and with (traced line) depletion is represented by the gray area in figures a–c. The depletion transients were measured with the pump and probed separated by 2 ps.


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The variation of the dynamics of the depletion signal is in agreement with the dynamics of the ultrafast IR bands observed in other experiments (Ikuta et al. 2006; Fujii et al. 2003). In these studies the trend of a systematic decrease of the lifetime of the IR bands with increasing number of conjugated double bonds was observed which corresponds to an equivalent decrease of the depletion duration with increasing number of conjugated double bonds in our experiment. These IR bands were assumed to be initially generated by S2 and by dark states (1Bu− and 3Ag−) few femtoseconds later. In this case, the lifetime of the S2 was shorter than 10 fs for the carotenoids here studied. Of course, within our instrument response, we cannot observe such time scale unambiguously. However, a 10 fs lifetime for S2 would be in disaccord with fluorescence lifetime studies for, e.g., M11, which is in the range of 190–150 fs (Bachilo 1995). It suggests that indeed the near-IR band absorbing state is the S2 state and, therefore, the observed variation of the depletion duration is a manifestation of the S2 lifetime variation. This point was recently confirmed by the fluorescence measurement of several carotenoids homologs (Kosumi et al. 2006): the rise of the depletion action traces matches astonishingly well the observed fluorescence lifetime of the S2 state. Therefore, the whole dynamics can be completely described by the three-level energy model comprising of the electronic levels S2, S1, and S0 and their respective vibrational structures. The assumption of an additional electronically excited state is not necessary with respect to the pump-probe and pump-deplete-probe results as well as to the known fluorescence lifetime, spectrum position, and the S* lifetime. However, an energy level system with an intermediate state with a much faster lifetime than S2 would also lead to similar depletion action traces but its relevance with respect to the kinetic network remains probably negligible. These results are a successful example of QCS, where the presence of a shorttimed depletion (control) pulse helped to disentangle the intricate kinetics of carotenoids and, as a result, led to a better understanding of the relaxation network of this molecule family.

3

Enhancement of vibrational coherence and population transfer

In the last section (Section 2), we interfered with the population which was already in the excited state by using an additional control pulse. In this section, we will expand our control repertoire and tailor the initial excitation of population between ground and excited state using the manipulation of vibrational wavepackets as control knob in a prototype molecule.

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3.1 Tailored excitation of vibrational wavepackets The control of the interaction between tailored pulses and matter waves represents an especially interesting subgroup of experiments in QCS. The reason for that is dual: firstly, matter waves may represent motion along a reaction coordinate, whose manipulation is the general aim of coherent control; secondly, the pulse shapes for optimizing those vibrations are very often found to be trains of pulses with a temporal spacing between the sub pulses equal to an integer multiple of the vibrational period (Weiner et al. 1990; Mukamel and Yan 1991; Yan and Mukamel 1991). The effect of pulse trains on molecular vibrations can be explained using the model of classical oscillator. Excitation with pulse trains, where the subpulse separation is in phase with a given molecular vibrational leads to partial or total suppression of other modes with different oscillation periods. However, in the case of non-resonant excitation, enhancement of the vibrational coherence (oscillations amplitude) cannot be achieved if the total excitation energy of the multipulse does not exceed the energy of the Fourier-limited pulse (Mukamel and Yan 1991; Yan and Mukamel 1991). Mode enhancement on the basis of coherence effects in individual molecules as required for efficiently controlling chemical reactions is therefore only to be expected for a pulse whose spectrum has some overlap with an electronic transition (near-resonant or resonant). In this context, we investigate two complementary experimental situations. Initially, we apply tailored degenerate four wave mixing (DFWM) and distinguish near-resonant from non-resonant excitation regarding vibrational coherence enhancement (Fig. 4a). In a second experiment (Fig. 4b), we expand our first findings by investigating the dependence of the enhancement of vibrational coherence and population in the excited state on the excitation wavelength by detuning the excitation from near-resonant to resonant interaction. Here, we aim to separate the role of resonant and near-resonant excitation on the enhancement effect using equal total excitation energies for the Fourier- and the multipulse-excitation. By applying these two modified time-resolved techniques it

Fig. 4 (a) Absorption spectrum of all-trans-β-carotene and the near-resonant and the nonresonant DFWM excitation spectra. (b) Absorption spectrum of Nile Blue (LC6900) and the pump excitation spectra used in the TA experiment.


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is possible to investigate the role of tailored excitation from the nonresonant case, obviously not possible to measure in a TA setup, to the completely resonant situation. The general control mechanism is based on the precise control of the transient absorption.

3.2 Experimental setup 3.2.1

Tailored degenerate four-wave mixing setup

All pulses in the degenerate four wave mixing sequence were generated in a single non-collinear parametric amplifier (nc-OPA), pumped by a regeneratively amplified mode-locked Ti:sapphire laser (Clark-MXR Model CPA-1000), delivering pulses at 1 kHz repetition rate with 795 nm central wavelength. The nc-OPApulse was subsequently split into the three FWM-beams: pump, stokes and probe pulse. To shape the spectral phase of the first two pulses, a 4f-pulse-shaper with a 640 pixel spatial-light-modulator (Jenoptik SLM) (Stobrawa et al. 2001) was used. In the case of transform-limited pulses, a temporal resolution of 22 fs was achieved as determined by an instantaneous DFWM-signal in glass. By the same means, it was confirmed that the temporal shape of the pulse adjusted in the SLM was maintained after passage through the sample. Therefore, reshaping of the pulse could be excluded. To obtain the desired pulse trains for pump and stokes in the DFWMsequence, a sinusoidal phase function was applied on the SLM,

φSLM (ω ) = a ⋅ sin ( b ⋅ ω + c ) ,

(1)

with parameters a, b and c. Parameter a was set to 1.23 according to Zeidler et al. (2001), this values guarantees the best contrast between the subpulses. Parameter b, which defines the separation between the sub-pulses was varied to coincide with an integer multiple of the vibration period to be excited (see Section 3.3). Parameter c was set to zero in all measurements discussed below. The probe pulse remained unmodified and transform-limited in all measurements. A 250 μm thick cell was filled with all-trans-β-carotene in HPLC-grade cyclohexane, which was used as received (Sigma–Aldrich). Optical densities of about 0.7 at 480 nm were used in order to obtain feasible signals at excitation in the red wing of the S0→S2 transition of all-trans-β-carotene (Fig. 4a). 3.2.2

Tailored pump-probe setup

The pump and the probe pulses used in the transient absorption experiment were generated in two home-built, single-stage nc-OPA, pumped by a regeneratively


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amplified mode-locked Ti:sapphire laser (Clark-MXR Model CPA-1000). The spectral phase of the pump pulse was modulated with a liquid-crystal shaper with 2 × 128 independently tunable pixels (CRI LCM 256). In principle, the shaper is capable of amplitude and phase modulation, but in this experiment, amplitude shaping was not used. The spectral phase φSLM(ω) applied to the transform-limited pump pulse was a sinusoidal function with three parameters a, b and c, as discussed in Section 3.2.1. In this experiment, the enhancement due to multipulse excitation was measured with b = 56 fs. This chosen sub-pulse spacing corresponds to the vibrational period of the ring-breathing mode of Nile Blue at about 590 cm−1. The probe pulse remained transform limited (TL) in all measurements. Nile Blue was acquired from Sigma-Aldrich and dissolved in methanol (HPLCgrade). No further purification was performed. The sample holder was a 1 mm thick fused silica cell. Therein, solutions were prepared with 1 optical density at about 625 nm (Fig. 4b), the absorption maximum of Nile Blue in methanol. The samples were carefully checked for any possible photo-degradation in the course of the experiment.

3.3 Results 3.3.1

Enhancement of vibrational coherence

A typical near-resonant DFWM-transient consists of a fast rise of 160–200 fs followed by slow dynamics typical for the S1 electronic state of all-trans-βcarotene (Sugisaki et al. 2007). Superimposed on this slow dynamics is an oscillatory component containing information on wavepacket dynamics. Such signal can be analyzed using a fast Fourier transformation (FFT) after the slow dynamics is fitted and subtracted from the signal. The typical vibrational modes of all-trans-βcarotene measured with unshaped excitation for near- and non-resonant excitation spectrum are shown in Fig. 5a and b. The FFT spectra of the non-resonant transients with excitation at 550 nm are shown in Fig. 5a (Hauer et al. 2006b). The phase of all pulses was left unshaped. At 529 nm detection wavelength, three vibrational modes at 1,004, 1,156 and 1,532 cm−1 can be seen. These are the ground state modes of all-trans-β-carotene as known from frequency domain spectroscopy (Polivka and Sundstrom 2004). If the first two pulses in the DFWM-sequence are phase shaped, one of these modes can be excited exclusively over the others, as seen in Fig. 5a for the mode at 1,156 cm−1. The multipulses show excellent filtering capabilities. Undesired modes can be totally suppressed. A direct comparison of Raman intensities after shaped and unshaped non-resonant excitation is given in the same graph. A pulse train matching the fifth harmonic of the strongest mode seen with transform-limited pulses


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Fig. 5 FFT spectra with unshaped (red dots) and shaped (black line) (a) non-resonant and (b) near-resonant DFWM excitation. Excitation with multipulses with b = 144 fs, matching the 1, 156 cm−1 fifth’s harmonic, leads to selective excitation of this vibrational wavepacket and complete suppression of other vibrational wavepackets out-of-phase with this period. However, just with near-resonant excitation (b), the vibrational coherence at 1,156 cm−1 can be enhanced. In both graphs, the data was normalized by the FFT amplitude at 1,156 cm−1 measured with unshaped excitation (Hauer et al. 2006b).

was chosen. The shaped and the unshaped beams had equal energies of 60 nJ. The shaped spectrum shows only the desired vibrational wavepacket. Its intensity however, is less than 60% of that in the Fourier-limited (unshaped) case. In agreement with theory for non-resonant excitation (Loring and Mukamel 1985; Mukamel and Yan 1991; Yan and Mukamel 1991), the Fourier-limited case is the upper boundary for all transition probabilities. The multipulse filters out the desired vibrational wavepacket but does not enhance it. Like under non-resonant conditions, adequately spaced pulse trains allow for full control when near-resonant excitation is used. Figure 5b shows an FFT-spectrum after near-resonant excitation with Fourier-limited pulses. The oscillations are again in perfect agreement with frequency domain data after excitation with a multipulse matching the period of the respective mode (1,156 cm−1). Like in the non-resonant case, the pulse trains show excellent filtering capabilities. All detected vibrational wavepackets can be selectively excited while the others are suppressed completely (not shown) (Hauer et al. 2006b). However, Fig. 5 shows the most striking difference between multipulse excitation near and off resonance. In the former case, Raman intensities can be enhanced in comparison to a Fourierlimited pulse of equal energy. 3.3.2

Excited-state population and vibrational coherence

In order to understand in more detail the vibrational enhancement observed in Section 3.3.1, it is necessary to explain whether the enhancement is due to a pure vibrational coherence effect or if it the population enhancement is also participating. Near-resonant DFWM is a technique where diverse molecular contributions and different forms of signal generation mechanisms (e.g. wavepackets in both electronic

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states) overlap, leading to a time-resolved signal of complex analysis. Transient absorption in simple systems, on the other hand, can give signals where population and vibrational coherence can be easily separated. Thus, as a model system, we investigate Nile Blue as prototype molecule with tailored transient absorption. The transient absorption signal of Nile Blue and related molecules was already studied extensively (Braun et al. 2006; Wise et al. 1987; Florean et al. 2006). The maximum absorption of the S0–S1 transition is around 625 nm in methanol (Fig. 4b). The corresponding bleach signal lives about 1 ns. At about 650–670 nm the transient fluorescence signal shows a strong stimulated emission at earlier delay times. The first excited state absorbs at about 550 nm and has a long life time (~1 ns) which depends on the solvent. The TA signal at 550 nm has almost no contribution (less than 1%) from the ground-state bleach and is only due to ESA (Florean et al. 2006). Thus, the TA signal detected at this wavelength can be used as a probe for the population transfer to the excited state. Figure 6a shows the transient signal measured at 550 nm. Overlapped to the slow population dynamics of the excited electronic state, there are strong oscillations with a main oscillatory component exhibiting a period of 56 fs (about 590 cm−1). As discussed in Section 3.2.2, these oscillations originate from a ring-breathing mode of Nile Blue and dephase in methanol in 1.7 ps. The effect of the multipulse excitation on the level of the ESA and on the oscillatory amplitude was investigated with fixed sub-pulse spacing for several excitation wavelengths (see Fig. 4b). The amplitude of the ESA signal and of the FFT spectrum obtained with shaped excitation was compared to TL excitation. The enhancement the FFT amplitude and of the ESA signal level is shown in Fig. 6b and c respectively. It has to be pointed out that the total pulse energy was kept constant for every pulse shape. The enhancement effect under discussion is therefore of a non-trivial nature as mentioned in the introduction. The ESA signal (Fig. 6c) is enhanced when the pump is blue detuned from the absorption centre of Nile Blue. For a resonant excitation and slightly red detuned spectrum at about 640 and 655 nm, respectively, the multipulse cannot enhance the ESA signal. In this case, the multipulse actually transfers 10% less population to the excited state than the TL pulse with same total energy. When the oscillatory amplitude is analyzed (Fig. 6b), a similar wavelength dependence is obtained. For blue-detuned multipulse excitation, the amplitude of the oscillations is enhanced up to 80%. Excitation with a completely resonant pulse spectrum at about 640 nm generates weaker oscillations than excitation with a TL pulse does. If excited at the low energy wing of the absorption spectrum at about 655 nm, the enhancement curve for the FFTamplitude distinguishes from the ESA signal: In this case, the oscillation amplitude is enhanced almost by a factor 2, while a multipulse cannot optimize the population transfer. It is interesting to note that the enhancement curve obtained for the coherence in the excited state (Fig. 6b) resembles the linear absorption spectrum of Nile Blue shown in Fig. 4b. The lowest point of the enhancement curve coincides with the absorption maximum of Nile Blue in methanol (625 nm).


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Fig. 6 (a) Vibrational component of the unshaped TA signal (black line) after the slow population dynamics was subtracted. The fitting (red line) was adjusted using four vibrational components (330, 550, 600 and at 650 cm−1). (b) Vibrational coherence enhancement factor measured with b = 56 fs at 550 nm at several pump wavelengths. (c) Excited state enhancement factor measured with b = 56 fs at 550 nm at several pump wavelengths. In (b) and (c), the black line is a guide to the eye and the dotted line shows the values measured with unshaped (TL) excitation.

3.4 Discussion The two experiments discussed in the previous two sections complement each other in their results. Firstly, the filtering of vibrational modes can be achieved in both molecular systems. Secondly, the enhancement of such vibrational coherence happens in both systems just when the excitation is near-resonant. For non-resonant excitation as well as for completely resonant excitation spectra, the vibrational coherence enhancement is absent. The third aspect is related to the population enhancement. Vibrational coherence does not occur without population in the involved vibrational states. A higher vibrational coherence (i.e. more oscillation amplitude in the signal) may be achieved because the process is more resonant, and therefore excites more population, or because of a more favorable wavepacket formation. In the case of enhancement of vibrational coherence for near-resonant DFWM, where the signal amplitude is mainly due to the population difference between the involved states, it is then not possible to affirm a priori if the enhancement is the result of the optimized manipulation of vibrational coherence (leading to more oscillation amplitude in the signal), or just a population effect, or a combination of both. The results obtained by tailored TA in Nile Blue clarifies

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that: Enhancement of vibrational coherence is not just due to population enhancement. For example, for blue shifted excitation wavelengths the population in the excited state is enhanced just 10% (Fig. 6b), while the vibrational coherence is enhanced almost 90% (Fig. 6a). This is more dramatic for red shifted excitation where population in the S1 of Nile Blue is decreased why the vibrational coherence is enhanced by 90% again. In general, the degree of control over a quantum mechanical (QM) process depends on the decoherence effects, such as incoherence properties of the laser source and the interaction of our QM system with the bath. Assuming perfect coherence properties of our laser, the coupling between the system and the solvent is ultimately the main upper limitation. This interaction leads to a rapid dephasing of the oscillating scatterers, i.e. erasing the phase memory of the QM ensemble and diminishing the macroscopic signal. Here, the enhancement effect suffers from similar limits. As we have shown in Buckup et al. (2008), the enhancement effect depends on the oscillatory behavior of the electronic coherence during the near-resonant excitation. The central control parameter is the timing between the sequential pulses (parameter b, see Equation (1)). In this context, one would expect neither the enhancement of vibrational coherence nor excited states’ population using large sub-pulse separation in a system with strong coupling with the bath (Fig. 7). For example, in the case of b = 56 fs, a fast electronic relaxation would bring the coherence of the system almost to zero and the next sub-pulse could not benefit from latter interaction. Obviously it does not happen if the coherence time is long enough between the sub-pulses. This leads to an upper limit to the enhancement effect (Fig. 7a) and allow us to give a quantitative rule for the effect amplitude: If the electronic decoherence time is of the magnitude of the sub-pulse separation

Fig. 7 (a) Simulation (Buckup et al. 2008) of the maximization of the enhancement factor for the coherence (circles) and population (squares) in the excited state, normalized by values obtained with unshaped excitation. Graph was calculated using b = 56 fs. (b) Scheme showing a multipulse structure with separation b = 56 fs and two electronic coherence decays. Note that for this separation, in the case of the fast electronic coherence relaxation (dotted line), the coherence has almost relaxed between each sub-pulse. This does no happen for the slow electronic coherence relaxation (continuous line).


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(parameter b), only half of the maximum enhancement is achieved. From another perspective, the variation of the multipulse separation is a way to determine the electronic decoherence times.

4

Conclusions

The combination of coherent control methods as an additional degree of freedom in spectroscopy, which we name Quantum control spectroscopy (QCS), is a powerful experimental concept to disentangle complex dynamics. Among all its application scenarios, the systematic unraveling of electronic relaxation networks in biological molecules and the precise control of population using excitation of tailored vibrational wavepackets in the excited as well as in the ground state exemplify very well the valuability of this novel spectroscopic method. These approaches have been already employed to gain a deeper insight into the ultrafast dynamics of several polyatomic molecules and biological molecules systems. Particularly, in respect to the excitation of vibrational wavepackets as discussed in this work, the energy flow in natural as well as in artificial light-harvesting complexes have been manipulated by the excitation of low frequency vibrational wavepackets in the ground state via ISRS (Impulsive Stimulated Raman Scattering) (Herek et al. 2002; Savolainen et al. 2008b). The filtering of those wavepackets, associated with the internal conversion within the embedded chromophore (carotenoids), allowed for the control of the energy transfer. On one hand, this is surprisingly remarkable, since the artificial system differs on several aspects from the natural one, like structure of the complex. On the other hand, both systems have carotenoids as first chromophores and, since carotenoids are well know for their Raman scattering efficiency, one could expect that a similar process should be observed in both systems. In this work, we employed QCS to investigate system of less complexity. Firstly, we have shed more light into the relaxation network of all-trans-β-carotene and its homologs using an excitation scheme with two excitation pulses. Comparison between dynamics obtained with one and two excitation pulses allows to distinguish which target states are populated via internal conversion. Applying the pump-depletion-probe experiment, we have shown that all-trans-β-carotene and its homologs show a dynamics compatible with a three energy level scheme (S2, S1 and S0) and their vibrational manifold without the inclusion of additional intermediate electronic states. This experiment corroborates the results observed for the natural and artificial light harvesting complexes. Also in this work, we have modified the excitation pulse to investigate the enhancement of vibrational coherence in systems in solution. We have shown that vibrational coherence can be enhanced in biological relevant molecules with tailored pulses. Moreover, concomitantly with the enhancement of vibrational coherence, electronic population can be more efficiently excited from the electronic ground- to

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the excited-state. However, enhancement depends on the coupling of the system with its environment and is thus limited by loss of coherence. Enhanced vibrational coherence as well as population in the excited state can only be achieved in molecular system with electronic coherence times of the magnitude or larger than the sub-pulse separation.

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Photonic Integration Enables Single-Beam Nonlinear Spectroscopy for Microscopy and Microanalytics Bernhard von Vacano Physikalische Chemie, Philipps-Universität Marburg, Hans-Meerwein-Straße, D-35043 Marburg, Germany Current affiliation: BASF SE, Polymer Physics Department, Carl-Bosch-Str. 38, D-67056 Ludwigshafen, Germany

Marcus Motzkus Physikalische Chemie, Philipps-Universität Marburg, Hans-Meerwein-Straße, D-35043 Marburg, Germany Current affiliation: Physikalisch-Chemisches Institut, Universität Heidelberg, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany [email protected]

Abstract Using femtosecond pulse shaping techniques, nonlinear spectroscopy can be performed in an integrated approach, which implements different optical functions in respectively designed pulse shapes. A successful example is singlebeam coherent anti-Stokes Raman Scattering, allowing chemically selective analytics and imaging on the sub-micrometer scale. Here, pulse shaping is used to create multi-colour pulse sequences at defined temporal delays, and also to integrate a local oscillator for interferometric detection. Furthermore, the possibility of adaptive pulse shaping allows the versatile utilization of photonic crystal fibers as a welldefined broadband light source. The complex structure of such white light source is characterized by an integrated pulse characterization technique which is based on the same pulse shaper and particularly suited for microscopic applications.

Keywords: Pulse shaping, CARS, coherent control, nonlinear spectroscopy, microscopy, photonic crystal fiber, white light continuum

1

Introduction

With the availability of high intensity pulsed laser sources, nonlinear optical phenomena such as two-photon fluorescence (TPF) or coherent anti-Stokes Raman


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Scattering (CARS) have become usable for new approaches in spectroscopic analytics and microscopy. This includes stand-off detection of hazardous materials (Scully et al. 2002; Katz et al. 2008; Li et al. 2008), chemical detection in microfluidic devices (von Vacano et al. 2008), fluorescence imaging with highly reduced photobleaching of the labelling dyes, bright images with very high three-dimensional resolution, deep-tissue imaging and novel contrast mechanisms, based on intrinsic molecular properties such as vibrational energy levels (Zipfel et al. 2003; Xie et al. 2006). The challenge for any application is to master the laser systems and optical setup necessary. Generally, for a specific nonlinear process a dedicated experimental setup has to be established. In the case of CARS, this normally means synchronizing different pulsed laser sources, multi-path optical setups with additional nonlinear frequency conversion steps and consequently a high degree of experimental complexity (Fig. 1a).

Fig. 1 (a) Schematic depiction of a possible nonlinear optical CARS setup, demonstrating the various required elements of an experiment, including synchronization, frequency conversion, modulation, beam splitting and recombination. (b) Using a photonically integrated approach, the experimental complexity is replaced by a pulse shaping unit, which implements all necessary linear transformations of the pulses (modulation, creation of pulse sequences and delays, pulse compression), replacing a multitude of optical elements usually performing these tasks. Nonlinear frequency conversion is not necessary any more, if all needed laser frequencies are provided by a broadband laser source. Also, signal selection in the collinear scheme is performed by computer control of the pulse shaper (e.g. by phase-cycling).

PHOTONIC INTEGRATION ENABLES SINGLE-BEAM

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However, in the approach presented here, this experimental complexity is reduced to complex laser pulses, which nowadays can be created with a high degree of flexibility and control in a simple setup. The requirement is a coherent laser source spectrally broad enough and therefore at the same time emitting short enough pulses to sculpt from them all pulses needed for the experiment with the right spectrum, temporal envelope and timing. Such ultrabroadband femtosecond lasers are increasingly available, and new microstructured fiber technologies such as photonic crystal fibers (Russell 2003) can help to boost laser bandwidth dramatically (Ranka et al. 2000), at low pulse energies and with extremely high efficiencies. Pulse shaping is then employed to correct for all possible phase distortions of the laser system and additional elements along the optical path. Therefore, the experiment is integrated into a single beam of shaped pulses in an approach which can be termed “photonic integration” – reminiscent of integrated circuits in microelectronics, which themselves replaced conventionally wired circuits of vast amounts of electronic elements. Along these lines, the different required ingredients of a photonically integrated experiment will be discussed in this contribution. First of all, this is the broadband laser source, providing the “raw material” of coherent photons of different colors. Here, we present the approach to use nonlinear spectral broadening in a photonic crystal fibre, which can upgrade almost any standard fs-laser source to the necessary bandwidths. Secondly, the requirement of perfect control over the pulses means tight phase management of the coherent femtosecond pulses. This is simplified by novel tools presented here to measure and compensate phase distortions, which themselves make use of photonic integration. Finally, the “construction of the optical spectroscopic setup” is now transferred into designing shaped pulses and pulse sequences to perform the desired experiment simply by computer controlling the pulse shaper accordingly. In this context, we will give some practical examples mainly regarding CARS microspectroscopy as a prototypically demanding nonlinear spectroscopy, and finally discuss further developments and the prospects of these approaches.

2

Creation of broadband laser pulses in photonic crystal fibres

Photonic crystal fibers (PCF) are made from microstructured glass (Fig. 2a) and feature very interesting nonlinear optical properties tunable by the geometric design of the fibre layout (Russell 2003). Quite soon it was shown that PCF allow very efficient spectral broadening of unamplified femtosecond pulses (Ranka et al. 2000) to continua of frequencies easily extending over more than one optical octave. These remarkable supercontinua have led to a growing number of applications in optical coherence tomography, (Hartl et al. 2001) linear scanning confocal microscopy (McConnell 2004) and especially optical metrology (Udem

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et al. 2002). In all these examples, the continuum does not need to be compressed to pulses. In fact, extremely broad continua often show intensity and phase fluctuations on the pulse-to-pulse scale and therefore prevent viable ultrashort pulse compression. However, the still relatively high peak intensities of the spectrally broadened output pulses have also been employed for nonlinear spectroscopy, including pump- and probe transient absorption (Nagarajan et al. 2002) and already a variety of CARS experiment (Paulsen et al. 2003; Kano and Hamaguchi 2004; Kee and Cicerone 2004; Konorov et al. 2004). In all applications mentioned so far, the phase of the supercontinuum was left unmodified. Due to the nonlinear effects in the broadening process and the dispersion in the fiber, the spectral phase of PCF output generally possesses a complicated structure (Dudley et al. 2002), translating into pulses far from the Fourier-transform limit of shortest duration. However, as broadband laser source in conjunction with pulse shaping in a photonically integrated single-beam experiment, this spectral phase distortions can be compensated for and extremely short pulses generated. In this context, it is important to operate the PCF source in a highly stable regime which does not show the pulse-to-pulse fluctuations known from extreme broadening to supercontinua. PCF with micrometer sized cores and a high fraction of air-filling around them allow a dramatic shift of the dispersion curve, so that the fibres exhibit anomalous dispersion already in the visible spectrum (Fig. 2b). Additionally, the reduced effective mode area in such fibres leads to highly enhanced Kerr nonlinearites in the PCF. Both facts together are the foundation of broadband supercontinuum generation with weak ultrashort pulses. For practical applications, the ability to tune the transition of normal to anomalous dispersion, characterized by the zerodispersion wavelength ZDW, is of utmost importance. Pumping a fibre in the region of anomalous dispersion leads to the broadest spectra generated at a given pump energy due to a multitude of nonlinear processes possible in this regime, including soliton dynamics (Dudley et al. 2006). However, these processes are also highly susceptible to noise amplification and therefore not the prime choice for a most stable compressible laser source. Matching the ZDW directly to the laser pump wavelength leads to highly diminished contributions of such processes, as only a fraction of the laser spectrum overlaps with the anomalous dispersion region. At the same time, the gross dispersion upon propagation is very low, which is of course advantageous for maintaining the high peak intensity of the pump femtosecond pulses in the process of broadening. Pumping entirely in the normal dispersion region restricts nonlinear processes almost exclusively to selfphase modulation, resulting in very stable spectra, but not reaching the enormous octave-spanning optical bandwidths of typical supercontinua generated by solitonic processes in the anomalous pumping regime. Another important factor is the fibre length. Even when pumping in the anomalous dispersion region, a very short fibre does not allow solitonic processes to come into play and emphasises only the initial self-phase modulation. Generally, spectral broadening of femtosecond pulses typically already saturates at lengths of several centimetres, whereas spectral


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Fig. 2 (a) Schematic drawing of an index-guiding PCF. Two filled holes (or alternatively larger air-filled holes) introduce birefringence, and thus make this PCF polarization maintaining. (b) Group velocity dispersion curves for a set of PCF commercially available from Crystal Fibre A/S (Denmark). The fibres shown have zero-dispersion wavelengths (ZDW) at 750 nm (dashed curve), 775 nm (dash-dotted curve), 800 nm (solid curve) and 830 nm (dotted curve).

coherence (and as such compressibility) diminishes markedly for long fibres. Thus, to achieve as high coherence as possible, the fibre length has to be kept at a minimum for a given spectral broadening required. The experimental realization of supercontinuum generation as broadband laser source is straight forward (Fig. 3). It consists of a 100 fs laser oscillator as pump source protected from reflections by a Faraday isolator (FI). The fs-laser light has to be coupled into the micrometer sized core of the PCF. This requires lenses with focal distances in the millimetre range. To avoid unnecessary dispersion of the pump pulses and yet achieve a diffraction-limited spot without spherical aberrations, the input coupling can be chosen as single aspheric lens. For highest coupling efficiency it is important to match both the spot size and the numerical aperture to the PCF. The PCF itself is mounted on a three axis micrometer stage (XYZ), which can be operated manually and fine-tuned automatically by piezo actuators. Coupling the continuum out of the fibre is performed with a microscope objective (40x, achromatic), again mounted on a three-axis stage. Due to the increased bandwidth, an aspheric lens cannot be used here due to its chromatic aberration, and the additional dispersion of a microscope objective has to be put up with. This describes a research setup, to exchange for different fibers. Once an

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Fig. 3 Experimental setup of PCF continuum generation. The fs-laser oscillator is sent through a Faraday isolator (FI) and coupled into the PCF by an aspheric lens (L1). The PCF is mounted on a piezo-driven three-axis stage with active tracking on highest coupling efficiency. The generated broadband light is coupled out of the fibre by a microscope objective (L2). To accommodate for the birefringence of the polarization-maintaining fibre, λ/2 wave plates are placed on either side of the PCF.

optimal PCF is chosen, the supercontinuum generation can be simplified, e.g. by splicing the PCF directly into robust fibre-based fs-oscillators. Also, if a polarization maintaining PCF is mounted in the right geometry, no other polarization optics are needed. To assess the pulse-to-pulse intensity stability of the 80 MHz PCF source, output pulse trains of several µs duration were recorded with a fast photodiode and a GHz-oscilloscope. A statistical analysis resulted in 1–2% fluctuations of the output, while the fs-oscillator itself showed ~1% fluctuations. This proves that the insertion of the PCF does not introduce significant instabilities, which is the reason why it will successfully be used as broadband spectroscopy source. Extensive trials showed the main experimental parameters to optimize PCF continuum generation under the pretext of the pump energy of ~3 nJ in our system clearly are the fibre length and the relative spectral position of the pump with respect to the zero-dispersion wavelength. Both parameters are examined in Fig. 4.

Fig. 4 PCF continuum generation dependence on experimental parameters. The fibre used for this comparison was always of the type NL-PM-750. (a) Dependence on the length of the PCF, pumped in the anomalous dispersion region at 795 nm, starting from pure self-phase modulation at 6 mm length (solid curve) to broader spectra at 15 mm (dash-dotted curve) and 22 mm (dashed curve). In the latter cases, clearly solitonic broadening processes have begun to play a major role generating the huge increase in bandwidth. (b) Dependence of continuum generation in a 22 mm long PCF on the pump wavelength used. Shifting the pump wavelength closer to the zero dispersion wavelength (ZDW) results in reduced bandwidth until directly at the ZDW of 750 nm even at this length broadening purely occurs by self-phase-modulation (solid curve).


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The experimental findings substantiate that the desired, highly coherent yet moderately broadened SPM spectra can be achieved in short fibre lengths and with pumping close to, or at the zero-dispersion wavelength. Consequently, if pumping in the anomalous dispersion region, the choice of very short pieces of PCF is mandatory.

3

Characterization, compression and shaping of broadband pulses for microspectroscopy

In order to perform nonlinear spectroscopy with unamplified femtosecond lasers, the necessary light intensities have to be created by tight focusing. This fact and the vast field of applications in microscopic imaging make the pairing with a microscope ideal. Broadband femtosecond pulses, however, experience dispersion when propagating through optical elements such as microscope objectives. This leads to a distortion of the spectral phase and hence to broadening in time. Furthermore, the use e.g. of PCF supercontinuum does not even provide perfectly compressed pulses to start from. For optimizing any nonlinear signal, the pulses need to be compressed in situ, at the site of the experiment. Compression is generally achieved by applying a suitable correction to the spectral phase of the pulses. This can be done with utmost flexibility employing fs-pulse shaping techniques (Weiner 2000), which intrinsically are the central element of photonically integrated experiments already. The most straight-forward approach to obtaining correction phases is measuring the distorted phase of broadband pulses in situ and applying it sign-inverted as correction.

Fig. 5 Schematic depiction of the SAC-SPIDER pulse compression scheme. Here, due to constraints in the available laser energy, the chirped reference was deduced from the 100 fs-oscillator directly and not from the PCF output. However, full photonic integration also of this reference is possible for just slightly increased laser power.

Pulse characterization for experiments employing microscope objectives of high numerical aperture has to be performed in collinear geometry. There are several schemes which are generally applicable under these constraints (Fittinghoff et al. 1998; Monmayrant et al. 2003; Lozovoy et al. 2004; Sung et al. 2008), however, none of them allows real-time measurements. A pulse characterization technique which on the contrary is well-known for its real-time capabilities is spectral phase

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interferometry for direct electric-field reconstruction (SPIDER) (Iaconis and Walmsley 1998; Shuman et al. 1999): By sum-frequency mixing a doublet of the pulse to be characterized with a chirped reference pulse, spectrally sheared replicas are obtained which reveal the spectral phase through their interference pattern. The rapidly measured phase can immediately be fed back to a pulse shaper to achieve online compression. The pulse shaper in the setup at the same time provides photonic integration, as it serves for the dispersion-free creation of the test pulse doublet in this shaper-assisted collinear SPIDER (SAC-SPIDER) (von Vacano et al. 2006a). It is capable of characterization and simultaneous compression of broadband pulses in situ and greatly facilitates nonlinear microscopy with tight online phase management (Fig. 5). In our first implementation of SAC-SPIDER, the chirped reference beam was not yet fully integrated into the pulse shaping. With more laser energy available, this would easily be possible. In the meantime, shaper-based SPIDER experiments have been shown which are fully photonically integrated (Sung et al. 2008). Alternatively, we had earlier used an integrated iterative scheme based on evolutionary learning optimization (von Vacano et al. 2006a) while others have reported methods relying on scanning reference phase functions (Lozovoy et al. 2004). Common to these mentioned variants is the proof of complete pulse characterization and compression with a single shaped beam. To perform SAC-SPIDER measurements, the shaped beam of double pulses with chirped reference is focused into a nonlinear crystal at the sample position of the microscope. The frequency shifted replicas generated by sum-frequency generation now interfere in a spectrometer to yield the SPIDER interferogram. This interferogram is processed directly and the strongly varying spectral phase of the uncompressed supercontinuum pulse is obtained in situ (Fig. 6, dashed curve). This phase is now sign-inverted and automatically fed back to the pulse-shaper, where it is applied in addition to the double-pulse mask function. After this first iteration, the spectral phase is already almost flat (Fig. 6, dash-dotted curve). Second (Fig. 6, dotted curve) and third (Fig. 6, solid curve) iteration finally yield

Fig. 6 SAC-SPIDER retrieves the distorted phase in situ (dashed black curve), and allows on-line compression to achieve a flat spectral phase (solid curve).


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an almost perfectly flat spectral phase. Iteration is not intrinsically necessary and only makes SAC-SPIDER robust against deviations in the calibration, and also extends the validity of SPIDER from pulses with hundreds of femtosecond duration down to few-cycle pulses (von Vacano et al. 2007). In time domain, SAC-SPIDER compresses stable self-phase modulation continua from PCF broadening (such as shown in Figs. 4 and 6) from initially over 400 fs (FWHM) in the focus of the microscope setup to clean 14.4 fs (FWHM) pulses less than 4% over the calculated Fourier limit.

Fig. 7 SAC-SPIDER verification of additional phase shaping on top of perfectly compressed pulses in the focus of a microscope: (a) quadratic phases, (b, c) sinusoidal phases with a phase shift of 90°, (d) phase indent (von Vacano et al. 2007).

Fig. 8 Confirmation of compressed two-color pulses created in the focus of a microscope by pulse shaping. (a) Spectrally resolved cross-correlation (XFROG) of compressed pulse with 100 fs reference, (b) reconstructed temporal intensity profile. (c) XFROG trace of two color double pulse with 1 ps temporal spacing, (d) reconstructed temporal pulse shapes (von Vacano and Motzkus 2008).

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Starting from a flat spectral phase for compressed pulses, the pulse shaper can be used to add desired phase patterns to create tailored pulses and perfectly timed pulse sequences. Again, the additional phase functions can be directly verified by SAC-SPIDER measurement, as shown in Fig. 7. To be able to verify also pulse sequences extending over a range of picoseconds, e.g. of two-color pulse pairs created by pulse shaping, independent verification was performed with a spectrally resolved cross-correlation measurement, as shown in Fig. 8 (von Vacano and Motzkus 2008). Note that such two-color pulse pairs with adjustable delay already constitute a huge degree of photonic integration, and are the basis for typical pump-probe type time-resolved experiments.

4

Single-beam nonlinear spectroscopy with broadband shaped pulses

With everything in place: broadband laser source, pulse shaper and compressed ultrashort pulses in the microscope, single-beam CARS can be implemented as an exciting application of photonically integrated broadband nonlinear microspectroscopy. In contrast to other multiphoton techniques such as TPF or harmonic generation, CARS provides spectroscopic, chemically selective information without the need for any labelling (Fig. 9) (Cheng and Xie 2004; Volkmer 2005; Müller and Zumbusch 2007). The experimental setup to perform such a rather demanding nonlinear spectroscopy as CARS is very simple (Fig. 1b), if the powerful capabilities of pulse shaping are used to integrate photonic function usually performed by extensive optical layouts into complex pulse shapes. This “reduces” the experiment mostly to appropriate software for computer-controlling the pulse shaper.

Fig. 9 Energy level diagrams of the coherent anti-Stokes Raman scattering (CARS) process. (a) Pump (pu), Stokes (st) and probe (pr) photons generate a blue-shifted signal at the anti-Stokes frequency of pr. This process is resonant with a vibrational level at energy spacing ΔE. (b) Principle of time-resolved CARS experiments: Pump and Stokes excite molecular vibration, which is probed after a time delay τ by a probe pulse.


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Conventionally, the pulse shaper blocks the blue wing of the excitation pulses in order to detect blue-shifted CARS signal in that spectral region, while interferometric schemes use exactly this light as local oscillator (Cui et al. 2006; von Vacano et al. 2006b), as will be shown later. All single-beam CARS schemes in time- or frequency domain have to rely on pulse shaping. This ensures that the CARS signal can spectrally be resolved and separated from nonresonant background. The seminal single-beam CARS approach by Dudovich, Oron and Silberberg (Dudovich et al. 2002, 2003) used a multipulse sequence (Fig. 10a) and later analyzing the signal as a function of the multipulse temporal spacing. Direct Fourier-transform CARS is implemented with identical double pulses (Fig. 10b), while defined twocolor double pulses can be used to exclude ambiguities (Fig. 10c) and for additional polarization shaping and control of the excitation (Fig. 10d).

Fig. 10 Time-domain single-beam CARS schemes. The blue wing of the excitation spectrum is blocked for ω > ωcut to detect the weaker, blue-shifted CARS signal. (a) Multipulse approach with a sine modulation of the spectral phase ϕ(ω) leading to a pulse train with temporal spacing τm = 1/Ωm. (b) Fourier-CARS implemented with identical double pulses. (c) Two-color double pulses increase the experimental flexibility. (d) Even more complex pulse sequences allow for selective excitation (sinusoidal phase) and a time-delayed probing.

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The experimental setup consists of the three building blocks sketched in Fig. 1b, presented in more detail in Fig. 11. Femtosecond laser pulses (3 nJ) of a standard 100 fs laser oscillator at 795 nm are broadened in a photonic crystal fiber (PCF, NL-PM-800) of 23 mm length. The pulse shaper consists of a 640 pixel liquid crystal spatial light modulator (SLM) and 1,200 grooves/mm gratings placed at the focal length (200 mm) of a pair of spherical mirrors. In the shaper, a knife-edge (KE) can block the blue part of the excitation spectrum. Detection of CARS signal is accomplished by a spectrograph and CCD camera. Excitation light is rejected by a short-pass interference filter (passing λ < 760 nm). As an example, high-resolution single-beam CARS microspectroscopy performed on Toluene is shown in Fig. 12. Here, the double-pulse scheme (Fig. 10b) has been used. One can see the clear oscillatory pattern (Fig. 12a) of the timeresolved molecular vibrational response, which can be Fourier-transformed into highly resolved, chemically relevant CARS spectra (Fig. 12b). This already demonstrates the power of single beam microspectroscopy, as such detailed spectra can be achieved at sub-micrometer resolution in the focus of a microscope. These capabilities immediately bring to mind applications in chemically selective microscopic imaging, but also for a rapid identification of minute amount of substances. A very prominent example, where such non-invasive, rapid chemical analysis of tiny amounts of sample could be of great importance, is the detection of hazardous materials. For example, rapid assessment of materials is necessary at security controls at the airport, e.g. to detect trace amounts of explosives. Also, the occurrence of “white powder” sent in mail needs to trigger the right countermeasures. As an example, a mixture of two white crystalline powders was examined with photonically integrated single-beam CARS (Fig. 13). One component was potassium benzoate (KBenzoate), a widely used preservative, while the other was calcium dipicolinate, which is a marker substance for dangerous anthrax bacterial spores. (Scully et al. 2002) As can be seen, CARS microspectroscopy performed on individual crystallites is able to unambiguously discriminate the two substances even on the basis of single grains.

Fig. 11 Experimental setup, consisting of a photonic crystal fiber (PCF) based broadband femtosecond laser source, a 4f pulse shaper and a microscope setup with detection.


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Fig. 12 Single-beam CARS spectrum recorded of Toluene with 14.4 fs double pulses, created with pulse shaping at different delays τ. (a) The resulting signal shows oscillations versus τ. (b) These yield a high-resolution CARS spectrum after Fourier transformation (von Vacano and Motzkus 2008).

Fig. 13 Demonstration of single-beam CARS microanalytical capabilities on the example of discrimination of potassium benzoate crystals (KBenzoate, a harmless preservative) from calcium dipicolinate (CaDPA, a marker substance found in high concentrations in dangerous anthrax bacterial spores). This analysis is performed on the basis of single crystals from a white powder sample.

As mentioned before, photonic integration in single-beam CARS can go even further. It has been shown, that interferometric detection of the coherent CARS signals gives a drastic increase in measurement sensitivity and a linear dependence of the CARS signal on the concentration of analyte molecules, which is highly desired for chemical analytics of minority components. Normally, interferometric detection requires the integration of an interferometer into the CARS setup, with the need for perfect interferometric stability. Also, the local oscillator laser light has to be supplied at the blue-shifted wavelength of CARS signal, which makes necessary synchronizing another laser source or employing additional frequency

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conversion steps. Using the single-beam approach, the interferometric detection can again be photonically integrated by pulse shaping. Instead of blocking the blue wing of the excitation spectrum for spectroscopy, it is used as local oscillator, with full phase control over the interferometry by the pulse shaper (Fig. 14).

Fig. 14 Photonic integration of a blue-shifted local oscillator (LO) for interferometric CARS detection. The LO is derived as the blue wing of the CARS excitation spectrum, and tightly phase-controlled by pulse shaping. Hence, it intrinsically has interferometric stability.

Fig. 15 Experimental setup of fiber laser-based heterodyne single beam CARS. The laser source employed here is a commercial polarization mode-locked (PML) Er3+-fs-fibre laser, frequencydoubled with a periodically poled lithium niobate crystal (PPLN). “C” signifies collimator packages. The laser output is sent through a 4f-pulse shaper with a liquid crystal spatial light modulator (SLM). The red part of the spectrum is chopped (CH) for lock-in detection. CARS spectroscopy is performed in a flow cell; the signal is detected after an interference bandpass filter (IF) and a monochromator with a photodiode (von Vacano et al. 2008).

The power of interferometric detection, introduced in single-beam CARS without any increase in experimental complexity (again, it practically only means leaving out the knife-edge in the shaper and a modification in software) becomes visible when combined with novel, compact and robust fiber-based laser sources. These turn-key systems are now available at considerably lower cost and practically maintenance-free. They cannot yet compete with the bandwidth and power of conventional fs-oscillators, but the use of PCF broadening can mitigate the bandwidth issue, while interferometric detection gives the necessary boost in detection sensitivity to cope with lower excitation energies in the picojoule range (von Vacano et al. 2008). An exemplary setup is shown in Fig. 15, equipped with a microfluidic capillary liquid flow cell for on-line analytics of minute liquid streams, as they might occur in chromatography or “lab-on-a-chip” applications (Fig. 16).


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Fig. 16 Experimental results of interferometric single-beam CARS spectroscopy performed with an unamplified femtosecond fibre laser. (a) Fourier spectra of CH2I2 (black dashed curve) and CHBr3 (grey solid curve). (b) Detection of a chemical concentration gradient in a microfluidic capillary, showing the full exchange of CHBr3 (signal fraction shown as open squares and averaged black solid curve) to CH2I2 (filled grey squares and averaged grey dashed curve) (von Vacano et al. 2008).

5

Summary and outlook

Photonic integration has been shown to drastically reduce experimental complexity of nonlinear spectroscopic experiments, such as CARS. Instead of building huge optical layouts with numerous elements, everything is integrated into complex broadband laser pulses, which are created by pulse shaping. Therefore, the complexity of the experiment is transferred to the design of pulse sequences and the respective control and analysis software. This development will certainly be an attractive route for establishing nonlinear spectroscopies for sensing and imaging applications. First examples in the case of hazardous material detection and online chemical sensing have been given. The use of compact fiber lasers and integration of single-beam CARS microanalytics with microfluidics is another step towards integrated devices, which in the case of electronics only unleashed the full power of the new technologies. With even more powerful, compact lasers of high bandwidth, advances in fibre technology, pulse shaping and integrated optics, the prospects look promising also in the case of nonlinear spectroscopy.

References Cheng, J.X., Xie, X.S.: Coherent anti-Stokes Raman scattering microscopy: Instrumentation, theory, and applications. J. Phys. Chem. B 108(3), 827–840 (2004) Cui, M., Joffre, M., Skodack, J., Ogilvie, J.P.: Interferometric Fourier transform coherent antiStokes Raman scattering. Opt. Express 14(18), 8448–8458 (2006)

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Dudley, J.M., Gu, X., Xu, L., Kimmel, M., Zeek, E., O’Shea, P., Trebino, R., Coen, S., Windeler, R.S.: Cross-correlation frequency resolved optical gating analysis of broadband continuum generation in photonic crystal fiber: Simulations and experiments. Opt. Express 10(21), 1215–1221 (2002) Dudley, J.M., Genty, G., Coen, S.: Supercontinuum generation in photonic crystal fiber. Rev. Mod. Phys. 78(4), 1135–1184 (2006) Dudovich, N., Oron, D., Silberberg, Y.: Single-pulse coherently controlled nonlinear Raman spectroscopy and microscopy. Nature 418(6897), 512–514 (2002) Dudovich, N., Oron, D., Silberberg, Y.: Single-pulse coherent anti-Stokes Raman spectroscopy in the fingerprint spectral region. J. Chem. Phys. 118(20), 9208–9215 (2003) Fittinghoff, D.N., Squier, J.A., Barty, C.P.J., Sweetser, J.N., Trebino, R., Muller, M.: Collinear type II second-harmonic-generation frequency-resolved optical gating for use with highnumerical-aperture objectives. Opt. Lett. 23(13), 1046–1048 (1998) Hartl, I., Li, X.D., Chudoba, C., Ghanta, R.K., Ko, T.H., Fujimoto, J.G., Ranka, J.K., Windeler, R.S.: Ultrahigh-resolution optical coherence tomography using continuum generation in an air-silica microstructure optical fiber. Opt. Lett. 26(9), 608–610 (2001) Iaconis, C., Walmsley, I.A.: Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses. Opt. Lett. 23(10), 792–794 (1998) Kano, H., Hamaguchi, H.: Femtosecond coherent anti-Stokes Raman scattering spectroscopy using supercontinuum generated from a photonic crystal fiber. Appl. Phys. Lett. 85(19), 4298–4300 (2004) Katz, O., Natan, A., Silberberg, Y., Rosenwaks, S.: Standoff detection of trace amounts of solids by nonlinear Raman spectroscopy using shaped femtosecond pulses. Appl. Phys. Lett. 92(17), 171116 (2008) Kee, T.W., Cicerone, M.T.: Simple approach to one-laser, broadband coherent anti-Stokes Raman scattering microscopy. Opt. Lett. 29(23), 2701–2703 (2004) Konorov, S.O., Akimov, D.A., Ivanov, A.A., Alfimov, M.V., Zheltikov, A.M.: Microstructure fibers as frequency-tunable sources of ultrashort chirped pulses for coherent nonlinear spectroscopy. Appl. Phys. B 78(5), 565–567 (2004) Li, H., Harris, D.A., Xu, B., Wrzesinski, P.J., Lozovoy, V.V., Dantus, M.: Standoff and armslength detection of chemicals with single-beam coherent anti-Stokes Raman scattering. Appl. Opt. 48(4), B17–B22 (2008) Lozovoy, V.V., Pastirk, I., Dantus, M.: Multiphoton intrapulse interference. IV. Ultrashort laser pulse spectral phase characterization and compensation. Opt. Lett. 29(7), 775–777 (2004) McConnell, G.: Confocal laser scanning fluorescence microscopy with a visible continuum source. Opt. Express 12(13), 2844–2850 (2004) Monmayrant, A., Joffre, M., Oksenhendler, T., Herzog, R., Kaplan, D., Tournois, P.: Timedomain interferometry for direct electric-field reconstruction by use of an acousto-optic programmable filter and a two-photon detector. Opt. Lett. 28(4), 278–280 (2003) Müller, M., Zumbusch, A.: Coherent anti-Stokes Raman scattering microscopy. ChemPhysChem 8(15), 2156–2170 (2007) Nagarajan, V., Johnson, E., Schellenberg, P., Schellenberg, P., Parson, W., Windeler, R.: A compact versatile femtosecond spectrometer. Rev. Sci. Instrum. 73(12), 4145–4149 (2002) Paulsen, H.N., Hilligsoe, K.M., Thogersen, J., Keiding, S.R., Larsen, J.J.: Coherent anti-Stokes Raman scattering microscopy with a photonic crystal fiber based light source. Opt. Lett. 28(13), 1123–1125 (2003) Ranka, J.K., Windeler, R.S., Stentz, A.J.: Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm. Opt. Lett. 25(1), 25–27 (2000) Russell, P.: Photonic crystal fibers. Science 299(5605), 358–362 (2003) Scully, M.O., Kattawar, G.W., Lucht, R.P., Opatrny, T., Pilloff, H., Rebane, A., Sokolov, A.V., Zubairy, M.S.: FAST CARS: Engineering a laser spectroscopic technique for rapid identification of bacterial spores. Proc. Natl. Acad. Sci. USA 99(17), 10994–11001 (2002)


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Applications of Coherent Raman Scattering Alexei V. Sokolov Physics Department and Institute for Quantum Studies, Texas A&M University, College Station, Texas 77843, USA [email protected]

Abstract Recent explorations based on the concept of molecular coherence have lead to exciting developments, extending the ideas of coherence from atomic physics to more complex, molecular systems. Atomic coherence lies at the core of such fascinating phenomena as electromagnetically induced transparency, ultraslow light propagation, and lasing without inversion, which all have been subjects of research and debate over decades. In turn, macroscopic molecular coherence allows broadband collinear generation of Raman sidebands, opening possibilities for compression of optical sub-cycle pulses, and for non-sinusoidal field synthesis; increased coherence also enables improvements in optical detection and sensing applications.

Keywords: Molecular modulation, coherence, Raman, attosecond, ultrafast, spectroscopy

1

Background: Atomic coherence and EIT

Coherence effects have been a subject of intense studies in atomic and molecular physics. One important example from atomic physics is electromagnetically induced transparency (EIT) (Harris 1997; Scully and Zubairy 1997). EIT relies on negation of refraction and absorption by quantum interference, and allows unperturbed propagation of resonant laser beams through optically thick media. Adiabatic preparation of a single coherent superposition-state is at the heart of EIT; it typically works in the transient regime when the coherence does not have time to decay. The coherent transient is possible if the dephasing time of the transition of interest is reasonably long (or, in other words, if the transition has a narrow homogeneous linewidth, characteristic of for example Raman transitions). Lasers used in the experiments must then (generally) have pulse lengths shorter than this dephasing time. Applications of EIT include control of refractive index and efficient frequency conversion at maximal atomic coherence (Harris et al. 1990; Jain et al. 1996). One spectacular example is the demonstration of how EIT can be used to eliminate optical self-focusing in a near-resonant atomic medium (Jain et al. 1995) (Fig. 1).


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Fig. 1 Elimination of self-focusing using EIT (adapted from Jain et al. 1995; please see that paper for detail). Three pictures show the image of a 3.2-mm aperture through which the probe laser beam propagates: (left) a weak near-resonant laser beam (283 nm wavelength) propagates through Pb vapor unperturbed; (center) a strong laser beam, however, sees a strong nonlinear refractive index and self-focusing and filamentation of the beam occur. The experiment is performed at a 10-Hz repetition rate and the filamentation pattern changes from shot-to-shot; (right) when the coupling laser (406 nm wavelength) is applied the propagation of the beam is almost perfectly restored.

EIT has originally been studied, and experimentally demonstrated, for singlephoton detunings which are sufficiently small that the rotating wave approximation applies. The more recent work on molecular systems may be viewed as broadening the concept of EIT to include an arbitrary number of spectral components, which are spaced by a Raman frequency and far detuned from electronic states.

2

Novel light sources utilizing maximal quantum coherence in molecular gasses and solids

Gradual development of the solid-state mode-locked (Ti:Sapphire) laser technology resulted in pulse length reduction from 6 fs in late 1980s to just under 4 fs in recent years (Baltuska et al. 1997; Cavalieri et al. 2007a; Fork et al. 1987; Keller 2003; Nisoli et al. 1997; Verhoef et al. 2006). With this technology it became possible to “see how atoms in a molecule move during a chemical reaction”.1 Generation of subfemtosecond pulses would extend the horizon of ultrafast measurements to the time scale of electronic motion (Corkum and Krausz 2007; Krausz 2001, 2002), but by the end of the last century it was obvious that ultrashort pulse production by the “traditional” techniques got stuck at the “fewfemtosecond barrier”, and new techniques were needed in order to break this barrier. The year 2001 saw a breakthrough in ultra-short pulse generation. Workers in the field of High Harmonic Generation (HHG) have measured subfemtosecond pulses in the soft X-ray spectral region (Drescher et al. 2001; Hentschel et al. 1

The 1999 Nobel Prize press release: http://www.nobel.se/chemistry/laureates/1999/press.html.

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2001; Paul et al. 2001). Since then remarkable progress has been made toward production and characterization of ever-shorter pulses in this short-wavelength spectral region (Kienberger et al. 2004; Mairesse et al. 2004; Niikura and Corkum 2007; Sansone et al. 2006), including generation of phase stabilized few cycle pulses in the extreme-ultraviolet regime (Mauritsson et al. 2006). The Nobelprize-winning optical comb technology (Udem et al. 2002), which played a crucial role in allowing production of single well-controlled attosecond x-ray pulses (Baltuska et al. 2003), has also been recently extended to the vacuum-ultraviolet region (Gohle et al. 2005; Jones et al. 2005; Ozawa et al. 2008). The attosecond precision of the applied laser field shape has already allowed controlling electron localization in molecular dissociation (Kling et al. 2006), and multiphoton photoemission from metal surfaces (Dombi et al. 2006). The X-ray pulses have been used to study attosecond atomic, molecular, and condensed matter spectroscopy, and the physics of electron tunneling (Cavalieri et al. 2007b; Drescher et al. 2002; Meckel et al. 2008; Uiberacker et al. 2007). HHG is a unique source of X-ray pulses, but by their very nature these pulses are difficult to control because of intrinsic problems of X-ray optics; besides the conversion efficiency into these pulses is very low (typically 10−5) (Antoine et al. 1996; Christov et al. 1997; Corkum et al. 1994; Farcas and Toth 1992; Harris et al. 1993; Schafer and Kulander 1997). The Raman technique is another promising candidate for subfemtosecond pulse generation (Belenov et al. 1992; Kaplan 1994; Kaplan and Shkolnikov 1996; Yoshikawa and Imasaka 1993); impulsive Raman scattering has produced pulses as short as 3.8 fs in the near ultraviolet (Zhavoronkov and Korn 2002). Recent experiments have shown that a weak probe pulse can be compressed by molecular oscillations, which are excited impulsively by a strong pump pulse (Zhavoronkov and Korn 2002; Belenov et al. 1994; Nazarkin et al. 1999; Wittmann et al. 2000, 2001; Kalosha and Herrmann 2000). In a related work it was suggested (Kalosha et al. 2002) and demonstrated experimentally (Bartels et al. 2002) that molecular wavepacket revivals can produce frequency chirp, which in turn would allow femtosecond pulse compression by normal group velocity dispersion in a thin output window. A disadvantage of the impulsive excitation method is that the generated Raman coherence is several orders of magnitude smaller than the maximal possible value. If the pump intensity is increased, the nonlinear response of the medium becomes substantial and ultimately limits the level of molecular excitation; as a result the spectral broadening is far from maximal. Harris’s group at Stanford University has demonstrated the Raman light source proposed by the same group earlier. From the comparison with theory, we inferred that we were able to synthesize single-cycle pulsed waveforms (or pulse trains, with repetition period of 11 fs and pulse duration of 2 fs). Our light source is based on what we have called “molecular modulation”. We have predicted that coherent molecular oscillations can produce laser frequency modulation (FM), with a total bandwidth extending over the infrared, visible, and ultraviolet spectral regions, and with a possibility of subfemtosecond pulse compression (Harris and

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Sokolov 1997, 1998; Sokolov et al. 1999; Kien et al. 1999; Sokolov 1999; Sokolov and Harris 2003). The technique utilizes ideas of EIT and relies on adiabatic preparation of maximal molecular coherence. This coherence is established by driving the molecular transition with two single-mode laser fields, slightly detuned from the Raman resonance so as to excite a single molecular eigenstate. Molecular oscillation, in turn, modulates the driving laser frequencies, causing the collinear generation of a very broad FM-like spectrum. In our experiment the generated spectrum extends over many octaves of optical bandwidth (Sokolov et al. 2000, 2001c), and the sidebands range in wavelength from mid-infrared to vacuum ultraviolet (Fig. 2). We have demonstrated that these sidebands are mutually coherent over their temporal and spatial profiles, and that their relative phases can be easily adjusted (Sokolov et al. 2001b). Generation of subfemtosecond pulses with a spectrum centered around the visible region opens an unprecedented opportunity: The pulse duration may be shorter than the optical period and will allow sub-cycle field shaping. Sub-cycle shaping will allow synthesis of waveforms where the electric field is an arbitrary predetermined function of time, not limited to quasi-sinusoidal oscillations (Fig. 3). As a result, a direct and precise control of electron trajectories in photoionization and high-order harmonic generation will become possible. By the very nature of the generation process, the pulses produced by molecular modulation are perfectly synchronized with the molecular motion in the given molecular system, and provide a unique tool for studying molecular and electronic dynamics (Sokolov 2003). The key idea is to use the same molecular species for pulse generation and for the target. We envision producing a coherent molecular oscillation in the target cell, applying a tightly focused train of perfectly timed

Fig. 2 Typical spectrum produced by molecular modulation. Frequency components are generated collinearly, and then separated by use of a prism. For ultraviolet sidebands, fluorescence is visible.

E(t)

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Fig. 3 Pulse shaping, old and new: (a) “traditional” pulse shaping; (b, c) sub-cycle pulse shaping.


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pulses, and studying electronic properties as function of molecular coordinates. Possible extensions of this general technique range from probing ultrafast electronic dynamics in atoms to studying complicated multimode motion of complex molecules. In the frequency domain, molecular motion corresponds to a coherent superposition of molecular states. The fact that the pulse train can be synchronized with the molecular motion means that there is a possibility for different multiphoton paths, starting from different molecular levels, to interfere constructively or destructively. In this frequency-domain representation the analogy to EIT is particularly clear: with appropriate sideband phasing (or rather “anti-phasing”) we can expect an EIT-like molecular stabilization against multiphoton ionization. Over the course of recent years, several groups have made substantial advances in molecular modulation. Harris’s group has demonstrated a nearly 100% conversion efficiency in rotational molecular modulation (Yavuz et al. 2002), and used the Raman light source to demonstrate single-cycle pulse compression (Shverdin et al. 2005); Katsuragawa’s group has implemented substantial technological improvements and has shown careful cross-correlation measurements of waveforms obtained through molecular modulation (Katsuragawa et al. 2005). Marangos’s group has performed detailed studies of molecular modulation in gasses driven by nanosecond, as well as femtosecond laser pulses, and has also investigated the possibility to combine the nanosecond excitation with a femtosecond probe (Gundry et al. 2005a, b; Sali et al. 2004). Of particular pertinence to our results described below, is the multiplicative technique proposed and demonstrated by Harris and coworkers (Harris et al. 2002; Yavuz et al. 2003). Their technique relies on using three laser pulses that can be tuned to simultaneously excite two independent Raman transitions in two molecular species. As a result, they obtain a denser comb of Raman sidebands, with the total number of sidebands equal to the product of sideband numbers that would be produced by each of the molecular species alone. We have discovered an intriguing interplay of stimulated rotational Raman generation and collinear molecular modulation in the same molecular ensemble (Burzo et al. 2006). Briefly, we have observed that under fixed experimental conditions, by only varying the Raman detuning from the vibrational transition in D2, we can enhance or completely suppress the (self-starting) rotational stimulated Raman scattering. Our further investigation (Fig. 4) has shown (by comparing the experiment with theory) that this behavior may be related to EIT-like quantum interference within the multilevel molecular system (Burzo et al. 2007). Such interference effects could be used to control the molecular motion in a Raman gas medium and to selectively access different degrees of freedom in molecules. Harris’s multiplicative technique led to production of quasi-periodic waveforms, since in the frequency domain it produced a non-equidistant comb of sidebands. But even when only one molecular transition is used and the comb is equidistant, the carrier-envelope (absolute) phase of the generated pulses will slip from one pulse to the next, if sideband frequencies are not equal to integer multiples of their frequency difference. Until very recently, this was the case in most molecular

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modulation experiments. In order to fix the absolute phase, one needs to tune both driving lasers so as to simultaneously satisfy the condition of Raman resonance, and to make the laser frequencies equal to integer multiples of their frequency difference. Experiments along these lines have now been reported (Chen et al. 2008; Suzuki et al. 2008).

(a)

(b) (c)

(e)

(d)

(f)

Fig. 4 Collinear Raman generation obtained by driving the fundamental vibrational transition in molecular deuterium (adapted from Sokolov et al. 2001a). Pictures (a) and (b) show the dispersed light generated under identical conditions of pressure and driving field intensities. Fine tuning of the frequency difference of the driving fields near the vibrational Raman resonance results in suppression [part (a)] or enhancement [part (b)] of rotational Raman generation, as detailed in part (c). We conduct a theoretical analysis that allows us to attribute this behavior to quantum interference among the probability amplitudes of the three molecular states involved. Parts (d), (e), and (f) show the results of a numerical simulation which are in agreement with our experimental results.


Laser system producing fields E0 and E-1 tuned to drive the Raman transition

f-to-2f interference observed here

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Fig. 5 Our experiment on generation of pulse trains with fixed carrier-envelope phases. We observe f-to-2f interference on the pulse-synchronized CCD detector (as shown in the insert on the right, where color-coded field intensity is plotted as a function of transverse coordinates x and y). The exact position of the interference fringes is determined by the carrier-envelope phase of that particular pulse train (on that particular shot). SLM is the proposed location of the spatial light (phase) modulator.

Our results (Fig. 5) show that we can obtain an “f-to-2f” (fundamental to second harmonic) interference within the generated Raman frequency comb, when we properly tune the two lasers that drive a rotational Raman transition in hydrogen gas. Constructive or destructive f-to-2f interference signifies the particular (sine or cosine) electric field shape under the single-cycle pulse envelope for the individual pulses in the train. Obviously, any observation of the f-to-2f interference is conditional on tuning the two applied laser frequencies to be precisely equal to an integer multiple of their frequency difference. We have achieved such tuning; however, (as expected) the interference fringes shift randomly from shot to shot, since the two applied lasers are not phase-locked to each other. In terms of the synthesized field shape, that means that the waveform changes between sine and cosine shapes randomly. In principle it may be possible to phase-lock the two diode lasers (that are, in our experiment, seeding the two Ti:Sapphire regenerative amplifiers) to the teeth of a femtosecond frequency comb, and then stabilize and control the synthesized pulse shape. Instead, we plan to use unstabilized pulse trains as they are, but sort them according to the f-to-2f interference, and look at shot-by-shot correlation between ion production and the position of the interference fringes. That will be equivalent to studying ion production as a function of

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the pulses’ carrier-envelope phase. Our next step will be to add real-time control of the synthesized non-sinusoidal field shapes, by using a computer-controlled spatial light modulator to adjust phases of the Raman sidebands (Sokolov et al. 2005). A particularly intriguing possibility is to have the same molecular species in the test cell as the one used in the Molecular Modulator, such that the synthesized pulse train is synchronized with the molecular motion of tested molecules, as shown in the insert on the lower left of Fig. 5 (Sokolov 2003, 2004). We now turn to discuss molecular modulation on a different time scale (femtosecond instead of nanosecond), and using a different state of medium (solid state). We have observed broadband sideband generation in Raman-active crystals. Such crystals have been used by others for making Raman lasers that extend the spectral coverage of solid-state lasers. We use 50 fs laser pulses tuned such that their frequency difference is approximately equal to the Raman frequency, and observe generation of sidebands covering infrared, visible, and ultraviolet spectral regions. This behavior is analogous to what we have seen in our experiments with molecular gases, where we used nanosecond driving pulses, with pulse durations somewhat shorter than the Raman coherence lifetime. Since coherence lifetime in a solid is typically shorter than in a gas, the use of femtosecond (or possibly picosecond) pulses, when working with room-temperature crystals, becomes inevitable. The laser system that we use in this experiment is similar to the one used for femtosecond CARS spectroscopy (see next section). We use two computercontrolled optical parametric amplifiers (OPerA, Coherent, with frequency-mixing options), pumped by an amplified femtosecond laser (Mira + Legend, Coherent). We overlap the laser beams at a small angle (a few degrees, adjustable). The use of a non-zero angle is critical: for collinear beams we observe weak generation of only lowest-order sidebands. When the beams cross at angles between 2° and 8°, we observe efficient nonlinear generation of many spatially well-separated sidebands. Apparently, phasematching plays a significant role here (as opposed to the experiments with low-pressure molecular gases); due to the material dispersion, phasematching is optimized when sidebands of different colors propagate at different angles. Figure 6 shows sideband generation in a 1 mm-thick lead tungstate (PbWO4) crystal, with two pulses (at 588 and 620 nm, with parallel polarizations) applied at an angle of 4° (parts a, d, and e), and three pulses applied for part (b). We observe up to 20 anti-Stokes (AS) and 2 Stokes (S) sidebands projected on a white screen (Fig. 6a), with the pump beams and the first 3 AS beams being attenuated by a neutral density filter (Zhi and Sokolov 2007). When we measure the spectrum of the anti-Stokes sidebands, we observe several interesting features: the spectra of the lower-order sidebands show a rich structure, possibly due to simultaneous excitation of several Raman lines (in PbWO4, there are strong Raman transitions at 901 and 323 cm−1). The spectra of higher-order sidebands are cleaner; however, the spacing between them is an unexpected 450 cm−1.


(a)

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(b) (c)

(d)

(e)

Fig. 6 Raman sidebands generated in PbWO4 with two (a) and three (b) pulses applied to the crystal. Part (c) shows improved generation when a pair of chirped pulses is used. Plot (d) gives peak frequency of the generated sidebands [under condition similar to those for (a)] plotted as a function of the output angle; one input frequency is fixed, while the other is tuned; the fact that higher order frequencies remain fixed confirms the Raman-resonant nature of the process. Histogram (e), compared with a theoretical prediction (dotted line) gives a confirmation of the mutual coherence of the generated sidebands. More detail can be found in Zhi and Sokolov (2007), Zhi and Sokolov (2008), Zhi et al. (2008).

In further experiments we demonstrate that we can use three input pulses to obtain a two-dimensional array of beams of varying colors (Fig. 6b). For example, we use pulses at 804, 730, and 604 nm to generate up to 50 beams of different colors. In addition, we show that improved generation can be obtained (Fig. 6c) when a pair of chirped pulses is used (Zhi and Sokolov 2008).

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Coherent multiple-order sideband generation in the femtosecond regime is not limited to PbWO4, but can be obtained in other solids. For example, we have repeated our above-described experiments with diamond (which has a strong and relatively narrow Raman line at 1,332 cm−1) (Zhi et al. 2008). We have obtained qualitatively similar results, with the main quantitative differences possibly coming from the fact that the Raman spectrum of diamond is less complicated, and also from the fact that diamond has a broader transmission window and correspondingly smaller dispersion. Nonlinear frequency conversion in diamond is very efficient (similar to PbWO4); for example, in one experiment we have measured 7% conversion efficiency into the first anti-Stokes sideband and 1% into the second antiStokes and first Stokes (Zhi et al. 2008). There must be substantial room for improvements in the efficiency of this experiment: we plan to investigate a possibility of using somewhat longer pulses (possibly obtained by pulse shaping) to closer match the coherence lifetime in these crystals (which we have measured to range from 1 to 10 ps, depending on the crystal). Finally, we have performed an interference measurement, which confirmed our expectation that the generated beams possess good mutual spatial and temporal coherence (Zhi and Sokolov 2007). Due to such coherence, in the future this broadband light source may be used for synthesis of subfemtosecond light waveforms. We propose to build a setup that will allow us to recombine the generated sidebands in space and time, and through spectral modification allow synthesis of subcycle waveforms. We then propose to use this light source in a photo-ionization experiment, using our ion-detection system developed for the nanosecond experiment. In addition, the technique of molecular modulation in gasses can be used to generate coherent radiation in the ultraviolet (UV) and infrared (IR) spectral regions, where laser sources are not readily available. One possibility is to use circular polarizations and produce a “rotating molecular wave-plate” in order to achieve single-sideband conversion (Sokolov et al. 2001a). Another possibility is the multiple sideband generation with controlled spectral power distribution. We have tested this possibility by a numerical simulation (Shon et al. 2002). A particularly promising idea is to use appropriate transverse modes in a glass capillary (Burzo et al. 2006), or to design dispersion in a hollow-core PCF, in order to produce single-sideband conversion similar to Shon et al. (2002). Eventually, applications to lithography, chemistry, biology, and astronomy may become practical.

3

FAST CARS for rapid identification of chemical and biological unknowns

Macroscopic molecular coherence, prepared and manipulated by pulsed laser fields, enables improvements in optical microscopy, detection and sensing applications. As one example, we have developed a laser spectroscopic technique for real-time detection of bacterial endospores (Scully et al. 2002). Our hybrid technique combines


85

the robustness of frequency-resolved coherent anti-Stokes Raman scattering (CARS) with the advantages of time-resolved CARS spectroscopy (Dogariu et al. 2008; Pestov et al. 2007b, 2008a). Instantaneous coherent broadband excitation of several characteristic molecular vibrations and the subsequent probing of these vibrations by an optimally shaped time-delayed narrowband laser pulse help to suppress the nonresonant (NR) background and to retrieve the species-specific signal. This pulse configuration mitigates the nonresonant four-wave-mixing noise while maximizing the Raman-resonant signal, allowing a rapid and highly specific detection even in the presence of multiple scattering. We have called this combination of broadband preparation and frequency-resolved detection Hybrid CARS. The same method of short-pulse excitation and time-delayed narrow-band probing was developed as early as 1980 by Zinth and coworkers with the aim of improving the spectral resolution of CARS beyond the limit of homogeneous linewidth (Zinth 1980; Zinth et al. 1982). Similar mixed time-frequency methods have recently been realized by other groups (Kano and Hamaguchi 2006; Nath et al. 2006; Prince et al. 2006). Our work on Hybrid CARS has been based on our earlier experience with IR, visible, and UV coherent Raman spectroscopy (Pestov et al. 2005, 2007a, 2008b). Figure 7 shows our experimental setup. We employ a Ti:Sapphire regenerative amplifier (Legend, Coherent) with two evenly pumped OPAs (OPerA-VIS/UV and OPerA-SFG/UV, Coherent). The output of the first OPA (λ1 = 712–742 nm, tunable; Δv1 ≈ 230 cm−1) and a small fraction of the amplifier output (λ2 = 803 nm, Δv2 ≈ 500 cm−1) are used as pump and Stokes beams, respectively. The output of the second OPA, the probe beam (λ3 = 578 nm), is sent through a home-made pulse shaper with an adjustable slit (see Fig. 7) that cuts the bandwidth of the pulse. As follows, the Stokes and probe pulses pass through delay stages (DS1,2) and then all the three beams are focused by a convex 2-in. lens (with the focal length f = 200 mm) on the rotated

Fig. 7 Schematics for the CARS experiment: DS1,2 are computer controlled delay stages; L1-3 are lenses; BPF+SPF is a set of bandpass and shortpass filters; M1,2 are alignment mirrors; CCD is a charge-coupled device attached to the spectrometer. The insets in the top-left corner show typical spectra of the pump, Stokes, and probe pulses on the sample. More detail can be found in Dogariu et al. (2008), Pestov et al. (2005, 2007a, b, 2008a, b).

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sample. The scattered light is collected with a 2-in. achromatic lens (f = 100 mm) and focused onto the entrance slit of the spectrometer (Chromex Spectrograph 250is) with a LN-cooled CCD (Spec-10, Princeton Instruments) attached. The CARS traces taken on NaDPA powder for different pump wavelengths are shown in Fig. 8. Streak-like horizontal lines are the signature of excited NaDPA Raman transitions while the broadband pedestal is the NR background. As expected, the tuning of the pump wavelength spectrally shifts the NR background leaving the position of the resonant lines untouched. Note also that the two contributions exhibit different dependence on the probe delay. The magnitude of the NR background is determined by the overlap of the three laser pulses and therefore follows the probe pulse profile. Relatively long decay time of the Raman transitions under consideration favors their long-lasting presence and makes them stand out when the probe is delayed. The same optimized hybrid CARS scheme can be applied to other problems requiring chemical specificity and short signal acquisition times; we explore possibilities for medical, industrial, and defense applications. We now demonstrate that conceptually the same pulse configuration can be obtained from a simpler (oscillator-only) laser system. The five-orders-of-magnitude increase in the pulse repetition rate accompanied by four-orders-of-magnitude decrease in the pulse energy (compared to Dogariu et al. [2008], Pestov et al. [2007b, 2008a]) makes our new system naturally suited for coherent Raman microscopy (Petrov et al. 2005, 2007), where lower-energy pulses are naturaly needed

Fig. 8 The CARS spectrograms recorded on NaDPA powder for different pump wavelengths, i.e. CARS spectrum as a function of the probe pulse delay for λ1 equal to: (a) 712 nm, (b) 722 nm, (c) 732 nm, (d) 742 nm. More detail can be found in Dogariu et al. (2008), Pestov et al. (2007b, 2008a).


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to avoid laser-induced sample damage. An ultrashort IR pulse from a laser oscillator is used for impulsive excitation of molecular vibrational modes, while its spectrallynarrowed second harmonic, generated in the nonlinear crystal and delayed with respect to the pump pulse, probes the induced coherent oscillations of the molecular ensemble (Peng et al. 2009). We use a mode-locked Ti:Sapphire laser oscillator (Kapteyn-Murnane Labs, TS laser kit, 85 MHz rep. rate) with the spectral full-width-at-half-maximum exceeding 100 nm and the center wavelength of 810 nm. The oscillator output is focused onto a 5-mm thick LBO crystal (CASTECH) to produce the probe light at 405 nm via second harmonic generation (SHG), where the SHG spectrum is narrowed down to 0.6

λmax, nm ε, M/cm

262 (5,100)a,b

259 (18,000)

348 (5,600)

τrelease

Microseconds

Sub-nanosecond

Sub-nanosecond

Suitable conditions and solubility

Solubility depends on the o-NB substituents. Release rate is highly solvent and pH dependentc

Limited to physiological conditions (aqueous, pH > 6)

Limited to physiological conditions (aqueous, pH > 6)

Fluorescence reporting

No fluorescent

PPG and photoproduct Fluorescence increase are both noncan be used to report fluorescent release

Photochemistry and Photochemistry involves photoproduct many intermediates and is not well understood.c Nitroso photoproduct is likely to have adverse physiological effects

Photochemistry is clean and well characterized. Photoproduct is photostable under irradiation conditions

Photochemistry is extremely clean and very well characterized. Photoproduct is photostable and exhibits low toxicity

a

Summarized in Goeldner and Givens (2005, chap. 1) Based on α-carboxy-2-nitrobenzyl protected glutamate c Pelliccioli and Wirz (2002) b

Fig. 7 Proposed photocaging of acyclovir via the amine (8) and the alcohol (9). Proposed photocaging of acyclovir including a ‘handle’ for attachment (10).

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Fig. 8 Proposed attachment of photocaged acyclovir to nanoparticles embedded in therapeutic contact lenses or artificial corneas. Acknowledgments The authors acknowledge the contributions of their coworkers whose names appear in the references to this review. This work has been generously supported by the Natural Sciences and Engineering Council of Canada, by the Canadian Institutes for Health Research, by the Province of Ontario and by the Canadian Foundation for Innovation.

References Abdullah, K.A., Kemp, T.J.: Solvatochromic effects in the fluorescence and triplet–triplet absorption spectra of xanthone, thioxanthone and N-methylacridone. J. Photochem. 32(1), 49–57 (1986) Blake, J.A., Gagnon, E., Lukeman, M., Scaiano, J.C.: Photodecarboxylation of xanthone acetic acids: C–C bond heterolysis from the singlet excited state. Org. Lett. 8(6), 1057–1060 (2006) Blake, J.A., Lukeman, M., Scaiano, J.C.: Photolabile protecting groups based on the singlet state photodecarboxylation of xanthone acetic acid. J. Am. Chem. Soc. 131(11), 4127–4135 (2009) Bosca, F., Miranda, M.A.: Photosensitizing drugs containing the benzophenone chromophore. J. Photochem. Photobiol. B 43(1), 1–26 (1998)

USE OF KETOPROFENATE AND XANTHONATE PHOTOCAGES

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Bosca, F., Marin, F.L., Miranda, M.A.: Photoreactivity of the nonsteroidal anti-inflammatory 2-arylpropionic acids with photosensitizing side effects. Photochem. Photobiol. 74(5), 637– 655 (2001) Cleator, G.M., Klapper, P.E.: Herpes Simplex. In: Zuckerman, A.J., Banatvala, H.E., Pattison, J.R. (eds.), Principles and Practice of Clinical Virology, 4th edn. Wiley, Chichester (2000) Coen, D.M., Safrin, S.: Herpes Simplex Virus and Varicella Zoster Virus. In: Richman, D.D. (ed.), Antiviral Drug Resistance. Wiley, London (1996) Costanzo, L.L., De Guidi, G., Condorelli, G., Cambria, A., Fama, M.: Molecular mechanism of drug photosensitization. II. Photohemolysis sensitized by ketoprofen. Photochem. Photobiol. 50(3), 359–365 (1989) Cosa, G., Martinez, L.J., Scaiano, J.C.: Influence of solvent polarity and base concentration on the photochemistry of ketoprofen: independent singlet and triplet pathways. Phys. Chem. Chem. Phys. 1(15), 3533–3537 (1999) Cosa, G., Llauger, L., Scaiano, J.C., Miranda, M.A.: Absolute rate constants for water protonation of 1-(3-benzoylphenyl) alkyl carbanions. Org. Lett. 4(18), 3083–3085 (2002) Goeldner, M., Givens, R.: Dynamic Studies in Biology; Wiley-VCH: Weinheim, Vol. 1. (2005) Heinz, B., Schmidt, B., Root, C., Satzger, H., Milota, F., Fierz, B., Kiefhaber, T., Zinth, W., Gilch, P.: On the unusual fluorescence properties of xanthone in water. Phys. Chem. Chem. Phys. 8(29), 3432–3439 (2006) Laferriere, M., Sanramé, C., Scaiano, J.C.: A remarkably long-lived benzyl carbanion. Org. Lett. 6(6), 873–875 (2004) Ley, C., Morlet-Savary, F., Fouassier, J.P., Jacques, P.: The spectral shape dependence of xanthone triplet–triplet absorption on solvent polarity. J. Photochem. Photobiol. A 137(2–3), 87–92 (2000) Liesegang, T.J., Melton, L.J., Daly, P.J., Ilstrup, D.M.: Epidemiology of ocular herpes simplex. Incidence in Rochester, Minn, 1950 through 1982. Arch. Ophthalmol. 107(8), 1155–1159 (1989) Llauger, L., Miranda, M.A., Cosa. G., Scaiano, J.C.: Comparative study of the reactivities of substituted 3-(benzoyl) benzyl carbanions in water and in DMSO. J. Org. Chem. 69(21), 7066–7071 (2004) Lukeman, M., Scaiano, J.C.: Carbanion-mediated photocages: rapid and efficient photorelease with aqueous compatibility. J. Am. Chem. Soc. 127(21), 7698–7699 (2005) Martinez, L.J., Scaiano, J.C.: Transient intermediates in the laser flash photolysis of ketoprofen in aqueous solutions: unusual photochemistry for the benzophenone chromophore. J. Am. Chem. Soc. 119(45), 11066–11070 (1997) Monti, S., Sortino, S., De Guidi, G., Marconi, G.: Photochemistry of 2-(3-benzoylphenyl) propionic acid (ketoprofen). Part 1. A picosecond and nanosecond time resolved study in aqueous solution. J. Chem. Soc., Faraday Trans. 93(13), 2269–2275 (1997) O’Brien, J.J., Campoli-Richards, D.M.: Acyclovir: an updated review of its antiviral activity, pharmacokinetic properties and therapeutic efficacy. Drugs 37(3), 233–309 (1989) Papageorgiou, G., Barth, A., Corrie, J. E. T.: Photochem. Photobiol. Sci. 4, 216–220 (2005) Pelliciolli, A. P., Wirz, J.: Photochem. Photobiol. Sci. 1, 441–458, 74 (5), 637–655 (2002) Pepose, J.S., Leib, D.A., Stuart, P.M., Easty, D.L.: Herpes simplex virus diseases: anterior segment of the eye. In: Holland, G.N., Wilhelmus, K.R. (eds.), Ocular Infection and Immunity. Mosby, St. Louis, MO, pp. 905–932 (1996) Scaiano, J.C.: Solvent effects in the photochemistry of xanthone. J. Am. Chem. Soc. 102(26), 7747–7753 (1980) Scaiano, J.C., Weldon, D., Pliva, C.N., Martinez, L.J.: Photochemistry and photophysics of 1-azaxanthone in organic solvents. J. Phys. Chem. A 102(35), 6898–6903 (1998) Toma, H.S., Murina, A.T., Areaux, R.G. Jr., Neumann, D.M., Bhattacharjee, P.S., Foster, T.P., Kaufman, H.E., Hill, J.M.: Ocular HSV-1 latency, reactivation and recurrent disease. Semin. Ophthalmol. 23(4), 249–273 (2008) Zimmerman, H.E.: J. Am. Chem. Soc. 117(35), 8988–8991 (1995)

Ultrasensitive Laser Analysis of Nanostructures: Theoretical Background and Experimental Performance Sergey V. Gaponenko B.I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, Minsk 220072 Belarus [email protected]

Abstract The role of incident electromagnetic field enhancement and photon density of states enhancement with respect to spontaneous scattering (Raman and Rayleigh processes) and spontaneous emission of photons is discussed. Field enhancement and density of states effects do manifest themselves in the same manner both in photoluminescence and scattering processes. Differences in scattering and luminescence enhancement are due to quenching processes which are less pronounced for scattering but crucial for luminescence because of finite internal relaxation processes between excitation and emission events. We consider recent experimental results on photoluminescence enhancement of organic molecules and quantum dots with polyelectrolyte spacers of the order of 10 nm to provide optimal balance of enhancement and quenching factors. A model of the so-called “hot point” in surface enhanced Raman spectroscopy is elaborated in terms of simultaneous incident field and density of states enhancements and high Q-factors at excitation and scattered frequencies.

Keywords: Nanoplasmonics, surface enhanced spectroscopy, Raman scattering, spontaneous emission

1

Introduction

Nanostructures with characteristic surface relief of the order of 10…100 nm are known to modify spatial distribution of incident electromagnetic field. Local field enhancement results in enhanced absorption of photons by molecules or nanocrystals adsorbed at the surface. The effect is extremely pronounced in metal – dielectric structures because of surface plasmon resonance. A systematic application of the field enhancement in Raman scattering enhancement (Surface Enhanced Raman Scattering, SERS) and in photoluminescence (PL) enhancement with respect to


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molecular probes is followed nowadays by application of the effect with respect to nanocrystals (quantum dots) adsorbed at metal-dielectric nanotextured surfaces. It is the purpose of the present contribution to review mechanisms of Raman scattering and photoluminescence enhancement factors in the context of their application to enhanced luminescence of molecules and quantum dots and Raman scattering. We consider not only the local field enhancement in terms of excitation process but also photon density of states enhancement effect on photon emission processes with Raman and Rayleigh scattering as specific photon emission processes. In this consideration, scattering of light experiences enhancement as spontaneous emission does. Therefore field enhancement and density of states effects should manifest themselves in the same manner in photoluminescence and scattering processes. Differences in scattering and luminescence enhancement are due to quenching processes which are crucial for PL and less pronounced for scattering. We consider recent experimental results on single molecule detection by means of SERS, PL enhancement of molecules and quantum dots and the approaches to efficient substrates fabrication for the purposes of ultrasensitive spectroscopy.

2

Spontaneous emission and scattering of photons in terms of quantum electrodynamics

In terms of quantum electrodynamics, spontaneous emission of photons by a really excited quantum system and spontaneous scattering of photons by a virtually excited system are considered alike in a sense that the number of emitted/scattered photons I0(ω΄) into a mode with frequency ω΄ per unit time for both type of processes is directly proportional to the incident photon flux I0(ω) with ω being the frequency of incident electromagnetic radiation, and density of photon states D(ω΄). In other words, one can write an expression (see, e.g. Barnes 1998; Berestetskii et al. 1989; Sushchinskii 1969) I0(ω΄) = I0(ω)[Interaction Term]D(ω΄).

(1)

Interaction term is to be explicitly calculated for every specific process taking into account interaction cross-section, quantum yield (for spontaneous emission) for a given quantum system and quantum process under consideration. In a specific case of resonant (Rayleigh) scattering one has always ω = ω΄. The role of photon density of states (DOS) in spontaneous emission of photons is well established and recognized (Barnes 1998; Bykov 1993). Since the pioneering paper by Purcell (Purcell 1946) predicting modification of spontaneous decay rates for radiofrequencies and first experiments in the optical range by K.H. Drexhage in 1970, modification of spontaneous decay rate because of spectral/spatial redistribution of photon density of states have been demonstrated for a number of mesoscopic structures with molecules, ions, and quantum dots used as elementary

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probe quantum systems. A non-exhaustive list includes microcavities (De Martini et al. 1987), photonic crystals (Petrov et al. 1998; Mogilevtsev and Kilin 2004), heterostructures and interfaces (Lavallard et al. 1996), dielectric slabs (Rikken 1995), biomembranes (Petrov et al. 1999). However photon density of states effects on spontaneous scattering of light in mesoscopic structures has been involved into consideration only recently (Gaponenko 2002; Zuev et al. 2005; Guzatov and Gaponenko 2009). Introduction of photon DOS effects into consideration of giant Raman signals in surface enhanced spectroscopy should noticeably contribute to the value of experimentally observed enhancement factors. It is reasonable to note one essential physical difference of elastic (Rayleigh) scattering of photons as compared to spontaneous photon emission and inelastic (Raman) scattering. Spontaneous emission and inelastic scattering result in photons with different frequency as compared to the incident radiation. For this reason elementary acts of photon emission/scattering can be understood only within the framework of quantum electrodynamics. Elastic scattering results in modification of direction and polarization of incident light and most probably can be completely understood in terms of classical wave theory as a result of multiple scattering and interference of scattered waves. Therefore the photon DOS concept is not necessary to describe elastic scattering in complex nanostructures. However certain intuition promoted by the DOS concept can be helpful in many complex structures for which correct calculation of multiple light scattering is difficult but tentative redistribution photon DOS is qualitatively understood.

3

Field enhancement in metal-dielectric structures

In metal-dielectric nanostructures like, e.g. nano-textured coinage metal surface, metal fractal clusters, dispersed metal colloids or isolated metal nanoparticles strong local enhancement of incident electromagnetic field is well established and documented. These structures are known to provide strong surface enhanced Raman scattering (SERS) (Chang and Furtak 1982; Kneipp et al. 1999, 2006; Gaponenko et al. 2001) which is typically interpreted in terms of local field enhancement factors and chemical enhancement mechanisms. In many cases the structures exhibiting SERS effect are found to exhibit noticeable enhancement of photoluminescence yield (Sokolov et al. 1998; Kulakovich et al. 2002 and references therein). Examples of metal-dielectric nanostructures showing enhancement of Raman scattering and/or luminescence are shown in Fig. 1. Surface plasmon resonances inherent in metal nanoparticles embedded in a dielectric ambient medium promote development of high local light intensity in the close vicinity of a particle. In coupled nanoparticles and in purposeful arrangement of particles as well as with purposeful shaping of particles the electromagnetic field enhancement can further be magnified. This phenomenon forms the basic prerequisite effect in linear and non-linear spectroscopies including metal nanotextured

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surface-enhanced and metal tip-enhanced fluorescence and Raman scattering among linear optical phenomena as well as nonlinear phenomena like, e.g. second harmonic generation, hyper-Raman scattering an others. Surface plasmon oscillations give rise to local increase in light intensity after some time the light enters the vicinity of a metal nanoparticle. The local areas of higher light intensity coexist with other areas where light intensity is not enhanced and even may be depleted as compared to propagation of light in a continuous dielectric medium. Even for a single isolated particle the thorough electrodynamical calculations to be performed are very cumbersome and can be realized only numerically. It is the general opinion that the ideal case would be a small particle (preferably a prolate ellipsoid or spheroid as well as cone-like or other sharp tip)

Fig. 1 Representative metal-dielectric structures exhibiting enhancement of Raman scattering and photoluminescence. (a) Gold colloidal particles in a solution, mean diameter is about 10 nm; (b) fractal-like silver clusters on dielectric surface, particle mean diameter is 15 nm; (c) gold deposited on an irregular nanotextured surface, image size is 380 × 380 nm; (d) gold deposited on a regular nanotextured surface formed by close-packed silica dielectric globules, globule diameter is about 200 nm.


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or, better, a couple of particles of material with low damping rate and minor interband transitions. Damping rate defines directly the sharpness of resonant response in terms of spectral width of the resulting real part of dielectric function of the composite metal-dielectric medium. Interband transitions give rise to finite imaginary part of the complex dielectric function and bring dissipative losses to the system in question. Because of the complexity of calculations we refer to the recent book (Kneipp et al. 2006) where computational techniques are discussed in detail. In what follows only final numerical results will be discussed. In the list of metals promising superior local field enhancement, the first place is given to silver, the second to gold and the third to copper in accordance with damping rates and interband transitions that diminish surface plasmon resonances. The crucial parameter is also the crossover point in the dielectric function where it passes through the zero value. For the above metals this point falls in the optical range. Alkali metals even in spite of the minor contribution from interband transitions are not suitable for local light intensity enhancement because their zero-crossover points in dielectric permittivity get deep in the ultraviolet region (additionally, alkali metals are actually not suitable for surface enhanced spectroscopy experiments in air because of the rapid oxidation). Because of the above arguments, silver is a typical material for all model calculations. For a single isolated silver particle local electric field enhancement factors E / E0 of the order of 10 are reported (Kneipp et al. 2006) for 30 nm particle 2

which corresponds to the intensity enhancement I / I 0 ∝ E / E0

2

≈ 102 . The

enhancement peaks at the wavelength of 370–400 nm i.e. close to the extinction peak typically observed for dispersed silver nanoparticles. The enhancement depends on a particle shape and gets higher for prolate particles. For example, for the bottom plane of a truncated tetrahedron with in plane size 167 and 50 nm height the intensity enhancement was calculated to be as high as 47,000 (Kneipp et al. 2006). Unlike smaller particles, in this case enhancement occurs at a wavelength of 646 nm. This red shift is from the scattering contribution to the extinction. For a prolate silver spheroid (length 120 nm, diameter 30 nm) intensity enhancement about 104 was obtained in calculations at a wavelength of 770 nm. Notably the extreme above enhancement numbers are inherent in very restricted space portions, typically measuring 1–5 nm in one dimension. These small values might be questionable in the context of the continuous electrodynamics applicability. Coupled particles (“plasmonic dimers”) show higher enhancement in the area between particles, the enhancement factors being strongly dependent on electric field orientation with respect to a dimer axis. 104 intensity enhancement was predicted between two silver spherical particles of 30 nm diameter and 2 nm spacing at wavelength of 520 nm (Kneipp et al. 2006). A transition from a single particle to dimers provides a hint towards engineering of plasmonic nanostructures with enormous enhancement of electromagnetic field. Further steps can be made based on more complex geometries. Sarychev and Shalaev (2000) computed surface field distribution for electromagnetic wave

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impinging the nanotextured surface formed by irregular particles with dense enough concentration to develop percolation clusters. They found in certain “hot spots” intensity may rise up by more than 104 times. Stockman with coworkers (Li et al. 2003) proposed an ingenious self-similar plasmonic “lens” using sequentially located nanoparticles with scalable size and spacing. This system consists of three silver nanospheres whose radii decrease by a factor of 1/3 from one sphere to another. This specific reduction factor should be significantly less than 1, but its precise value is not of principal importance. The larger nanoparticle’s radius R1 should be considerably smaller than the wavelength, whereas the minimum radius should be still big enough to use the continuous electrodynamics. The gaps between the surfaces of the nanosphere are chosen to be d12 = 0.3R2 , and d23 = 0.3R3 . This self-similar system exhibits superior enhancement based on transfer down the spatial scale. The external field with frequency close to the nanosphere surface-plasmon resonance excites the local field around the largest nanosphere enhanced by a factor of approx. 10. This local field is nearly uniform on the scale of the next, smaller nanosphere and plays the role of an external excitation field for it. This in turn creates the local field enhanced by an order of the magnitude. Similarly, the local fields around the smallest nanosphere are enhanced by a factor of 103 for the field amplitude. The “hottest” spot in the smallest gap at the surface of the smallest nanosphere was found to feature the local field enhanced by a factor of |E| ≈ 1,200. In the recent work (Guzatov and Gaponenko 2009) the enhancement of Raman scattering from the local incident field enhancement has been examined theoretically in the near field of a spherical and a spheroidal prolate silver particle (Fig. 2).

Fig. 2 Raman scattering cross-section enhancement factor due to local incident field enhancement as a function of spectral shift (Guzatov and Gaponenko 2009). Wavelength of incident light is 375 nm. See text for detail.


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Raman scattering cross-sections have been calculated for a hypothetical molecule with polarizability 103 Å3 placed in a close vicinity near a silver prolate spheroid with length 80 nm and diameter 50 nm and near a silver spherical particle with the same volume. Polarization of incident light has been chosen so as the electric field vector is parallel to the axis connecting a molecule and the center of the silver particle. Maximal enhancement has been found to occur for molecule dipole moment oriented along electric field vector of incident light. The position of excitation wavelength for maximal values of Raman cross-section is approximately determined by the position of maximal absolute value of nanoparticle’s polarizability. For selected silver nanoparticles it corresponds to 383.5 and 347.8 nm for spheroid, and 354.9 nm for sphere. It was shown that a prolate spheroid can give nearly 1011-fold enhancement of Raman response. Local field enhancement for incident light cannot be interpreted as surface redistribution of incident light, i.e. as a kind of local light “microfocusing” as commonly anticipated by many authors. Surface enhanced Raman scattering (as well as surface enhanced photoluminescence) are considered within the framework of linear light–matter interaction contrary to e.g. surface enhanced second harmonic generation. Therefore the total Raman signal harvesting from a piece of area containing statistically large number of molecules will be same independently of surface redistribution of light intensity because total incident light intensity integrated over the piece of area remains the same. Within the framework of linear light–matter interaction, Raman signal enhancement by means of incident field enhancement can only be understood in terms of high local Q-factors for incident light, i.e. in terms of light accumulation near the surface rather than light redistribution over the surface. Q-fold rise up of light intensity then occurs near hot points as it happens in microcavities and Fabry-Perot interferometers. However, accumulation of light energy needs certain time. Therefore huge Raman signals can develop only after certain time which is necessary for transient processes to finish resulting in steady increase of incident light intensity near hot points as compared to average light intensity in incoming light flux. Transient SERS experiments are therefore to be performed to clarify Q-factor effects in hot points formation.

4

Density of states effects on emission and scattering of photons

In a continuous medium photon density of states D can be defined as a number of electromagnetic modes per unit spectral interval in a unit volume. If spectral dependence is expressed in terms of wavenumber k, the D(k) function reads

D3 (k ) =

k2 1 k , D2 (k ) = , D1 (k ) = 2 2π 2π π

(2)

S.V. GAPONENKO

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where subscripts stand for dimensionality of space d = 3, 2, 1. These relations are universal and inherent also for acoustic waves and for quantum particles. If spectral DOS dependence is expressed in terms of frequency ω, then the corresponding functions can be derived from Equation (2) using a relation

D(ω ) = D(k )

dk dω

(3)

and the dispersion law for EM-waves in a continuous homogeneous medium

ω (k ) = ck / n

(4)

with refraction index n. For continuous homogeneous 3-, 2-, and 1-dimensional media with finite n one has respectively

D3 (ω ) =

ω 2 n3 ωn2 n . ω , D ( ) = , D1 (ω ) = 2 2 3 2 2π c 2π c πc

(5)

These examples show that in continuous dielectric media photon density of states effect results in dependence of scattering and emission rate upon dimensionality and refractive index of a medium. In heterogeneous medium and for complex dielectric-dielectric and metaldielectric nanostructures density of states should be calculated explicitly. In certain model structures DOS behavior is well known. For example, in photonic crystals one has redistribution of DOS over spectrum with inhibited DOS in the band gap region and enhanced DOS beyond the gap. In microcavities DOS value reduces to discrete set of cavity modes resulting in strong enhancement of spontaneous decay and Raman scattering rates. In anisotropic porous structures with subwavelength channel-like pores like, e.g. porous anodic alumina, photon DOS is expected to redistribute over solid angle to give maximal DOS along pore axes and minimal DOS within the plane normal to pore axes. These structures were found to possess anisotropic light scattering (Lutich et al. 2004) which can be understood in terms of DOS redistribution. In case of complex heterogeneous structures with big local variation in dielectric permittivity of materials involved like individual, coupled or aggregated metal nanoparticles or nanorods, explicit evaluation of DOS is not so obvious and straightforward. In such cases the discussion in terms of the DOS concept should be replaced by the local DOS (LDOS) value. The latter is currently under debates with respect to correct calculations and experimental determination. Imaging using scanning near-field optical microscope (SNOM) in the illumination mode seems to provide reasonable insight on local DOS over a surface under examination similar to tunneling current in scanning tunneling microscopy which is sensitive to local electron DOS for complex surfaces (Chicanne et al. 2002). In this context observation


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of “hot spots” in metal aggregations (Sarychev and Shalaev 2000) should be interpreted rather in terms of local DOS enhancement than in terms of incident field concentration and the results reported give more insight at DOS effects in these structures rather than at field enhancement factors if Raman or fluorescent probes are attached thereto. D’Aguanno et al. (2004) proposed a reasonable operational approach to the local DOS D(ω , r ) definition in terms of the factor in modified radiative decay rate γ (ω , r ) of a probe quantum system at the point under consideration with radius-vector r, i.e.

γ (ω , r ) = γ 0 (ω ) Here

γ 0 (ω ), D0 (ω )

D (ω , r ) . D0 (ω )

(6)

are radiative decay rate and density of states in vacuum,

respectively. In this context, a systematic evaluation of modified spontaneous emission rate near a metal nanobody (Klimov and Guzatov 2007; Guzatov and Klimov 2007) can be interpreted in terms of local DOS calculations. It is clear from the above consideration that in SERS-active metal-dielectric structures with complicate topology (fractal clusters, island films, granular structures of coinage metals) exhibiting drastic redistribution of EM-field in space, photon DOS will redistribute as well. This will result in angular and frequency redistribution of Raman scattering rates. Local enhancement factor for Raman signal is equal to local field enhancement factor (at frequency ω) multiplied by local DOS enhancement factor (at frequency ω΄). In other words, “hot spots” can develop at the surface both with respect to incident field amplitudes and with respect to emitted photon DOS at the same time. These two enhancement factors when combined may give rise to enormously big Raman signals observed in single molecule experiments (Kneipp et al. 1999). Recently a rationale for more than the 1014-fold rise-up in Raman scattering cross-section has been obtained (Guzatov and Gaponenko 2009) by means of simultaneous account for local field and local density of photon states enhancements in close proximity to a prolate silver nanoparticle. Local DOS effect has been introduced in the theory of surface enhanced Raman scattering as LDOS γ RS (ω , ω ′, r ) = γ RS (ω ,ω ′ )

D (ω ′, r ) D0 (ω ′ )

(7)

by analogy to Equation (6). The results of calculations of Raman scattering enhancement for a silver prolate and spherical particles with size and configuration as were in Fig. 2 are presented in Fig. 3. In calculations, simultaneous account for incident field enhancement and Raman scattering rate enhancement due to LDOS effect have been accounted for. One can see, LDOS account provide more than 103-fold additional factor to Raman signal enhancement.

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Fig. 3 Raman scattering rate for a molecule near a spheroidal and spherical silver nanoparticle. All parameters are the same as in Fig. 2 but additionally, the local density of states was included in calculations.

The above consideration is valid not only for Raman spectroscopy but for all versions of vibrational spectroscopies, e.g. it can be applied for single quantum dot vibrational spectroscopy. It is also valid for Mandelstam–Brillouin scattering as well as for Rayleigh scattering. Simultaneous action of incident field enhancement and local density of photon states enhancement does provide a reasonable rationale for single molecule Raman spectroscopy. The results are considered as a first step towards extensive theory for single molecule Raman detection. Furthermore, coupled metal nanoparticles which have been found to exhibit higher efficiency in Raman scattering enhancement (Michaels et al. 1999) have also been proven theoretically to possess superior LDOS enhancement in the spacing between spheres (Klimov and Guzatov 2007) and are believed their SERS efficiency can be described by simultaneous incident field and LDOS enhancements. The proposed model sheds light on the so-called “hot points” as such places on a nanotextured metal surface or near metal nanobodies where simultaneous spatial redistribution of electromagnetic field occurs both at the frequency of the incident radiation ω and at the frequency of scattered radiation ω ′ . Recalling the original Purcell’s idea on Q-fold enhancement of spontaneous emission rate in a cavity (Purcell 1946), local DOS enhancement can be treated as high Q-factor development in certain portion of space near a metal nanobody. Therefore, a hot point in SERS can be treated as a place where high Q-factor develops simultaneously at the incident light frequency and emitted light frequency. Local DOS enhancement in a sense accounts for concentration of electromagnetic field at the scattered frequency. This statement unambiguously implies probe, non-existing field. However, concentration of real field by many authors


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was anticipated to offer E (ω ′) enhancement factor by analogy to E (ω ) factor for input light intensity. That anticipation is by no means justified because for both incident and scattered light enhancement occurs only in the close subwavelength-scale vicinity of a nanobody. Therefore enhancement of scattered field by no means can contribute to light harvesting in typical far field experiments. LDOS enhancement means unmeasurable concentration of vacuum electromagnetic field rather than emitted light concentration. The latter can actually contribute to SERS but only as induced Raman scattering (I0 term in Equation (1)). 2

5

2

Experimental performance of enhanced photoluminescence

Though spontaneous emission rate is proportional, similar to Raman scattering, to the product of incident field and photon DOS, it does not mean that structures showing big SERS signal will be at the same time efficient in photoluminescence enhancement if probe SERS molecules are replaced by a fluorescent probe (molecules, ions, quantum dots). There is principal difference in photon scattering and photon spontaneous emission events. The latter unlike instantaneous scattering is characterized by a finite internal relaxation rate and a finite excited state lifetime. Proximity of a fluorescent probe to metal particles at metal surface promotes rapid non-radiative relaxation path which in most cases predominates over radiative lifetime and manifest itself as strong luminescence quenching. Quenching overthrows enhancement in most experiments which is well known from SERS studies since 1980s. To make use of enhancement factors one has to engineer the optimal topology of probe-metal nanostructure to get positive balance of competing enhancement/ quenching effects. In other words, luminescent probe is to be displaced at certain distance of metal nanobodies at a point were quenching is yet negligible but field and DOS enhancement still present. This is the case if probes are dispersed in a matrix with nanosize metal colloids at low concentration. Another case is using a dense metal surface but adjusting a spacer between a probe and the surface. Such experiments have been performed by many groups for luminescent ions and molecules during last decades. The representative examples are given in Fig. 4 of photoluminescence enhancement of semiconductor nanocrystals and organic fluorophores. Semiconductor nanocrystals (so-called quantum dots) are novel luminescent species whose absorption and emission spectra as well as transition probabilities are essentially controlled by quantum confinement of electrons and holes (Gaponenko 1998). These factors offer tunability of emission spectra simply by size variation. Simultaneously, superior stability of luminescent properties have been found which is more than

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Fig. 4 Photoluminescence intensity as a function of the number of polyelectrolyte layers forming a dielectric spacer over a metal surface. (a) Core-shell CdSe/ZnS semiconductor nanocrystals with the mean diameter 4 nm on a nanotextured gold surface (Kulakovich et al. 2002); (b) Albumin-fluorescein isothiocyanate conjugate of bovine serum on a nanotextured silver surface (Kulakovich et al. 2006).

100 times higher than that of traditional organic luminophores like rhodamines or fluorescein. Quantum dots are therefore promising candidates for novel commercial luminophores as well as for bioluminescent labels at the single molecule level. For fluorescein-labeled protein (Fig. 4b) ninefold enhancement has been observed. It is clearly seen that luminescence intensity is sensitive to the distance between the silver island film and the fluorophore and exhibits a maximum at around three polyelectrolyte layers, which corresponds to a spacer thickness of about 4.2 nm. This value is perfectly agrees with the recent report on tip-enhanced single molecule spectroscopy using a spherical gold ball (Anger et al. 2006). For semiconductor core-shell quantum dots the optimal distance has been found to be about 10 nm for CdSe/ZnS dots adsorbed on a gold colloidal film. This discrepancy could be due to the different metal – fluorophore system, in particular due to a different length scale of surface roughness and also due to the relatively large size of labeled protein molecules (which are oblate ellipsoids with dimensions of 140 × 4 nm) in comparison with the 4 nm size of nanocrystals. Progress in synthesis of semiconductor nanoparticles in various ambient environments, understanding of their optical properties combined with an idea of using quantum dots as efficient luminophores in light emitting devices and as fluorescent labels in high sensitive biospectroscopy do stimulate extensive experiments on purposeful application of field enhancement and DOS effects for quantum dot based nanostructures.


6

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Conclusions

In conclusion, there is definite progress in the last decade toward consistent theory of modified spontaneous emission and scattering of photons near metal nanobodies and on nanotextured surfaces along with far-reaching experimental observations of single-molecule Raman scattering and single-molecule and single quantum dot luminescence enhancement. This is important for further progress in novel luminophores development as well as in ultrasensitive biomolecular analyses. Optimal combination of electromagnetic field and photon density of states enhancement factors under condition of moderate quenching for proper probe–metal spacing is considered to give rise to efficient light emitting devices and ultra-high-sensitive spectral analysis using semiconductor quantum dots as light sources. Acknowledgments Stimulating discussions and cooperation with U. Woggon, S. Maskevich, N. Strekal, E. Petrov, M. Artemyev, O. Kulakovich, A. Lutich, and D.V. Guzatov are greatly acknowledged. The work has been supported by the National Program “Molecular and Crystalline Structures” as well as by the European Network of Excellence “PHOREMOST”.

References Anger, P., Bhardwaj, P., Novotny, L.: Enhancement and quenching of single-molecule photoluminescence. Phys. Rev. Lett. 96, 113002 (2006) Barnes, W.L.: Fluorescence near interfaces: the role of photonic mode density. J. Mod. Opt. 45, 661–699 (1998) Berestetskii, V.B., Lifshitz, E.M., Pitaevskii, L.P.: Quantum Electrodynamics. Nauka, Moscow (1989) Bykov, V.P.: Radiation of Atoms in a Resonant Environment. World Scientific, Singapore (1993) Chang, R.K., Furtak, T.E. (Eds.): Surface Enhanced Raman Scattering. Plenum, New York (1982) Chicanne, C., David, T., Quidant, R., Weeber, J.C., Lacroute, Y., Bourillot, E., Dereux, A.: Imaging the local density of states of optical corrals. Phys. Rev. Lett. 88, 097402 (2002) D’Aguanno, G., Mattiucci, N., Centini, M., Scalora, M., Bloemer, M.J.: Electromagnetic density of modes for a finite-size three-dimensional structure. Phys. Rev. E 69, 057601 (2004) De Martini, F., Marrocco, M., Mataloni, P., Crescentini, L., Loudon, R.: Spontaneous emission in the optical microcavity. Phys. Rev. A 43, 4280–4297 (1991) Gaponenko, S.V.: Optical Properties of Semiconductor Nanocrystals. Cambridge University Press, Cambridge (1998) Gaponenko, S.V.: Density of states effects on photon scattering in mesoscopic structures, Phys. Rev. B 65, 140303(R) (2002) Gaponenko, S.V., Gaiduk, A.A., Kulakovich, O.S., Maskevich, S.A., Strekal, N.D., Prokhorov, O.A., Shelekhina, V.M.: Raman scattering enhancement using crystallographic surface of a colloidal crystal. JETP Lett. 74, 309–313 (2001) Guzatov, D.V., Gaponenko, S.V.: Combined effect of local field and density of states enhancement near metal nanobodies in single molecule Raman spectroscopy. In: Borisenko, V.E., Gaponenko, S.V., Gurin, V.S. (eds.), Physics, Chemistry and Applications of Nanostructures. World Scientific, Singapore (2009)

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Guzatov, D.V., Klimov, V.V.: Spontaneous emission of an atom placed near a nanobelt of elliptical cross section. Phys. Rev. A 75, 052901 (2007) Klimov, V.V., Guzatov, D.V.: Strongly localized plasmon oscillations in a cluster of two metallic nanospheres and their influence on spontaneous emission of an atom. Phys. Rev. B 75, 024303 (2007) Kneipp, K., Kneipp, H., Itzkan, I., Dasari, R.R., Feld, M.S.: Ultrasensitive chemical analysis by Raman spectroscopy. Chem. Rev. 99, 2957–2975 (1999) Kneipp, K., Moskovits, M., Kneipp, H. (eds.): Surface-enhanced Raman scattering. Springer, Berlin (2006) Kulakovich, O., Strekal, N., Yaroshevich, A., Maskevich, S., Gaponenko, S., Nabiev, I., Woggon, U., Artemyev, M.: Enhanced luminescence of CdSe quantum dots on gold colloids. Nano Lett. 2, 1449–1452 (2002) Kulakovich, O., Strekal, N., Artemyev, M., Stupak, A., Maskevich, S., Gaponenko, S.: Improved method for fluorophore deposition atop a polyelectrolyte spacer for quantitative study of distance-dependent plasmon-assisted luminescence. Nanotechnology 17, 5201–5206 (2006) Lavallard, P., Rosenbauer, M., Gacoin, T.: Influence of surrounding dielectrics on the spontaneous emission of sulforhodamine B molecules. Phys. Rev. A 54, 5450–5460 (1996) Li, K., Stockman, M.I., Bergman, D.J.: Self-similar chains of metal nanospheres as an efficient nanolens. Phys. Rev. Lett. 91, 227402 (2003) Lutich, A.A., Gaponenko, S.V., Gaponenko, N.V., Molchan, I.S., Sokol, V.A., Parkhutik, V.: Anisotropic light scattering in porous materials: a photon density of states effect. Nano Lett. 4, 1755–1759 (2004) Michaels, A.M., Nirmal, M., Brus, L.E.: Surface enhanced Raman spectroscopy of individual Rhodamine 6G molecules on large Ag nanocrystals. J. Am. Chem. Soc. 121, 9932–9938 (1999) Mogilevtsev, D.S., Kilin, S.: Probing the atom–field bound state. Phys. Rev. A 69, 053809 (2004) Petrov, E.P., Bogomolov, V.N., Kalosha, I.I., Gaponenko, S.V.: Spontaneous emission of organic molecules in a photonic crystal. Phys. Rev. Lett. 81, 77–80 (1998) Petrov, E.P., Kruchenok, J.V., Rubinov, A.N.: Effect of the external refractive index on fluorescence kinetics of perylene in human erythrocyte ghosts. J. Fluoresc. 9, 111–118 (1999) Purcell, E.M.: Spontaneous emission probabilities at radio frequencies. Phys. Rev. 69, 681 (1946) Rikken, G.L.J.A.: Spontaneous emission from stratified dielectrics. Phys. Rev. A 51, 4906–4611 (1995) Sarychev, A.K., Shalaev, V.M.: Electromagnetic field fluctuations and optical nonlinearities in metal-dielectric composites. Phys. Rep. 335, 275–371 (2000) Sokolov, K., Chumanov, G., Cotton, T.M.: Enhanced fluorescence of organic molecules using silver nanoparticles. Anal. Chem. 70, 3898–3906 (1998) Sushchinskii, M.M.: Raman Scattering Spectra of Molecules and Crystals. Nauka, Moscow (1969) Zuev, V.S., Frantesson, A.V., Gao, J., Eden, J.G.: Enhancement of Raman scattering for an atom or molecule near a metal nanocylinder: quantum theory of spontaneous emission and coupling to surface plasmon modes. J. Chem. Phys. 122, 214726 (2005)

Laser-Matter Interaction in Transparent Materials: Confined Micro-explosion and Jet Formation Ludovic Hallo, Candice Mézel Centre Lasers Intenses et Applications, UMR 5107 CEA – CNRS – Université Bordeaux 1, 33405 Talence, Cedex, France [email protected]

Antoine Bourgeade, David Hébert CEA/CESTA, 33114 Le Barp, France

Eugene G. Gamaly Laser Physics Centre, Research School of Physical Sciences and Engineering, the Australian National University, Canberra ACT 0200, Australia

Saulius Juodkazis CREST-JST and Research Institute for Electronic Science, Hokkaido University, N-21-W10, CRIS Bldg., Kita-Ku, Sapporo 001-0021, Japan

Abstract High intensity laser beam was tightly focussed inside bulk of dielectrics at adjustable distance from the outer boundary (1–15 µm). Laser– matter interaction region is thus confined inside a cold and dense material, with and without boundary effects. In what follows we first describe self-consistently the relevant laser–matter interaction physics. At high intensity of the laser beam in a focal region (> 6 × 1012 W/cm2) the material is converted into a hot and dense plasma. The shock and rarefaction waves propagation, formation of a void inside the target are all described. Then, a model was developed to predict size of the voids in the bulk of materials, i.e. without boundary effects. Results were compared to experimental observations. The size of a void formed by 800 nm 150 fs laser pulses is ~0.2 µm3. Finally we present new results in confined geometries and we show that jets can develop sizes and expansion velocities depending both on energy laser and distance from the rear surface. This jet formation regime, apparently new, can be related to some LIFT process, with submicrometer diameter jets. T.J. Hall and S.V. Gaponenko (eds.), Extreme Photonics & Applications, © Springer Science + Business Media B.V. 2010

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Keywords: Nanoscale material processing, dielectrics, femtosecond laser interaction

1

Introduction

Recent studies demonstrated that short pulse laser tightly focused inside the bulk of a transparent solid could produce three-dimensional (3D) structures with a controlled size less than half of micron (Juodkazis et al. 2002, 2003; Schaffer et al. 2003; Glezer et al. 1996; Watanabe et al. 1998; Qiu et al. 1998). It has been also demonstrated that these structures can be formed in different spatial arrangements (Glezer et al. 1996; Watanabe et al. 1998; Qiu et al. 1998; Juodkazis et al. 2002). This technique could be used for formation of photonic crystals, waveguides and gratings for application in photonics. A single structure can also serve as a memory bit because it can be detected (read) by the action of a probe laser beam (Juodkazis et al. 2003). There are different ways for inducing the changes in optical properties in a bulk solid by laser action. First, non-destructive phase transitions can be induced by lasers at the intensity below the damage threshold. Second, irreversible structural changes may be produced at high intensity above the optical breakdown threshold. We concentrate onto this latter regime in this paper. There is a fundamental dependence of the laser–matter interaction on the focusing conditions: either the laser beam is tightly focused inside a transparent material or it is focused onto the surface. In the former case the interaction zone containing high energy density is confined inside a cold and dense solid. For this reason the hydrodynamic expansion is insignificant if the energy density is lower than the structural damage threshold, and above this threshold it is highly restricted. This results in a change in the optical and structural properties in the affected region. Now if the interaction zone is close to a boundary, there is also an interaction of hydrodynamic process with this boundary which may results in the boundary deformation. The interaction of a laser with matter at intensity above the ionization threshold proceeds in a way similar for all the materials (Gamaly et al. 2002). The plasma generated in the focal region increases the absorption coefficient and produces a fast energy release in a very small volume. A strong shock wave is generated in the interaction region and it propagates into the surrounding cold material. The shock wave propagation is accompanied by compression of the solid material at the wave front and decompression behind it leading to the formation of a void inside the material. In some solids chemical decomposition may occur at a relatively low temperature or beam intensity.

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Transparent dielectrics have several distinctive features. Firstly, they have a wide optical band-gap (it ranges from 2.2–2.4 eV for chalcogenide glasses up to 8.8 eV for sapphire) that ensures the transparency in the visible or near infrared spectral range at low intensity. In order to induce material modification with moderate energy pulses, the laser intensity should be increased to induce a strongly non-linear response from the material and to assure the plasma formation. This requires the intensities in excess of 1014 W/cm2 where most dielectrics can be ionized early in the laser pulse. A second feature of dielectrics is their relatively low thermal conductivity characterized by the thermal diffusion coefficient, D, which is typically ~10-3 cm2/s (compared with a few cm2/s for metals). Therefore micron-size regions will cool in a rather long time t ~ l2/D ~ 10 µs. This opens an opportunity of a multipulse effect: the energy deposed by a sequence of several laser pulses focused into the same point in a dielectric will accumulate if the period between the pulses is shorter than the cooling time. In what follows we first present some experimental results obtained in silica and sapphire. Second, we describe the laser–solid interaction physics for the case where a laser beam is tightly focused inside a transparent dielectric at high intensity well above the ablation threshold. We model the micro-explosion in glass by two-fluid hydrodynamic computer simulations and discuss the material modifycations produced by the shock, heat and rarefaction waves. We then make a comparison of our model with experimental results in silica and sapphire. Finally we address the possibility to observe back surface deformation if laser interaction zone is close to the boundary.

2

Experimental evidence of void formations in solid dielectrics

A laser beam (800 nm, 130 fs, average intensity ≤6.6 TW/cm2) was focused into a glass sample by the infinity-corrected oil-immersion objective lens with numerical aperture NA = 1.35 and magnification × 100 using the microscope Olympus IX70 (Juodkazis et al. 2003). The experimental set-up is presented in Fig. 1. Waist diameter of the Gaussian beam was estimated as 2r0 = 0.48 μm while the length of a focal volume (doubled Rayleigh length) comprises 2z0 = 0.65 μm giving the focal volume of ~0.3 μm3. The lateral and axial sizes of the void produced in silica and sapphire at intensities higher than damage threshold are of few tens of nanometers, as it is shown in Fig. 2. The detailed description of the experimental set-up and measurements can be found in Juodkazis et al. (2003).

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Fig. 1 Experimental set-up (Juodkazis et al. 2003).

Fig. 2 Typical voids obtained in sapphire produced by single pulses. On the left, array of voids in the transverse direction and, on the right, axial structures along the propagation axis of the laser.

In what follows we describe the processes that occur in the laser-affected material during the pulse duration and after the pulse end.

3

Laser–matter interactions inside a bulk of a solid at high intensity

In order to produce some detectable structures inside the material one must transport the laser beam over a given distance and depose the energy in a small volume. That means that the absorption length should be large and the energy needs to be focused to the smallest possible volume, with dimensions of the order


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of the laser wavelength ~λ. Two particular properties of transparent dielectrics, the large absorption length and the low thermal conductivity, make them very suitable for that purpose. The major mechanism of absorption in the low intensity laser–solid interaction is the inter-band electron transition. Since the photon energy is smaller than the band-gap energy, the electron transitions are forbidden in linear approximation, which corresponds to a large real and small imaginary part of the dielectric function. The optical parameters in these conditions are only slightly changed during the interaction in comparison to those of the cold material. The absorption can be increased for shorter wavelengths where the photon energy becomes larger than the band-gap value or if the incident light intensity increases to the level where the multi-photon processes become important. We are interested in the second possibility. Under such conditions the properties of the material and the laser– material interaction change rapidly during the pulse. As the intensity increases above the ionization threshold, the neutral material transforms into plasma, which absorbs the incident light very efficiently. A localized deposition of the laser light creates a region of high energy density thereby allowing a formation of various three-dimensional structures inside a transparent solid in a controllable and predictable way. The full description of the laser–matter interaction process and laser-induced material modification from the first principles embraces the self-consistent set of equations that includes the Maxwell’s equations for the laser field coupling with matter, complemented with the equations describing the evolution of energy distribution functions for electrons and phonons (ions) and the ionization state. Resolution of such a system of equations is a hard task even for modern supercomputers hence some preliminary estimations are required. This complicated problem is usually split into a sequence of simpler interconnected problems (Hallo et al. 2007; Mézel et al. 2008): the absorption of laser light, the ionization and energy transfer from electrons to ions, the heat conduction, and hydrodynamic expansion, which we are describing below. Let us consider first the delivery of laser beam inside a solid and the intensity distribution in a focal volume under tight focusing conditions using high numerical aperture optics.

3.1 Delivery of the laser beam to the focal area inside a solid: limitations imposed by the self-focusing The power in a laser beam aimed to deliver the energy to a desirable spot inside a bulk transparent solid should be kept lower than the self-focusing threshold for the medium. Indeed, the aim is to deposit the laser energy in the smallest possible volume, and not to transport and distribute it inside the whole material. The critical value for the laser beam power depends on the non-linear part of refractive index, n2, ( n = n 0 + n 2 ⋅ I ), as follows (Hallo et al. 2008):

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Pcr =

λ2

(1)

2π ⋅ n 0 ⋅ n 2

The beam self-focusing (balance between diffraction and Kerr effect) begins when the power in a laser beam, P0, exceeds the critical value, P0 > Pcr. The Gaussian beam under above condition self-focuses after propagating along the distance, Ls–f (Hallo et al. 2008):

Ls− f

−1/ 2 2π ⋅ n 0 ⋅ r02 ⎛ P0 ⎞ = −1⎟ ⎜ λ ⎝ Pcr ⎠

(2)

Here r0 is the minimum waist radius of the Gaussian beam. For example, in a fused silica (n0 = 1.45; n2 = 3.54 × 10-16 cm2/W) for λ = 1,000 nm, the critical power comprises 3 MW, while the self-focusing distance (assuming P0 = 2Pcr and r0 ~ λ) equals to ~9λ. Therefore one can obtain the intensities above 1013 W/cm2 in the focal plane and stay below the self-focusing threshold by using highaperture optics.

3.2 Laser interaction zone It is well-known that the minimum diffraction-limited focal spot radius obtained with the lens of numerical aperture NA expresses as (Born and Wolf 1999):

rmin =

0.61⋅ λ 0 NA

Then one defines the Rayleigh length for the Gaussian beam by

(3)

z0 =

πr02 . λ

Thus the volume that is confined inside the surface of intensity minima can be found as follows: 2 V foc = π ⋅ rmin ⋅ 2 z0

Finally, we get the final formula for the volume near the focus embracing the area inside the boundaries of the minimum intensity:


VI min = 0.87 ⋅

λ03

( NA)

127

(4)

4

In conditions of the experiments at λ = 800 nm ; NA = 1.35 one obtains VI,min = 0.13 μm3, rmin = 0.32 μm; πr2min = 0.32 μm2. The electric field in the focal region represents a complex interference pattern where the notion of polarisation is hard to introduce. Therefore, in what follows we consider only the effects of the total intensity on the matter in a focal volume.

3.3 Absorbed energy density Let us assume that dielectric became absorbent by any physical process once ionization is completed (few tens of femtoseconds). The plasma in the focal volume has a free-electron density comparable to the ion density of about 1022 cm-3. Hence, the laser interaction proceeds with plasma during the remaining part of the pulse. One can consider the electron number density (and thus the electron plasma frequency) as being constant in order to estimate the optical properties of laser-affected solid. The effective collision frequency in this dense non-ideal plasma is approximately equal to the plasma frequency, νeff ~ ωp (Gamaly et al. 2002). The real and imaginary parts of dielectric function and refractive index,

N ≡ ε = n + ik , with help of (Kruer 1988)

ε = 1−

ω 2p ω 2p ν eff + i ≡ ε′ + iε ′′ 2 2 ω 2 + ν eff ω 2 + ν eff ω

are then expressed as follows: −1

ω ⎛ ω2 ⎞ ⎛ ε ′′ ⎞ ω2 ε ′ ≈ 2 ; ε ′′ ≈ pe ⎜1 + 2 ⎟ ; n ≈ k = ⎜ ⎜ ⎝ 2⎠ ω pe ω ⎝ ω pe ⎠ The absorption length equals to ls =

1/ 2

(5)

c , and absorption coefficient is ω .k

expressed by formula (6). The absorbed laser energy per unit time and per unit volume, is then related to the laser intensity I and to the skin depth labs. We assume that the electric field

⎫ ⎧ c . exponentially decays inside a focal volume, E = E0 exp ⎨− x ⎬ , with labs = ⎩ labs ⎭

ω.k

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A is the absorption coefficient defined by the Fresnel formula (Glezer et al. 1996) as the following A = 1− R =

4n

(n + 1)

2

+ k2

(6)

The optical parameters of the plasma obtained after the breakdown of silica glass (ωp = 1.45 × 1016 s-1) by 800 nm laser (ω = 4.7 × 1015 s-1) are expressed as follows: dielectric function ε′ = 0.095; ε″ = 2.79, ε = 2.8 and correspondingly the real and imaginary parts of refractive index n ~ κ = ε/√2 = 1.985, giving an absorption length of ls = 32 nm, and an absorption coefficient A = 0.62. Therefore, the optical breakdown converts silica into a metal-like medium. This result has to be compared to the absorption coefficient obtained for n = 1.45, and k = n, which gives A = 0.72.

3.4 Ionization thresholds It is conventionally suggested that the ionization threshold (or breakdown threshold) is achieved when the electron number density reaches the critical density corresponding to the incident laser wavelength:

ncr =

meε 0ω L2 e2

The ionization threshold for the majority of dielectrics lies at intensities in between (1013–1014) W/cm2 (λ ~ 1 μm) with a strong non-linear dependence on intensity. The conduction-band electrons gain energy in an intense short pulse much faster than they transfer energy to the lattice. Therefore the actual structural damage (breaking inter-atomic bonds) occurs after electron-to-lattice energy transfer, usually after the pulse end. It was determined that in fused silica the ionization threshold was reached to the end of 100 fs pulse at 1,064 nm at the intensity 1.2b × 1013 W/cm2 (Hallo et al. 2008). Similar breakdown thresholds in a range of (2.8 ± 1) ×1013 W/cm2 were measured in interaction of 120 fs, 620 nm laser with glass, MgF2, sapphire, and the fused silica (Linde and Schuler 1996). This behaviour is to be expected, since all transparent dielectrics share the same general properties of slow thermal diffusion, fast electron–phonon scattering and similar ionization rates. An illustration is the comparison of silica and water’s behavior when irradiated by a 100 fs, 800 nm laser beam.


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Estimates of the ionization thresholds in silica (Ith = 28.9 TW/cm2) and water (Ith = 29.3 TW/cm2) show that electrons are generated for almost the same laser intensity (Fig. 3). The number of free electrons in the absorption zone is higher in water because multi-photon ionization requires only four photons, instead of six in silica.

Fig. 3. Electrons generated after laser pulse in silica and water.

The breakdown threshold fluence (J/cm2) is an appropriate parameter for characterization conditions at different pulse duration.

3.5 Absorbed energy density, electron energy and pressure in the focal region The electron-to-ion energy transfer time and heat conduction time in a hot and dense plasma lies in a range of picoseconds as we will show later. Therefore in a sub-picosecond laser–solid interaction the deposited energy is confined to the electrons whilst the ions remain cold. Hence, the electron energy in the absorption volume at the end of the laser pulse can be estimated with all losses being neglected by the general formula (Gamaly et al. 2006)

ε e (t p ) =

2A ⋅ Fp ne ⋅ l abs

; Fp =

tp

∫ I (t )dt 0

Now one can estimate the electron energy with laser-modified optical parameters. For the interaction of 800 nm, 100 fs, F = 10 J/cm2 laser pulse with the fused silica assuming that all atoms are at least singly ionised (ne =7.98 × 1022 cm-3). The modified parameters of the absorption zone are: the absorption length labs = 32 nm, the absorption coefficient A = 0.62, the absorption volume,

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Vabs = π rl 2min labs = 1.02 × 10 −2 μm 3 (rImin = 0.32 μm, see Sect. 3.2). The absorption volume is much smaller than the focal volume, because labs rstop apparently not affecting the properties of material. One can apply the above formula to estimate the shockaffected area in experiments in question. Taking the absorbed energy of 1–10 nJ, and P0 ~ 10 GPa, one obtains rstop ~ 0.3–0.6 µm for a single pulse action.

4.3 Rarefaction wave: formation of void The experimentally observed formation of a hollow, or low-density, region within the laser-affected volume, the void, can be understood from the simple reasoning. Let us consider for simplicity spherically symmetric motion. The strong spherical shock wave starts to propagate outside the centre of symmetry compressing the material. Meantime, a rarefaction wave propagates to the centre of the sphere creating a void. The pressure behind the shock decreases as r-3, and the shock ceases to exist at rstop defined above. One can estimate the radius of the void under suggestion that material between the void boundary at rvoid and rstop is compressed to an average compression ratio of δ = ρ/ρ0. Then void radius follows from the mass conservation law:

rvoid = rstop (1 − δ −1 )

1/ 3

(8)

For example, if rvoid ~ 0.5 rstop then amorphous material shall have a density 1.14 times higher than the initial density. The maximum void radius during the high compression stage is just 20% less than the shock wave stopping radius. Another estimate of the void radius is based on the assumption of isentropic expansion (Juodkazis et al. 2003; Gamaly et al. 2004). The heated material can be considered as a dense and hot gas (absorbed energy per atom significantly exceeds the binding energy) that starts to expand adiabatically with adiabatic constant, γ, after the pulse end. Therefore, condition PVγ = const holds. The heated area stops expanding when the pressure inside the expanding volume is comparable with the pressure in the cold material, P0. The adiabatic equation takes a form:

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(9)

The energy deposition volume is estimated as follows

Vabs = π r12 ls = e

1.1×10−2 μm3 , and rabs = 0.32 µm. We assume that during the expansion phase the void attains a spherical shape. Then the void radius reads: 1/ 3γ

rvoid

⎛P⎞ = rabs ⋅ ⎜ ⎟ ⎝ P0 ⎠

(10)

Of course this is only a qualitative estimate because the equation of state of the laser-affected material undergoes dramatic changes as the material cycles from a solid to a melt, to a hot gas and back. Accordingly, the adiabatic constant (rather the Gruneisen coefficient (Zel’dovich and Raizer 2002)) changes in a range from 5/3 to 3. Nevertheless, estimate by (40) gives reasonably that the void radius lies in the sub-micron range. Taking P ~ 5,600 GPa, P0 ~ 10 GPa; γ ~ 5/3 – 2, one obtains rvoid ~ 0.5 µm close to the stopping radius estimated above. Note that this is a void size during the interaction, the final void forms after the reverse phase transition and cooling.

5

3D Maxwell and hydrodynamic modeling

Cavity formation results in a succession of physical process which can be studied separately because they occur on different time scales. First, electrons absorb the laser energy during the laser pulse, i.e. on about tens of femtoseconds. Intensities reached in the focal plane can be easily estimated, assuming that the focal area is ellipsoidal: Sfoc = π * w0 * z0, with w0 the beam waist and z0 the Rayleigh length. In conditions of experiments, w0 = 0.32 μm, z0 = 0.4 μm, and laser energy is about tens of nanoJoule, which leads to an intensity of about 100 TW/cm2 in the focal plane. At such intensities, electrons of the valence band can absorb several photons at a time, cross the dielectric’s band gap, and join the conduction band. A plasma is thus created in the focal plane, which enhances laser absorption. Free electrons are then accelerated in the laser field. Provided they get enough energy, they collide with neutrals and ionise them. The time taken for the plasma to return to local temperature equilibrium τeq is estimated as follow:

τ eq =

1

ν eq

=

μT

3

2

3.2 ×10 Z 2λD ni −9

,


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where m = mi/mp, with mi the ion mass and mp the proton mass, T = Te = Ti the equilibration temperature in eV, Z the ionization state of the material, λD the Coulomb logarithm and ni the ion density. In a bulk of silica fully ionised once, assuming that the electronic temperature does not exceed the band gap energy, the energy transfer from electrons to ions occurs 3 ps after the pulse end. This part of the problem is supported using a self-consistent 3D model solving Maxwell’s equations for the laser beam propagation, coupled with a ionization model dealing with electron generation in the material:

⎞ ⎛ r ⎟ ⎜ r r r ∂D 1 ⎜ = ∇ ∧ B − J e + J mpi ⎟ { { ⎟ ∂t μ0 ⎜ electronic ⎟ ⎜ current ionisation current ⎠ ⎝ r r ∂B = −∇ ∧ E , ∂t where B the magnetic field and D = εE is the electric displacement field. Full set of Maxwell’s equations are needed because laser focuses in a subwavelength dimension zone. Non-linear response of the matter is accounted for in the source terms. Je is the electric current, resulting of the conducting electron’s response to the laser field considered within the Drude’s model:

r r e 2 ne r ∂ t J e = −ν e J e + E, me with ne the density of free electrons and νe = νei + νen the electron collision frequencies with ions and neutrals respectively. The ionization current Jmpi is obtained by equating the laser energy losses due to

r r ∂ tU mpi = − J mpi .E and the corresponding electron = Wionν mpi nn : r r E J mpi = − r 2 Wion nnν mpi . E

the multiphoton ionization energy gain

∂ tU mpi

Wion is the band gap energy, i.e. the energy required for electrons in the valence band to be transferred in the conduction band, and νmpi is the rate, i.e. the probability, of multiphoton ionization:

ν mpi = σ k I k ,

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where σk is the effective ionization cross section with

k=

Wion the number of E ph

photon required to perform ionization, Eph being the energy of one photon. For the case of a fused silica target irradiated by a λ = 800 nm laser (Eph = 1.55 eV), Wion = 9 eV, k = 6 and σ6 = 9.8 10-70 s-1(cm2/TW)6 (Stuart et al. 1995). Maxwell’s equations are complemented with an evolution equation for the free electron density:

∂ t ne = ν mpi nn + ν col ne −

ne

τ rec

,

where the right hand side accounts for the multiphoton and collisional ionization and for the radiative recombination due to the electron trapping. The collisional ionization rate νcol is evaluated using a Maxwellian distribution function and depends on electronic temperature and density.

5.1 Discussion on Equations of State (EOS) parameters We are interested here in the warm dense regime of the ion equation of state, the electronic part and the electron ionization equilibrium model being obtained by the Thomas–Fermi model complemented by the semi-empirical bonding correction proposed by Barnes (see More et al. 1988 for details) or a parametrizable cold curve.

5.2 Computations with Quotidian Equations of State (QEOS ) Main features of the QEOS model are described in (More et al. 1988). The ion equation of state combines Debye, Grüneisen, and liquid scaling law theories and corrections. Input parameters are the solid density and the cold sound speed which are used to restitute the cold bulk modulus (β = ρc2) and a zero total pressure at 0 K. Silica is known to show a series of high-pressure transitions. The experimental shock Hugoniot changes its slope at each of these transitions. QEOS has a poor matching with Hugoniot data and does not predict condensed matter phase transitions as well as critical conditions (details are presented in More et al. 1988, Fig. 14). The cold Grüneisen coefficient is large (G = 1.5) compared to the experimental one (0.6 in quartz and 0.03 in fused silica) due to a large contribution of the dilatation coefficient.


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However, for pressures larger than Mbar, Hugoniot behaves correctly which explains its “quotidian” use in plasma physics. The density contours obtained at t = 50 ps using QEOS are presented in Fig. 4 for the 40 nJ laser energy. The cavity obtained is bounded by a low density level. It is 1.2 µm long and 200 nm wide (volume = 0.037 μm3) which must be compared to experimental measurements (800 nm long and 175 nm wide). The minimum density level is about 0.2 g/cm3 and negative pressures are reached in this region which produces a cavity collapse after t = 50 ps. The 100 nJ laser energy case presented in (Hallo et al. 2007) did not have negative pressure but overestimated the final cavity size. An appropriate description of the cavity formation requires an EOS in an extended domain of parameters from very high pressures of the GPa scale and high densities of a few g/cm3 to the very low densities and pressures reached inside the cavity. By construction, the QEOS provide accurate pressures at solid densities as well as a correct transition to the Thomas–Fermi-like plasma EOS at high temperatures. Moreover, the Hugoniot curves, obtained in compression experiments with shock waves are generally reproduced well with QEOS. But the Grüneisen coefficient and the cohesion energy are not appropriate and cannot be easily modified although their values may affect the hydrodynamic.

5.3 Computations with SESAME In order to show the effect of the negative pressure domain in the EOS on the process of cavity formation, we made a simulation similar to that described above with the EOS defined by the SESAME1 table 7387 (density contours presented in Fig. 5). It describes silica in an extended domain of densities and pressures which appears to fit to the cavity formation problem but has a 0.65 Grüneisen coefficient. A 80 nJ laser energy is used.

Fig. 4 Density contours at t = 50 ps in silica for a 40 nJ laser shot (axis in µm). In density contours figures, red is the solid density and blue is the low density after expansion process. 1

LANL Report No. LAPL-83-4, SESAME, Report on the Los Alamos Equation-of-State Library, 1983.

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Fig. 5 Density contours at t = 200 ps in silica for a 80 nJ laser shot with SESAME 7387 (axis in µm).

Fig. 6 Pressure contours at t = 200 ps in silica for a 80 nJ laser shot with SESAME 7387 (axis in µm).

Contrarily to the simulations carried out with QEOS, an asymptotic constant pressure is obtained in the low density region (Fig. 6) which constitutes the cavity. After 200 ps, only acoustic waves propagate with quasi-constant dimensions. Shock intensity at t = 200 ps shows that the shock affected zone will be larger than 1 µm (pressure ratio of 20 across the shock wave), with the larger shock intensity on the lateral section of maximum absorbed energy on axis. This shock affected region is a signature of the shock–acoustic wave transition. It is obviously ellipsoidal, which corresponds to experimental measurements and which was not obtained with QEOS. Despite this better behavior, this EOS produces a too large cavity compared to experiments. Its Grüneisen, 20 times larger than that of fused silica, may be responsible for this discrepancy. Since EOS are tabulated, they cannot be easily modified. This led us to use home-made EOS in order to isolate the contribution of parameters.


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5.4 Parametric study For the same 80 nJ laser case, a new cold curve has been designed. It has been adjusted to fit the Hugoniot curve showing a non-monotonic Bulk modulus, first decreasing and then increasing with pressure as SESAME tables. In the low density region, the generalized Lennard–Jones potential was used, allowing to choose the dissociation energy and a continuous sound speed at standard energy. The ion contribution to the EOS was described by a structural Cowan model as in the QEOS but with user supplied Debye–Grüneisen–Lindeman parameters corresponding to the fused silica. The electronic pressure part was calculated via the Thomas–Fermi model. In this EOS, with the Grüneisen coefficient equal to its known cold value (G = 0.03), a 2.203 solid density was imposed and the Hugoniot curved reproduced experimental data. The value of 17 MJ/kg fusion latent heat is different from 10 MJ/kg in SESAME 7387, but similarly to the previously used EOS, it does not account for liquid–vapor transition. This home-made EOS will be referenced as IL0005. SESAME 7387, QEOS and home-made EOS main parameters chosen are presented in Table 1. Table 1. SESAME 7387, QEOS and home-made IL type EOS parameters. EOS name

G

QEOS

1.5

Esub (MJ/kg) 0.01

SESAME 7387

0.65

10

IL0005

0.03

17

IL0006

0.03

28

IL0007

0.03

8

A comparison of the density contours between SESAME and IL0005 is presented in Fig. 7.

Fig. 7 Comparison of density contours at t = 200 ps in silica after a 80 nJ laser shot – SESAME 7387 (up), and IL0005 (down) (axis in µm).

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Fig. 8 Comparison of density contours at t = 200 ps in silica for a 80 nJ laser shot – SESAME 7387 (up), and IL0006 (down) (axis in µm).

The cavity obtained at t = 200 ps is slightly smaller than the cavity obtained with SESAME 7387. A new EOS, the IL0006 has been built (see Table 1). A comparison of the contours density is presented in Fig. 8. The cavity size obtained becomes very small compared to that obtained with SESAME 7387. A third EOS, IL0007, built with a smaller latent energy than IL0006 gives a cavity similar to that obtained with SESAME 7387 as shown in Fig. 9. The Figs. 4–9 clearly demonstrate that the cavity dimensions can be connected to the latent heat of sublimation: the larger is the sublimation energy the smaller is the energy available for hydrodynamic. Indeed, QEOS has a 0.01 MJ/kg sublimation energy, neglectable compared to the electron energy variation, leading to a large cavity size. Moreover, the cavity collapse can be connected to a bad choice of critical conditions: critical density seems to be located at a very low density compared with SESAME 7387 which gives an unphysical behavior in the low density zone. This explains also why the QEOS cannot be used for such an application.

Fig. 9 Comparison of density contours at t = 200 ps in silica for a 80 nJ laser shot – SESAME 7387 (up), and IL0007 (down) (axis in µm).


139

We also show here that the Grüneisen coefficient is not an important parameter for the cavity, but it affects the sound speed and the volume crossed by the diverging shock wave. This effect can be seen from a comparison of shocked zones obtained with the QEOS and with SESAME 7387. For large Grüneisen (G = 1.5 in QEOS), initial pressure is larger than for small Grüneisen (G = 0.6 for Sesame 7387) because initial pressure and energy are linearly connected via the Grüneisen coefficient. In the 80 nJ laser energy case, the longitudinal dimension of cavity at t = 200 ps with IL0005 EOS is 1.1 µm, while its half width is 220 nm. We have carried out the 40 nJ laser energy case with IL0005 EOS. The density contours are presented in Fig. 10 at t = 50 ps. Due to a small amount of absorbed energy, the cavity is smaller than in the 80 nJ case. The longitudinal dimension of cavity at t = 50 ps is 600 nm while the half width is 130 nm. The density level achieved in the cavity is about 0.1 g/cm3 which indicates that the 40 nJ case is near the threshold for cavity formation. The affected shock region is also smaller (few hundred nm).

Fig. 10 Density contours at t = 50 ps in silica for a 40 nJ laser shot with IL0005 EOS (axis in µm).

The cavity computed diameters can be compared to experimental ones. For comparison purpose, the 40 nJ laser energy cavity obtained in silica is presented in Fig. 11, where Dv is the cavity diameter and Rs the shock affected zone radius, from (Gamaly et al. 2006).

Fig. 11 Cavity observed experimentally after a 40 nJ laser shot in silica (from Gamaly et al. 2006).

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Fig. 12 Cavity diameter on the laser energy in silica, simple modelling (circles) (Gamaly et al. 2006), IL0005 EOS (diamond shapes) and experiment (triangles).

A good agreement is also obtained for 80 nJ (Fig. 12) which shows cavity diameter on laser energy obtained with a simple modeling, with the IL0005 EOS and experimentally. We can conclude that such experiments on an ultra-fast and calibrated energy deposition in transparent materials allow a high precision measurement of the material state after explosion and open the way for the tuning of equations of state in the low temperature and low density domain. It has been shown that latent heat of sublimation could be connected to cavity radius and that the cold Grüneisen coefficient could be determined via the shock affected volume.

5.5 EOS validation and design The choice of the Equation of State to be used depends on the type of regime you are investigating. Hugoniot curves are globally in a good agreement with the experimental values of the pressure above 10 GPa, (i.e. solide state). The experimental data of the Hugoniot of silica come either from gas-gun flyer plate, electron beam generator or laser generated shocks. Each of these facilities gives access to different part of the P-density diagram (Fig. 13). As an example, experiments on gas gun flyer plate (Malaise et al. 2006) (on the SYLEX – SYstème de Lanceur sur EXplosif – facility at CESTA – Centre d’Etudes Scientifiques et Techniques d’Aquitaine) allow the construction the Hugoniot of silica at pressure in the range of 7–8 GPa at room temperature, where silica goes from quartz to Stishovite phase (Fig. 14).


141

(b) (a)

(c)

(d)

Fig. 13 From a to d, Hugoniot of the silica for three different EOS, P-density diagram for Sesame, L0002 and L0005.

Fig. 14 SYLEX, Hugoniot obtained at SYLEX facility is in the range of a few GPa.

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5.6 EOS study using an ultra-short low energy laser We have seen that interaction of subpicosecond laser pulses with transparent dielectrics opens a way to release energy inside a material so that interaction takes place in a confined geometry. The deposited energy produces a micro explosion with a diverging shock wave in a bulk material and a cavity in the center. We have seen also that the accurate description of the phenomena requires knowledge of such parameters of the EOS as the binding energy, the bulk modulus and the Gruneisen coefficient. By using hydrodynamic model, we can study the influence of such parameters on the void size and the radius of laser-affected zone in dielectrics. A direct relation was found between the cavity size and the latent heat of sublimation. We can then experimentally fine tune and validate the EOS of silica and other transparent materials. The main advantage of this scheme is to use a high repetition rate, low energy laser. Typically, a 100 nJ pulse focused on a 300 nm waist, results in a focal volume of 0.15 µm3 and in a pressure in the order of 100 GPa (Fig. 15). Obviously, the pressure generated by the short pulse decreases really rapidly in time. The density thus decreases, leaving a cavity in the bulk of the material.

Laser domain High pressure

Fig. 15 Silica Hugoniot. A 100 nJ, 100 fs laser gives access to pressure above 60 GPa, with a density above 4,500 kg/m3.

6

Application to deposition of nano-objects

In addition to this work, a specific application concerning dielectrics is in view at CELIA laboratory, by using femtosecond lasers sources. This is the Laser Induced


143

Forward Transfer (LIFT) (Ringeisen et al. 2006) in the femtosecond regime. A high aperture laser wave is focused into a dielectric (silica, water…). Due to the high aperture angle, dielectric material begins to absorb only once critical power is reached. For silica and sapphire and water, the critical power is typically 2 MW, which produces ionization, i.e. absorption, in the waist region. Waist can vary from 300 nm to 1 µm, which can generate a specific energy release of several MJ/kg in a submicrometric region. This energy release is high enough, to produce an expanding shock wave and to form a cavity. When the expanding shock wave interacts with the rear surface, a jet is created by hydrodynamic reaction of the surface. The diameter of the jet as well as its expansion velocity are directly connected to laser energy and waist radius (Fig. 16). Since the laser energy release is very small, the nanojet ejection can be used as a way to transport matter on a collector located nearby, without spurious heating of matter. It can include living cells, proteins, human tissue and could be used for tissular reconstruction. Again, like nanocavity formation process, it is a very interesting mean for physical model testing, numerical code benchmarking and EOS tuning. Typical energy contours obtained with Maxwell simulations in water are presented (Fig. 17).

Fig. 16 Nanojet formation scheme within a femtosecond irradiation context.

Fig. 17 Normalized absorption coefficient in a water layer of 12 µm thickness, after a 55 nJ, 100 fs, laser shot (laser comes from the left).

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For a 55 nJ laser energy release in water, the focal region being located 1 µm from the back surface of the target, we present the void formation process and the jet formation process in Fig. 18 at time t = 500 ps, 5 ns, 7.5 ns and 13.2 ns. Compared to classical Laser Induced Forward Transfer process, the nano-LIFT enables to obtain jets of few tens nanometer diameters, which seems to be relatively new. This process is under development both from modeling and experimental points. The main process involved for jet formation seems to be related to the cavity collapse which produces an expansion wave. Interaction of this expansion wave with the back surface produces a strong and localized acceleration of the surface, which corresponds to the nanojet formation. 500ps

13.2 ns

1 μm

5 ns

7.5 ns

Fig. 18 Density contours at time t = 500 ps, 5 ns, 7.5 ns and 13.2 ns. The void formation produces the jet expansion at the backsurface of the target.

7

Conclusions

The presented experimental and theoretical studies of intense laser beam interaction inside a transparent solid allow drawing the following conclusions. A transparent solid is converted to plasma by the action of a single femtosecond pulse that results in a nanometer-sized void formation inside a bulk of a solid. The following stages of this transformation were identified. First, the laser beam has been tightly focussed inside of a transparent solid (up to 15 µm depth >> laser wavelength) into the focal volume ~0.3 μm3. The high intensity laser field (~1013 W/cm2) inside the focal volume swiftly transforms a material into plasma. It is well established that the interplay between electron avalanche and the multi-photon ionization is the major factor leading to the optical breakdown in a transparent solid. That results in dramatic modifications of optical properties of a material followed by strong laser absorption and high concentration


145

of energy in the absorption volume that decreases down to 1/6 μm3. As a result the huge absorbed energy density of 5.6 × 106 J/cm3 has been created in the absorption volume. A pressure almost three orders of magnitude higher than the strength of material has been created. The strong expanding shock wave was generated to compress the material surrounding the absorption volume and simultaneously (due to mass conservation law) creating a void in the focal volume center. The size of the void and laseramorphizied material has been observed and measured by the means of high resolution electron microscopy giving the void size of 500 nm. These phenomena were simulated by the two-fluid hydrodynamics calculations with a real equation of state that predicts the void size and laser-affected area size in a qualitative agreement with the experiments and simpler analytic models. On the basis of these studies one can predict the size of the void produced by the single and multiple laser pulses with a reasonable accuracy. These studies have been reproduced in confined geometries. In this case, interaction of the expansion wave produced during collapse with a surface can generate a jet with a diameter similar to the laser interaction region (few tens of nanometers diameter).

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Juodkazis, S., Rode, A.V., Gamaly, E.G., Matsuo, S., Misawa, H.: Recording and reading of three-dimensional optical memory in glasses. Appl. Phys. B 77(2–3), 361–368 (2003) Kruer, W.L.: The physics of laser plasma interactions. Addison-Wesley, New York (1988) Linde, D. von der, Schuler, H.: Breakdown threshold and plasma formation in femtosecond laser–solid interaction. J. Opt. Soc. Am. B 13(1), 216–222 (1996) Malaise, F., Chevalier, J.-M., Bertron, I., Malka, F.: Investigation fused silica dynamic behaviour. J. Phys. IV, 134, 929–934 (2006) Mézel, C., Hallo, L., Bourgeade, A., Hébert, D., Tikhonchuk, V.T., Chimier, B., Nkonga, B., Schurtz, G., Travaillé, G.: Formation of nanocavities in dielectrics: A self-consistent modeling. Phys. Plasmas 15(9), 093504 (2008) More, R.H., Warren, K.H., Young, D.A., Zimmerman, G.G.: A new quotidian equation of state (QEOS) for hot dense matter. Phys. Fluids 31(10), 3059–3078 (1988) Qiu, J., Miura, K., Inouye, H., Nishi, J., Hirao, K.: Three-dimensional optical storage inside a silica glass by using a focused femtosecond pulsed laser. Nucl. Instrum. Methods Phys. Res. B 141(1–4), 699–703 (1998) Ringeisen, B.R., Othon, C.M., Barron, J.A., Young, D., Spargo, B.J.: Jets-based methods to print living cells. Biotechnol. J. 1(9), 930–948 (2006) Schaffer, C.B., Garcia, J.F., Mazur, E.: Bulk heating of transparent materials using a highrepetition rate femtosecond laser. Appl. Phys. A 76(3), 351–354 (2003) Stuart, B.C., Feit, M.D., Rubenchick, A.M., Shore, B.W., Perry, M.D.: Laser-induced damage in dielectrics with nanosecond to subpicosecond pulses. Phys. Rev. Lett. 74(12), 2248–2251 (1995) Watanabe, M., Sun, H., Juodkazis, S., Takahashi, T., Matsumoto, S., Suzuki, Y., Nishi, J., Misawa, H.: Three-dimensional optical data storage in vitreous silica. J. Appl. Phys. 37(12B), L1527–L1530 (1998) Zel’dovich, Y.B., Raizer, Y.P.: Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Dover, New York (2002)

Influence of the Cut-Off Wavelength on the Supercontinuum Generation in a Highly Non-linear Photonic Crystal Fiber Rim Cherif, Mourad Zghal Cirta’Com Laboratory, Engineering School of Communication of Tunis (Sup’Com), Ghazala Technopark, 2083 Ariana, Tunisia [email protected]

Luca Tartara, Vittorio Degiorgio CNISM and Department of Electronics, University of Pavia, 27100 Pavia, Italy

Abstract We report on the influence of the cut-off wavelength on the supercontinuum generation in which the spectral broadening occurs only on the blue side of the pump wavelength. As a consequence a limit to the extent of the supercontiuum is set and thus a way for tailoring the broadened spectrum according to a peculiar application is provided. The experiment consists of launching a train of femtosecond pulses into a 45-cm-long span of a photonic crystal fiber (PCF) by means of an offset pumping technique that can selectively excite higher-order modes. The PCF presents a wide range of wavelengths in which the fundamental mode experiences normal dispersion, whereas higher-order modes, LP11 and LP21 propagate in the anomalous dispersion regime, generating a supercontinuum based on the soliton fission mechanism. When exciting LP21 we are able to generate an almost purely visible supercontinuum even with pulse energies below 100 pJ. Our experimental results are compared with the numerical solutions of the nonlinear Schrödinger equation.

Keywords: Photonic crystal fiber, nonlinear optics, pulse propagation and temporal solitons, supercontinuum generation

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The recent development of highly-nonlinear photonic crystal fibers (PCF) has opened new horizons in the generation of wide-band supercontinuum radiation


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from visible to infrared by launching short pulses in a PCF length in the tens of centimetres range (Dudley et al. 2006). Unique dispersion characteristics and enhanced nonlinearity make the small-core PCF an ideal candidate for nonlinear applications. Supercontinuum generation (SCG) is a nonlinear optical phenomenon which is characterized by the appearance of several new spectral components on both sides of the pump wavelength (Alfano and Shapiro 1970). Soliton dynamics, Raman shift and coupling with dispersive waves, modulation instability, four wave mixing (FWM), are the main effects leading to the generation of a broad spectrum starting from a narrow laser line. Such a broadened spectrum can be profitably exploited in several applications with many examples coming from the field of optical communications. A multi-wavelength source covering all the telecom spectral range can be easily obtained from one single-line laser diode by broadening its spectrum. All-optical signal processing also can be performed by means of SCG: in a wavelength-division-multiplexed system a signal at a given carrier wavelength can be switched to a single destination or multicast to several destinations exploiting the wavelength-conversion capability offered by the filtering of the broadened spectrum of the input signal. Since the first observation of SCG in PCF (Ranka et al. 2000), several experiments and numerical simulations were performed in order to investigate the dynamics of SCG in normal and anomalous regime (Husakou and Herrmann 2001; Dudley et al. 2002; Gaeta 2002; Wadsworth et al. 2004; Champert et al. 2004; Cristiani et al. 2004; Genty et al. 2004; Gorbach et al. 2006). Almost all the reported experiments were performed by exciting the fundamental fiber mode LP01 and observing SCG in the same mode. Dudley et al. (2002) describe situations in which the first higher-order mode, LP11 is predominantly excited, but the role of the cut-off wavelength is not discussed. In a few cases, third-harmonic generation (Efimov et al. 2003; Tartara et al. 2003) and four-wave-mixing (Tartara et al. 2003; Konorov et al. 2004) processes involving energy exchange among different modes were studied, but, as discussed by Tartara et al. (2003), the efficiency of such processes is low because the overlap integrals involving two different fiber modes are intrinsically small. Since higher-order modes present dispersion characteristics different from those of the fundamental LP01 mode, new opportunities for applications in nonlinear fiber optics can be opened. As an example, a recent work by Lee et al. (2007) presents the generation of Čerenkov radiation, obtained by exploiting the fact that higher-order modes allow for anomalous-dispersion propagation in a wavelength range in which LP01 propagates in normal dispersion. A further reason why it is interesting to study the nonlinear propagation of higherorder modes is that they present a cut-off wavelength above which the attenuation becomes very large, so that the mechanism of growth of the supercontinuum must be significantly different from that of LP01, when the pump wavelength is close to the cut-off wavelength. In this work we investigate the SC generated either by the fundamental mode or by the two first higher-order modes (Cherif et al. 2008) and focus our attention on the nonlinear propagation of the third-order mode, presenting not only experimental

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spectra, but also a detailed simulation aimed at investigating the main mechanisms producing the asymmetric spectral broadening. Our results are obtained by coupling a train of femtosecond pulses from a Ti: Sapphire laser into a 45-cm-long span of a PCF. For input wavelengths below 810 nm, the fiber allows for the propagation of higher-order modes. We are able to achieve a selective excitation of each mode by an offset pumping technique. When exciting the third-order mode we are able to generate a mainly visible supercontinuum even with pulse energies below 100 pJ. We find that the existence of a cut-off wavelength λco for the higher-order modes makes the spectral broadening asymmetrical when the pump wavelength is close to λco. The mechanism behind the spectral broadening is mainly ruled by soliton propagation leading to the generation of a blue-shifted dispersive wave. The paper is organized as follows. In Section 2 we describe the experimental set-up as well as the linear optical properties of the used PCF. Chromatic dispersion and effective area are calculated using the finite-element-method (FEM). The experimental results are presented in Section 3. Section 4 is devoted to the description of the numerical solutions of the nonlinear Schrödinger equation (NLSE) and to the comparison between experiment and simulation. Conclusions are drawn in Section 5.

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The experimental set-up is depicted in Fig. 1. The pump source consists of a cwmode-locked Titanium-Sapphire laser (Spectra-Physics Tsunami) emitting a 110fs pulse train at the repetition rate of 80 MHz, tunable over the wavelength range 700–870 nm. A Faraday isolator, inserted to block the back-reflection from the input tip of the fiber into the laser cavity, broadens the pulse width up to 190 fs. The beam is coupled into a PCF span of 45-cm length by means of an aspheric lens having a numerical aperture of 0.65. The fiber is fixed at both ends and held straight in order to avoid bending losses. A three-axis piezoelectric translation stage allows positioning the input cross-section with a resolution of 20 nm. By moving the input tip of the fiber in the focal plane, we can exploit an offset pumping technique in order to excite selectively different fiber modes at the expense of the coupling efficiency. At the output end of the fiber the light is collected by a 100x objective with a numerical aperture of 0.95. The spectral properties of the output radiation are monitored by an optical spectrum analyzer (Ando AQ-6315A) having a resolution of 0.1 nm. We also measure the total average power at the fiber output, which is only slightly smaller than the power coupled inside the fiber, considering that the linear attenuation coefficient of the fiber is about 0.3 m−1 in the investigated range of wavelengths. The used PCF is made of fused silica and is fabricated by means of a conventional stack-and-draw process. The fiber cladding consists of a hexagonal lattice of holes with average diameter of 2.5 μm and holeto-hole spacing of 2.7 μm. The resulting ratio d/Λ equal to 0.9259 indicates that the fiber is multimode.

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We have performed a numerical study of the modal properties of the fiber by using a commercial finite-element-method mode-solver. As input profile for the simulation we inserted a discretized scanning-electron-microscope image of the fiber cross-section. The calculated cut-off wavelengths of the LP11 and LP21 modes are, respectively, around 1,300 and 830 nm. The dispersion curves of the three modes are shown in Fig. 2. The zero dispersion wavelengths are found to be 835, 665 and 600 nm, respectively. The spatial patterns of the three modes are shown in the insets of Fig. 2. The effective area of the three modes, calculated at 785 nm, is 3.46 μm2 for the LP01 mode, 3.26 µm2 for the LP11 mode, and 3.82 µm2 for the LP21 mode.

Fig. 1 Experimental set-up for SCG.\

Fig. 2 Numerically calculated dispersion curves for the propagation modes of the PCF. Insets (bottom to top): fundamental, second and third order mode intensity distribution numerically obtained by means of FEM.


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The experiments are performed by changing both wavelength and average power of the laser beam coupled inside the PCF, and detecting the optical spectrum and the average power of the output signal. Using a train of femtosecond pulses at a wavelength above 810 nm, the light is observed to propagate only in the fundamental LP01 mode. When the pump wavelength is tuned below 810 nm we are able to excite several modes. By choosing, by trial and error, the appropriate positioning of the fiber in the focal plane, the excitation turns out to be highly selective so that at the output of the fiber we can easily detect a single mode, either LP01 or LP11 or LP21. The proof of higher-order mode selective excitation is provided by the far-field pattern, showing clear evidence of the second- and thirdorder mode propagation. In all cases we find that the spatial pattern at the output is unchanged even when the input power is increased to the point at which nonlinear interactions give rise to a large spectral broadening. The phenomena observed when the fundamental mode LP01 is excited are similar to those already reported in previous works (Ranka et al. 2000; Husakou and Herrmann 2001; Dudley et al. 2002; Gaeta 2002; Wadsworth et al. 2004; Champert et al. 2004; Cristiani et al. 2004; Genty et al. 2004; Gorbach et al. 2006). We show in Fig. 3 the output spectrum observed with excitation at 790 nm and average power of 200 mW. The supercontinuum extends from 540 to 1,430 nm.

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The LP11 mode can be easily excited at input wavelengths lower than 810 nm, with a coupling efficiency comparable to that of the fundamental mode. Some output spectra are presented in Fig. 4a. As the propagation occurs in the anomalous dispersion regime, soliton fission dynamics rules the spectral evolution (Husakou and Herrmann 2001; Cristiani et al. 2004; Gorbach et al. 2006; Cherif et al. 2008). The red shift of the Raman solitons is coupled to the emission of blue-shifted dispersive waves leading to the broadening of the spectrum on both sides of the pump wavelength. Consequently several peaks, appearing both in the visible and in the infrared range, characterize the spectral evolution. However it is important to notice that while the extent of the broadening towards the short wavelength side is comparable to the one observed with the fundamental mode, the generation of redshifted spectral components stops at wavelengths shorter than 1,300 nm with a progressive decaying intensity above 1,100 nm. In Fig. 3 we compare the output spectra obtained by exciting the two modes LP01 and LP11 with the same input wavelength and similar coupled power, in order to show clearly the effect of the cut-off wavelength on the LP11 mode. The fact that the observed broadening does not extend beyond 1,300 nm is consistent with the value of the cut-off wavelength predicted by the numerical analysis. It remains unclear why we are not able to excite the LP11 mode by tuning the input wavelength above 810 nm. The excitation of the LP21 mode is instead rather difficult. The focal spot of the pump beam has in fact to be critically positioned on the input cross section of the fiber somewhat displaced from the point yielding the highest coupling efficiency. Such a strong offset severely reduces the amount of the power coupled into the fiber, in comparison with the LP01 mode and even with the LP11 mode. The maximum average power we detect at the output of the fiber span is about 20 mW. The exact value could be slightly larger because of the limited collection efficiency provided by the objective. Figure 4b shows the output spectrum as a function of the average power at the input wavelength of 805 nm for the LP21 mode. While the spectrum broadens down to the blue side, no spectral components on the long wavelength side are generated above 830 nm. Our finding is consistent with the value of the cut-off wavelength computed using numerical analysis. We also find that, even when the pump wavelength is tuned down to 705 nm, the spectral broadening is prevalently developed towards the shorter wavelengths, except for an isolated low-intensity peak arising near 800 nm. The experimental results show that, notwithstanding the fact that the spectral broadening on the long-wavelength side is not possible because the mode is not guided at wavelengths above 830 nm, still a considerable broadening on the shortwavelength side can be developed. This result is not obvious because the dynamics leading to continuum generation should imply some kind of coupling between redand blue-shifted spectral components. As the input pulse experiences the anomalous dispersion regime, soliton propagation is expected to play a significant role in the


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Comparison between experiment and simulation

We have carried out a numerical simulation of the pulse propagation in the fiber, in order to better understand the dynamics leading to SCG and validate the proposed explanation. The numerical tool we have developed, is based on the resolution of the NLSE by the split-step Fourier method (Cristiani et al. 2004). In order to take into account the modal cut-off, the linear loss coefficient is made wavelength-dependent in our calculation by using a very steep function (errorfunction) growing around cut-off from very low to very high values. As it is not possible to characterize the dispersion for wavelengths longer than the cut-off wavelength, we have turned to the dispersion curve of the fundamental mode of a real fiber having the zero dispersion wavelength at 700 nm. The numerical results we obtain are in fair agreement with the experimental data, as shown, for instance, by the comparison between simulation and experiment reported in Fig. 5 for the case of the LP21 mode excitation at 780 nm. Not only the visible continuum but all the main features of the experimental spectrum are reproduced such as the multi-peak structure arising around the pump wavelength, some secondary peaks, and the main peak close to 520 nm. In order to better clarify the contributions of several mechanisms playing a role in the spectral evolution, we have also run the numerical simulation by excluding some specific terms of the NLSE and keeping fixed the value of the input

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wavelength and of the average power. The spectrum displayed in Fig. 6a is obtained by plugging in the simulation the whole nonlinear response made up of self-phase modulation, stimulated Raman scattering and self-steepening. The modal cut-off is instead neglected in order to reproduce the most common and investigated experimental situation. The spectrum broadens on both sides of the pump wavelength and a well-defined spectral peak clearly arises at the longwavelength edge of the supercontinuum radiation. The generation of such a peak can be explained according to the soliton fission mechanism. The corresponding pulse is in fact originated by the decay of the perturbed input pulse and is subsequently red-shifted because of the Raman effect. At the same time the resonant coupling between the soliton-like pulses and the dispersive waves experiencing the normal dispersion regime enhances the broadening on the blue side. 0

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The result displayed in Fig. 6b has been obtained by inserting the modal loss in the code and keeping into account all the nonlinear processes just like in the previous case. When running the numerical situation in such a way it is thus possible to study the same situation as the one experimentally investigated and discussed in this work. One can clearly see how the modal cut-off prevents the spectrum from broadening towards longer wavelengths. However a white-light radiation is generated on the blue side of the pump wavelength, even if the spectrum widens to a slightly lower extent than the one in the preceding figure. The reason can be easily ascribed to the modal cut-off hindering the propagation of infrared pulses. Consequently the resonant coupling to the visible dispersive waves fails weakening their growth.

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We have performed various simulations in order to identify which kind of perturbation turns out to be mainly responsible for the generation of the blueshifted continuum. Figure 7a shows the spectrum obtained when the nonlinear perturbations are neglected, that is to say when the terms accounting for the stimulated Raman scattering and the self-steepening effects are not included in the NLSE. The contribution of the higher-order linear dispersion is instead working as well as the wavelength-dependent loss. The values of the excitation parameters are the same as in the previously discussed situations. The result is highly similar to the one reported in Fig. 6b suggesting that the main reason for the growth of dispersive waves leading to SCG is the effect of the differential dispersion. As a proof of that one can consider the spectrum displayed in Fig. 7b, which is the result provided by the numerical tool when the only active perturbation is the one coming from the Raman effect. In such a case the broadening to the short-wavelength side has a much smaller extent as the contribution from the Raman scattering is severely limited by the impossibility for the Raman solitons to propagate. The results provided by the numerical simulations clearly show that the impossibility for the spectrum to broaden to the red side does not halt the generation of a continuum on the blue side. In the usual situation the Raman-induced broadening of the spectrum to longer wavelengths increases the extent and the intensity of the continuum also on the blue side because of the resonant coupling between dispersive waves and red-shifted pulses. However when the Raman shift is prevented by the fact that the propagation is forbidden because of the modal cut-off, there are other processes responsible for the generation of new spectral components at higher frequencies. Such blue-shifted components arise for the perturbative effect of higher-order linear dispersion and of nonlinear dispersion. According to our numerical results, when the pump wavelength is tuned close to the zero dispersion wavelength, the role of differential dispersion is predominant. When the pump wavelength is instead tuned far away from the zero dispersion wavelength, the self-steepening effect is the main reason for the spectral broadening.

5

Conclusions

We have investigated the dynamics leading to supercontinuum generation in a PCF by selectively exciting higher-order modes through an offset pumping technique. The properties of the PCF are such that there is a wide range of wavelengths in which the fundamental mode experiences normal dispersion, whereas the LP11 and LP21 modes propagate in the anomalous dispersion regime, generating, at sufficiently large input power, a supercontinuum based on the soliton fission mechanism. We find that the existence of a cut-off wavelength for the higherorder modes sets a limit to the spectral broadening on the long-wavelength side. This latter effect is particularly dramatic in the case of the LP21 mode, in which, by using a pump wavelength slightly below the cut-off wavelength, the spectral

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broadening occurs only on the blue side of the pump wavelength. Our experimental results are successfully compared to numerical solutions of the NLSE. We believe that the discussion presented in this work can be helpful also for the investigation of pulse propagation in photonic nanowires (Foster et al. 2008). Such a kind of waveguides has recently drawn much attention because of the sub-micron dimensions leading to a small footprint and enhanced nonlinear properties which make them very promising candidates for performing all-optical signal processing on photonic chips. SCG has in fact been demonstrated in photonic nanowires with characteristics similar to the ones reported in this work and the important role played by the wavelength-dependent loss has been highlighted (Foster et al. 2005). Acknowledgments This work was partially supported by the projects FIRB-MUR RBIN043TKY and FIRB-MUR RBIN04NYLH. The collaboration between the Tunisian and Italian groups was enabled with support from the Tempus Project JEP_31006.

References Alfano, R.R., Shapiro, S.L.: Emission in the region 4000 to 7000 Å via four-photon coupling in glass. Phys. Rev. Lett. 24(11), 584–587 (1970) Champert, P.A., Couderc, V., Leproux, P., Février, S., Tombelaine, V., Labonté, L., Roy, P., Nérin, P., Froehly, C.: White-light supercontinuum generation in normally dispersive optical fiber using original multi-wavelength pumping system. Opt. Express 12(19), 4366–4371 (2004) Cherif, R., Zghal, M., Tartara, L., Degiorgio, V.: Supercontinuum generation by higher-order mode excitation in a photonic crystal fiber. Opt. Express 16(3), 2147–2152 (2008) Cristiani, I., Tediosi, R., Tartara, L., Degiorgio, V.: Dispersive wave generation by solitons in microstructured optical fibers. Opt. Express 12(1), 124–135 (2004) Dudley, J.M., Provino, L., Grossard, N., Maillotte, H., Windeler, R.S., Eggleton, B.J., Coen, S.: Supercontinuum generation in air-silica microstructured fibers with nanosecond and femtosecond pulse pumping. J. Opt. Soc. Am. B 19(4), 765–771 (2002) doi:10.1364/ JOSAB.19.000765 Dudley, J.M., Genty, G., Coen, S.: Supercontinuum generation in photonic crystal fiber. Rev. Mod. Phys. 78(4), 1135–1184 (2006) Efimov, A., Taylor, A.J., Omenetto, F.G., Knight, J.C., Wadsworth, W.J., Russell, P.St.J.: Phasematched third harmonic generation in microstructured fibers. Opt. Express 11(20), 2567– 2576 (2003) Foster, M.A., Dudley, J.M., Kibler, B., Cao, Q., Lee, D., Trebino, R., Gaeta, A.L.: Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation. Appl. Phys. B 81(2–3), 363–367 (2005) Foster, M.A., Turner, A.C., Lipson, M., Gaeta, A.L.: Nonlinear optics in photonic nanowires. Opt. Express 16(2), 1300–1320 (2008) Gaeta, A.: Nonlinear propagation and continuum generation in microstructured optical fibers. Opt. Lett. 27(11), 924–926 (2002) Genty, G., Lehtonen, M., Ludvigsen, H., Kaivola, M.: Enhanced bandwidth of supercontinuum generated in microstructured fibers. Opt. Express 12(15), 3471–3480 (2004) Gorbach, A.V., Skryabin, D.V., Stone, J.M., Knight, J.C.: Four-wave mixing of solitons with radiation and quasi-nondispersive wave packets at the short-wavelength edge of a supercontinuum. Opt. Express 14(21), 9854–9863 (2006)


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Husakou, A.V., Herrmann, J.: Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers. Phys. Rev. Lett. 87(20), 203901 (2001) Konorov, S.O., Serebryannikov, E.E., Zheltikov, A.M., Zhou, P., Tarasevitch, A.P., von der Linde, D.: Mode-controlled colors from microstructure fibers. Opt. Express 12(5), 730–735 (2004) Lee, J.H., van Hove, J., Xu, C., Ramachandran, S., Ghalmi, S., Yan, M.F.: Generation of femtosecond pulses at 1350 nm by Cerenkov radiation in higher-order-mode fiber. Opt. Lett. 32(9), 1053–1055 (2007) Ranka, J.K., Windeler, R.S., Stentz, A.J.: Visible continuum generation in air silica microstructure optical fibers with anomalous dispersion at 800 nm, Opt. Lett. 25(1), 25–27 (2000) Tartara, L., Cristiani, I., Degiorgio, V., Carbone, F., Faccio, D., Romagnoli, M., Belardi, W.: Phase-matched nonlinear interactions in a holey fiber induced by infrared super-continuum generation. Opt. Commun. 215(1–3), 191–197 (2003) Wadsworth, W., Joly, N., Knight, J.C., Birks, T.A., Biancalana, F., Russell, P.St.J.: Supercontinuum and four-wave mixing with Q-switched pulses in endlessly single-mode photonic crystal fibres. Opt. Express 12(2), 299–309 (2004)

Symmetry and the Local Field Response in Photonic Crystals Jeffrey F. Wheeldon, Henry P. Schriemer Centre for Research in Photonics (CRPuO), University of Ottawa, 800 King Edward Ave., Ottawa, Ontario, ON K1N 6N5, Canada [email protected]

Abstract Singular states of photonic crystal local optical field are described based on group theoretic treatments of the fundamental system symmetries expressed through local representations of the singular character of the Bloch modes. The local field response is illustrated, in a purely two-dimensional system for clarity, by an analysis of the local polarization state. The fundamentals of group theory are introduced through the group representation, whose partner functions may be used to express the photonic crystal eigenmodes. We have extended the eigenmode k-group treatment to the derivation of the irreducible matrix representations of the transformation operators of the eigenmode space k-group, which may be applied with respect to any symmetry axis. This permits a comprehensive discussion of the local polarization state, and its symmetry transformation properties, through the set of crystallographic orbits. Transformation relations permit the nature and location of polarization singularities to be identified by the sites’ Wyckoff positions. Confirmation with analytic mode determination is presented.

Keywords: Photonic crystal, group theory, singularity, vortex, polarization, Wyckoff position, symmetry, transformation, disclination

1

Introduction

The control and manipulation of light is a subject of great interest, in many fields and applications, where the interaction of light with matter is the fundamental concern. Several decades ago, it was suggested (Yablonovitch 1987; John 1987) that microstructures with optical periodicity could control photonic properties in the same manner as crystalline semiconductor materials control electronic properties. In such photonic crystals, the transport of light is renormalized by means of distributed Bragg reflections. For sufficient dielectric contrast, photons within a photonic crystal will exhibit many of the same properties as electrons within a semiconductor, most notable forbidden energy gaps. Equally important is the fact


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J.F. WHEELDON AND H.P. SCHRIEMER

that the transport of electromagnetic energy within a photonic crystal can be tailored by modifying the underlying periodically-structured dielectric profile, and characterized by its dispersion surfaces. A renewed appreciation for the richness of the dispersion relations has led to a deeper consideration of the modal structure itself across a range of applications. Inherent to each of these studies is an interest in the properties of light within photonic crystals at length scales smaller than their lattice constant. Our investigation into the energy transport properties at the local level reveals optical features never before expounded in the photonic crystal community: The electromagnetic mode structure bears the hallmarks of singular optics, whereby the field becomes characterized by singularities – that is, spatial locations where mathematical descriptions of optical properties become undefined. It was not until the mid-1970s that the optics community realized that the subwavelength features of electromagnetic waves could be systematically described by examining the singular points within the electromagnetic field (Nye 1999). Optical singularities are locations in the electromagnetic field where a particular description of the electromagnetic field becomes undefined. In each case, the singular points are major features that greatly impact the behaviour of the electromagnetic field around the singular point. In electromagnetic fields that are adequately described using scalar waves (i.e., light which is linear polarized in a single direction), the singular points take the form of phase singularities (Wheeldon et al. 2007). For more complex vector fields, one must also consider the state of polarization. The state of polarization of the electromagnetic field is one of its most fundamental properties, quantifying the vector character of the optical response of systems ranging from the simple to the complex. The polarization state is circumscribed by three distinct types of singularities: pure circular polarization (indeterminate ellipse rotation angle); pure linear polarization (indeterminate polarization handedness); and disclinations (indeterminate polarization due to zero field intensity in all polarization directions). Polarization singularities in transverse electromagnetic fields were first discussed (Hajnal 1987), with generalization to random vector fields significantly later (Berry and Dennis 2001). We present the first formal application of group theoretic principles to the study of such singularities in systems of high symmetry, namely two-dimensional photonic crystals. Our goal is a systematic application of group theoretic methods to the optical modes of photonic crystals as a means of establishing fundamental relationships linking the symmetry of an optical system to the singular points of its electromagnetic field. Previous group theoretic studies focused on the global optical response of photonic crystals (Sakoda 2005). To explain the sub-wavelength character of the electromagnetic field, our application of site (local) symmetry to the optical response of a photonic crystal has foundationally expressed the local state of polarization within the context of singular optics (Wheeldon and Schriemer 2009). Polarization singularities reveal the complex nature of the vector character of the local state of the electromagnetic field in photonic crystals. A group theoretic

SYMMETRY AND THE LOCAL FIELD RESPONSE

163

representation is used to express its complementary polarization singularity representations, where fundamental transformation operations permitted by the system’s space group are quantified. As a result, the entire electromagnetic field is determined from the fundamental domain of the system’s space group, using derived transformation properties of the local state of polarization. Furthermore, it is found that local symmetry requires the electromagnetic field to become singular at particular Wyckoff positions. We first review the fundamentals of group theory and its application to the Bloch modes of a two dimensional photonic crystal, noting that they are the partner functions of the irreducible representations of the symmetry transformation operators. We then introduce the concepts of site symmetry and establish the necessary transformation properties of the optical Bloch modes. We do this by first illustrating the local transformation properties of the dielectric profile within crystallographic orbits, classifying the nature of high symmetry locations by their Wyckoff positions. The relationship between specific space k-groups and the point k-group is established in terms of their irreducible matrix representations, permitting translations within the unit cell to be expressed. The fundamentals of the local polarization state are then given, followed by group theoretic expressions describing the nature of their local symmetry transformations. Finally, it is revealed, through a constraint analysis predicated on these local symmetry transformation relations, that singularities of particular type must manifest at the special Wyckoff positions. Comparison is made with the polarization response of an analytically determined Bloch mode, fully validating our group theoretic predictions.

2

Periodic symmetry and the photonic crystal point group

2.1 Group theory fundamentals The symmetry properties of the system under consideration may be established through its group G, which is a set of elements g that is closed under group multiplication and subject to the associative property; in addition, for every g ∈ G , g −1 ∈ G is its inverse and gg −1 = g −1 g = E is the identity element. We note that gi and g j are conjugate to one another if there exists q ∈ G such that

gi = qg j q −1 . All of the elements that are mutually conjugated form a conjugacy class (Tinkham 1992). To express the symmetry properties, we consider the representation of G provided by the homomorphic group GD , whose elements are linear operators Dˆ ( g ) = gˆ in a vector space L of dimension m and basis {e } ( i = 1, 2, … m ). m

i

The basis elements are the partner functions of the representation, thus matrix

164


v formulations for Dˆ ( g ) are provided by D ( ) ( g ) , whose matrix elements

D (ji ) ( g ) are defined by v

l Dˆ ( v ) ( g ) e i = ∑ j D (jiv ) ( g ) e j ,

(1)

where l is the number of degenerate partner functions. The set Γ( v ) of unitary v matrices D ( ) ( g ) , for all g ∈ GD , is the matrix representation, v, of GD in the basis ei ( i = 1, 2, … m ). An equivalent representation may be constructed by ˆ ˆ ( g ) Aˆ −1 , for all g ∈ G , means of a similarity transformation, Dˆ ′ ( g ) = AD D

ˆ = deriving from the choice of a new basis defined through Ae ∑ Aji e j i j

( i = 1, 2, … m ) in Lm . Clearly, there is an infinite number of equivalent representations of GD . Since similarity transformations preserve matrix trace, we may define the character χ ( g ) of a matrix representation as

{

}

χ ( g ) = Tr D ( g ) = ∑ Dii ( g ) .

(2)

i

The set of characters, χ ( g ) for all g ∈ GD , identically labels all equivalent representations and permits group characterization independent of the basis. The fundamental representations of the group are those for which no invariant subspaces L p ⊂ Lm exist when acted upon by all Dˆ ( g ) . For a finite group, there will be a finite number of irreducible representations (equal to the number of conjugacy classes), each with dimension mv , such that h = ∑ mv2 is the number v

of elements in the group. All other representations are reducible, and their vector spaces Lm can be decomposed into a direct sum of invariant sub spaces, as

Lm = ∑ L(mv) , of the irreducible representations L(mv) . This implies that reducible v

v

matrix representations Γ ( red ) can be decomposed into a direct sum of irreducible matrix representations, Γ ( red ) = ∑ av Γ ( v ) , where the constituent matrix elements possess block diagonal form for a particular similarity transformation. It is thus the choice of basis that determines the reducibility or irreducibility of a representation. The complete system symmetry is thus fully articulated by a character table, where for each independent representation the set of characters is provided for each conjugacy class. This is ordinarily done for the irreducible representations.


165

2.2 Transformation operators and the eigenvalue problem A photonic crystal is described by the periodicity of its lattice for a particular distribution of material properties within a unit cell, from which the full space group may be determined. Restricting ourselves to a purely dielectric response, where the material variation is solely in the permittivity, the symmetry properties of the photonic crystal are described by transformation operators, R, that leave the dielectric profile ε ( r ) invariant:

Rε ( r ) = ε ( r ) .

(3)

We limit our consideration to the symmorphic space group, comprising the periodic translation symmetry subgroup and the photonic crystal point subgroup. The former arises from the translational invariance of the Bravais lattice, whence we may define transformation operators in this subgroup as periodic translation operators T ( R ) , where R is an integer sum of lattice vectors. The latter expresses the maximal symmetry of the unit cell with respect to local invariance, where the point associated with the principle axis of symmetry remains unchanged under the application of rotations and reflections. The point group transformation operators are the rotational symmetry operators Cn , where n specifies the 2π n rotation angle about the principal axis ( n = 1 equivalent to the identity operator E), and the mirror reflection operators σ v , where v identifies the vertical mirror plane through the principal axis.

yˆ σ y′ C3 C2

xˆ

σx

σ d′ C2

C4

xˆ

σy

C6−1 C3−1

(a)

σ x′ C6

σy

yˆ

σ d′′

σ y′′

C4−1

σ x′′

(b)

σx

Fig. 1 Two-dimensional photonic crystal lattices, top down view of the dielectric profile and the set of symmetry operators that leave the dielectric profile invariant: (a) hexagonal lattice, and (b) square lattice.

166


There may be multiple axes of rotation and planes of mirror reflection symmetry that satisfy (3), but the principal axis will be the axis of highest rotational symmetry and it will be coincident with the largest number of mirror reflection planes (Evarestov and Smirnov 1993). For two-dimensional photonic crystals, the dielectric structure is periodic in the x–y plane and infinite in the zˆ direction. Thus we focus on transformation operators acting in the x–y plane (the principal axis will be parallel to the zˆ axis). For a system that is periodic in two-dimensions, the only possible rotational symmetries in the x–y plane are Cn , n = 1, 2,3, 4, 6 . Due to this limitation on rotational symmetry, the main lattices of interest are of hexagonal and square symmetry; the dielectric profiles of their extended unit cells are shown in Fig. 1 for lattices of circular columns. For a hexagonal lattice, the point group whose symmetry operations leave the dielectric profile invariant is

C6 v = { E , C6 , C6−1 , C3 , C3−1 , C2 , σ x , σ x′ , σ x′′, σ y , σ y′ , σ y′′} .

(4)

For a square lattice, the point group is given by C4 v = { E , C4 , C4−1 , C2 , σ x , σ y , σ d′ , σ d′′} .

(5)

Note the non-uniqueness of principal axis location (the unit cell origin being a column center), as a consequence of the periodicity expressed by T ( R ) . Shifting the coordinate system associated with the principal axis by any transformation operation leaves all position vectors invariant, but alters their decomposition. Such coordinate transformations may be represented by R , as r ′ = Rr ,

(6)

where r and r ′ are the respective position vectors described by the two coordinate systems. Since R is an orthogonal matrix, the inverse of this operation is R −1 = R † , where R † is the transpose of R , and RR −1 = E is the identity matrix. An equivalent perspective, leaving the coordinate system invariant, has R transforming r into a new vector r ′ . In this study, the latter perspective will be used for the transformation of any object in space, including scalar functions and vector fields. Comparison of (1) and (6) implies that R r ≡ R r = r′ ,

(7)

identifying R as the matrix form of the operator R in the fixed coordinate system. We further note that any scalar function f ( r ) may be transformed by R, as

( )

R f R r = f ( r ) , or equivalently


(

)

R f ( r ) = f ′ ( r ) = f R −1r ;

167

(8)

that is, the application of R to f ( r ) produces a new scalar function, f ′ ( r ) . Finally, we note that if R transforms a vector field F ( r ) , the scalar part is transformed as (8) while the vector portion transforms as (7), so that (Sakoda 2005)

(

)

RF ( r ) = RF R −1r .

(9)

The same transformation operators that leave the dielectric profile invariant also leave the Maxwell operators, which describe the photonic crystal eigenvalue problem, invariant. The complex eigenfunction E ( r ) must satisfy LE E (r ) ≡

1 ω2 ∇ × {∇ × E ( r )} = 2 E ( r ) , ε (r ) c

(10)

where L E is the Maxwell operator for the electric field, ω 2 c 2 the eigenvalue, and c = 1

ε 0 μ0 is the speed of light in free space (Sakoda 2005). The operators

L E and R thus commute:

R L E R −1 = L E .

(11)

A similar prescription holds for L H , the Maxwell operator for the magnetic

field H ( r ) . We note that the time harmonic response is assumed, and that for brevity we will continue to solely present treatment of the electric field. Since the photonic crystals are periodic, as defined by (3), Bloch’s theorem can be applied. We may thus express the electric field eigenvector as

E ( r ) ≡ Ekn ( r ) = ukn ( r ) eik ⋅r ,

(12)

The Bloch mode Ekn ( r ) is composed of two parts: a periodic part that has the periodicity of the Bravais lattice, given by

ukn ( r + R ) = ukn ( r ) ,

(13)

and a space harmonic term eik ⋅r , where k is the Bloch wave vector. For any Bloch wave vector q = k + G , where k is a wave vector in the first Brillouin zone and G is the appropriate reciprocal lattice vector, (12) and (13) yield

168


E q n ( r + R ) = u qn ( r + R ) e

iq ⋅ ( r + R )

= e iq ⋅ R E q n ( r ) ,

(14)

Since eiG ⋅R = 1 , we find

Ekn ( r + R ) = ukn ( r + R ) e (

i k + G ) ⋅( r + R )

= eik ⋅R eiG ⋅R Ekn ( r ) = eik ⋅R Ekn ( r ) ,

(15)

so q and k cannot be distinguished on the basis of translational symmetry, and are thus equivalent Bloch modes. Thus any wave vector q can be mapped back into the first Brillouin zone. We may therefore identify each Bloch mode by its Bloch wave vector k in the first Brillouin zone and its frequency (energy) band index n. For a purely two-dimensional photonic crystal, the eigenmodes of (10) are linearly polarized, and hence may be treated as scalar functions; the TM (transverse magnetic) Bloch modes are given by

Ekn ( r ) = zˆ Ekn ( r ) = zˆ ukn ( r ) eik ⋅r ;

(16)

the associated magnetic fields H kn ( r ) = xˆ H k(nx ) ( r ) + yˆ H k(ny ) ( r ) may be determined from the relevant Maxwell’s equation. Complementary forms exist for the TE (transverse electric) modes. As a consequence of (11), the Maxwell and transformation operators commute and so must possess simultaneous eigenmodes (Sakoda 2005). Hence, applying L(ETM ) to (16) and employing (11) in two dimensions yields RL(ETM ) R −1 REkn ( r ) = L(ETM ) Rukn ( r ) eik ⋅r =

ω2 c2

Rukn ( r ) eik ⋅r .

(17)

The TE case follows in like manner. From (17) since ukn ( r ) eik ⋅r is an eigenmode,

Rukn ( r ) eik ⋅r must also be an eigenmode with the same eigenvalue ω 2 c 2 (i.e., they are degenerate).

Since the eigenmode is a scalar function, applying R to it yields, via (8),

(

)

Rukn ( r ) eik ⋅r = ukn R −1r eik ⋅ R

−1

r

(

)

= ukn R −1r eiRk ⋅r = uR′ kn ( r ) eiRk ⋅r

(18)

since applying the same transformation to both vectors in a scalar product does not change its consequent value (i.e., k ⋅ R −1r = R k ⋅ R R −1r = R k ⋅ r ) (Tinkham 1992). This new eigenmode is degenerate, having the same band index n, but different wave vector R k ; that is, Bloch wave vectors k and Rk are degenerate in frequency, and the solution space may be reduced to an irreducible Brillouin zone.


169

This fundamental degeneracy within the Brillouin zone is shown in Fig. 2 with respect to the directions of high symmetry. In Fig. 2a, the irreducible Brillouin zone of a hexagonal lattice is shown by the grey triangle. It will cover the first entire Brillouin zone when transformed by the operators of the C6v point group. For each operation, the eigenmode will change from ukn ( r ) eik ⋅r to uR′ kn ( r ) eiRk ⋅r . Figure 2b is the comparable presentation for a square lattice. kyˆ

(a)

(b)

Κ

kyˆ

Μ

Μ Γ

Μ

Σ

Σ kxˆ

Κ

Μ Κ

kxˆ

Γ

Rk = k + G

Χ

Χ

Μ

Μ

Fig. 2 Reciprocal space description of the two-dimensional photonic crystal lattices, with the irreducible Brillouin zone (shaded regions) and points of high symmetry labeled for the (a) hexagonal lattice, and (b) the square lattice.

More specifically, the action of the periodic translation operator T ( R ) upon a function f ( r ) , from (3) and (8), is T ( R ) f (r ) = f (r + R ) .

(19)

If we apply T ( R ) to the eigenmode of (16), this yields T ( R ) ukn ( r ) e

ik ⋅ ( r )

= eik ⋅R ukn ( r ) eik ⋅r .

(20)

Equation (20) has the same form as (1), revealing the irreducible matrix representation of T ( R ) to be eik ⋅R for the one-dimensional partner function ukn ( r ) eik ⋅r . Subsequently applying T ( R ) to the degenerate eigenmode of (18),

whose Bloch wave vector is R k , we obtain

(

)

(

)

(

)

T ( R ) ukn R −1r eiRk ⋅(r ) = ukn R −1r + R −1R eiRk ⋅(r + R ) = eiRk ⋅R ukn R −1r eiRk ⋅r , (21)

170


(

)

(

)

where ukn R −1r + R −1R = ukn R −1r , since R −1R is a lattice vector and (18)

(

revealed ukn R −1r

)

to be periodic in the unit cell (Tinkham 1992). Hence, the

(

)

irreducible matrix representation of T ( R ) in the ukn R −1r eiRk ⋅r basis is eiRk ⋅R . When (15) holds, the wave vectors R k = k′ = k + G

(

(22)

)

are equivalent, ukn ( r ) eik ⋅r and ukn R −1r eiRk ⋅r must be associated with the same irreducible matrix representation of T ( R ) and are thus equivalent eigenmodes of the Maxwell operator. The point group transformation operators that satisfy (22) form the k-group, a subgroup associated with the Maxwell operator that reveals the symmetry of the eigenmodes (Sakoda 2005). The k-group establishes whether two eigenmodes are symmetry degenerate. For example, in Fig. 2a, the various wave vectors symmetry equivalent according to (22) are identified in the Brillouin zone of the hexagonal lattice. The K point, which has two other equivalent points, is associated with C3v = { E , C3 , C3−1 , σ y , σ y′ , σ y′′} . The M point has only one other equivalent point

and is associated with C2 v = { E , C2 , σ y′′, σ x′′} . Finally, the Σ point has no other equivalent points and is associated with C1h = { E , σ x′′} . Thus, while the dielectric

profile sets the maximum symmetry of the photonic crystal system, it is the subgroup established by the wave vector that determines eigenmode symmetry. The irreducible matrix representations D ( v ) ( R ) of the k-group operators R may be given via (1) as REk( vni) ( r ) = ∑ j D (jiv ) ( R ) Ek( vnj) ( r ) , l

(23)

where D (jiv ) ( R ) is a matrix element of D ( v ) ( R ) , v is the irreducible representation label, j is the degeneracy index, and l is the total number of degenerate partner functions (which is determined by the dimension of the basis) (Tinkham 1992). The partner functions are the photonic crystal eigenmodes Ek( vni) ( r ) , here given by (16). Since there are an infinite number of frequency bands, but only a finite number of irreducible representations, many of the photonic crystal eigenmodes will be associated with the same irreducible representation. In the same way, there are many different irreducible matrix representations, as defined by (23), which are equivalent representations, as earlier described. These equivalent representations have the same set of characters, defined by (2).

SYMMETRY AND THE LOCAL FIELD RESPONSE Table 1

171

The C3v character table. Conjugacy classes

v

E

2C3

3σ v

A1

1

1

1

A2

1

1

−1

E

2

−1

0

The set of characters collectively identify all equivalent irreducible representations. Labels for the sets of characters are used for conciseness, as indicated in Table 1 for the irreducible representations of the C3v point group and their associated characters. Three distinct irreducible representations here exist: nondegenerate A1 and A2 representations, and the doubly-degenerate E representation (NB: E as employed here is not the identity operator, context makes the choice clear). The degeneracy of an irreducible representation is equal to the character, χ ( v ) ( E ) = l , of the identity operator, E . The irreducible labels from the independent periodic translation and point subgroups may be combined to generate an overall label for the photonic crystal eigenmode. Since for all Bloch modes, the irreducible representations of the periodic subgroup are l( periodic ) = 1 , from (20), the overall degeneracy of the purely two-dimensional photonic crystal eigenmodes will be entirely determined by the dimension of the irreducible representations of its k-group. The character tables of all two-dimensional point groups are well known and may thus be used to determine the properties of eigenmodes associated with particular irreducible representations.

3

Site symmetry and point subgroups of the photonic crystal

3.1 Transformation operators of the site symmetry groups We have examined the periodic symmetry subgroup and the photonic crystal point group, introducing the notion of irreducible representations with respect to the principal axis as a means of determining eigenmode properties. To more comprehensively determine such properties, it will be necessary to examine the irreducible representations of transformation operators that act with respect to any axis located a distance d away from the principal axis (Evarestov and Smirnov 1993). We may establish the general properties of such transformation operators for the Euclidean group E3 , in particular, how these transformation operators act

172


upon spatially dependent functions. The site symmetry may then be extracted by constraining the dielectric profile to be invariant, as per (3). The operators of E3 in the Cartesian coordinate system take the form of a combined transformation operator, gˆ = ( R a ) = ta R ,

(24)

which represents an operation R, such as a rotation or a mirror reflection, followed by a translation by an arbitrary vector a. If we apply the transformation operator ( R a ) to a point in space we obtain r ′ = ( R a ) r = Rr + a .

(25)

The formal composition law of two successive transformation operations is given by

(R

2

a 2 )( R1 a1 ) = ( R2 R1 R2 a1 + a 2 ) .

(26)

If we let the second element equal the inverse of the first element

( R a ) = ( R a ) , the right-hand side of (26) is equal to the identity element ( E 0 ) = ( R R R a + a ) . Then R R = E and R a + a = 0 , and the inverse −1

2

2

1

2

1

1

2 1

2

2

1

2 1

2

operator is given by

(R a ) 1

1

−1

(

)

= R1−1 − R1−1a1 .

(27)

Conjugacy classes for space group operators can be established in the same manner as for the point group operators. Two operators ( R a ) and ( R a )′ are

( ) ( )

conjugate if there exists a third operator R d , R d ∈ E3 , such that

( R a )′ = ( R d ) ( R a ) ( R d )

−1

(

)

= RRR −1 d + Ra − RRR −1d .

(28)

If we examine (28) for the case of a = 0 and R% = E , we find

( R 0 )′ = ( E d )( R 0 )( E d )

−1

= ( R d − Rd ) .

(29)


173

As shown in Fig. 3, if we define R = RO as a rotation about an axis passing through the primary origin O, then the operator ( R 0 )′ can be viewed as an O

identical rotation, but with the axis of rotation at a secondary origin O′ a distance d from the first (Tinkham 1992):

( R 0 )′ = ( R

0 ) = ( RO d − RO d )

O′

O

(30)

(restricting d to translations in the plane). O′

RO d − d d

RO d

O Fig. 3 The displacement of a point transformation operation R from the principle axis O to a secondary axis O′ a distance d away, with the accompanying translation d − RO d .

Application of the composite transformation operator (29) to a scalar function f ( r ) proceeds in the same manner as (8), whence

( R d − Rd ) f ( r ) = f ( ( R d − Rd ) = f

(( R

−1

−1

−1

r

)

))

(

(31)

)

d − R d r = f R −1r + d − R −1d .

Likewise, the composite transformation operator (29) acts on a vector field F ( r ) in the same fashion as (9) for a single operator R. The scalar part transforms as (31), and the vector portion transforms as (25), so that

(R

(

)

(

)

d − R d ) F ( r ) = ( R d − R d ) F R −1r + d − R −1d = F ′ R −1r + d − R −1d . (32)

3.2 Space group: Crystallographic orbits and Wyckoff positions We now extend our symmetry considerations of the photonic crystal to its space group G, explicitly including all symmetry elements, by recasting (3) as

( R a ) ε (r ) = ε (r ) ,

(33)

174


where ( R a ) ∈ G . Every point ri in a photonic crystal’s dielectric profile ε ( r ) may be classified into sets of symmetrically equivalent points known as crystallographic orbits. Beginning with any point r1 in a particular crystallo-

graphic orbit for ε ( r ) , the entire crystallographic orbit can be obtained from

( R a) for all

( R a)

−1

−1

r1 = ri ,

(34)

∈ G and ri ∈ Q , where Q is the set of all points in the

crystallographic orbit. This follows from (33), where the complementary local transformation property of the dielectric profile may be written as

( R a ) ε (r ) = ε (( R a) 1

−1

)

r1 = ε ( r1 ) ,

(35)

whence the symmetry constraint

ε ( ri ) = ε ( r1 ) ,

(36)

for all ri ∈ Q , is then established. Hence, the crystallographic orbit determines the

set of points with identical local symmetry, ε ( ri ) , when transformed by

( R a ) ∈ G . Since the space group contains an infinite number of lattice translations, there will be an infinite number of points in every orbit. Not all transformation operators ( R a ) produce unique elements from the generation point r1 . Instead, some transform r1 onto itself,

( R a) r

1

= r1 ,

(37)

whence

( R a) ε (r ) = ε (r ) . 1

1

(38)

The elements of the photonic crystal point group derive from (38), for a = 0 and all R ∈ Cnv , as

( R 0) ε (r ) = ε (r ) 1

1

(39)


175

where r1 locates the principal axis of the photonic crystal. The set of transformation operators that satisfy (37) form the site symmetry group Gr1 of

ε ( r1 ) with respect to G. The site symmetry group Gr will be a subgroup of the 1

photonic crystal point group. For a given crystallographic orbit, all site symmetry groups Gri will be conjugate subgroups of Gr1 because ( R a ) Gr1 ( R a ) = Gri for −1

all ( R a ) ∈ G with ( R a ) ∉ Gr1 , ( R a ) ∉ Gri . Consequently, the points of a given crystallographic orbit have comparable site symmetry groups Gri , but operators of each group are with respect to their individual points ri . Points are termed special points if the site symmetry groups Gri contain at least one transformation operator

( R a)

in addition to the identity element

( E 0 ) ; otherwise they are known as

general points. A photonic crystal has infinitely many crystallographic orbits, but they may be classified according to the nature of their site symmetries. As there are a finite number of such symmetries, there will be a finite number of labels, Wx , termed Wyckoff positions. The alphabetical subscript indicates the hierarchy of the positions, with a denoting positions of highest site symmetry. The site symmetry groups associated with any given Wyckoff position are conjugate as defined by

( R a) G ( R a) r1

−1

= Gri , where ( R a ) ∈ E3 , ( R a ) ∉ Gri , ( R a ) ∉ Gr j (Hahn 1995).

This is a more general statement of conjugacy than articulated above because ( R a ) ∈ E3 for Wyckoff positions, not just the space group. Note that, for any two points r1 and r2 associated with the same Wyckoff position, (36) must hold only if these points are of the same crystallographic orbit; if they are of different orbits, then r1 and r2 are not necessarily symmetry equivalent points in ε ( r ) . The international tables for crystallography list the Wyckoff positions and their site symmetry groups within the primitive unit cell for every possible crystal space group (Hahn 1995). Since the number of site symmetry groups associated with a Wyckoff position is infinite, only the conjugate site group Gr (a subgroup of the point group Cnv ) is listed for simplicity. The number of points for a given crystallographic orbit Gr1 that appear in the primitive unit cell is equal to the order hCnv of the photonic crystal point group divided by the order hGr of the site symmetry group associated with the Wyckoff position (Evarestov and Smirnov 1993): t = hCnv hGr .

(40)

176


Since all the crystallographic orbits associated with a Wyckoff position have conjugate subgroups, (40) holds for all the crystallographic orbits associated with a particular Wyckoff position. Let us consider a lattice structure with C3v point symmetry (i.e., for a particular eigenmode of a hexagonal photonic crystal), as illustrated in Fig. 4a. Its first three Wyckoff positions are listed in Table 2. The primitive unit cell of the hexagonal lattice is a rhombus (composed of two equilateral triangles). A general point in this unit cell has a site symmetry group containing just the identity operator, thus t = 6 / 1 = 6 elements of crystallographic orbit appear in the unit cell, as shown by the black dots in Fig. 4a. At the vertices of the equilateral triangles, the site symmetry group is C3v = { E , C3 , C3−1 , σ y , σ y′ , σ y′′} . The vertices are part of the crystallographic orbit where the principal axis at ( 0, 0 ) is the generating point; they are associated with the Wa Wyckoff position and there is t = 6 / 6 = 1 per unit cell, as anticipated. At the middle of the equilateral triangles, whose locations with respect to the indicated origin are symmetries are C3 = { E , C3 , C

−1 3

}.

1 2

(

a −1,1

3

)

(

)

and a 0,1

3 , the site

These two points are part of the same

crystallographic orbit (associated with the Wb Wyckoff position), and there are clearly t = 6 / 3 = 2 per unit cell. The mirror reflection planes coincide with the equilateral triangle line segments, on which the site symmetries are C1h = { E , σ v } . There are an infinite number of points on each line segment, where each point generates a separate crystallographic orbit, all associated with the Wc Wyckoff position, t = 6 / 2 = 3 elements of each orbit (from the line segments terminating on the principle axis) occur within the unit cell. (a)

(b)

σ y′

y x

( 0, 0 )

O ′′ d′′

O′ d′

O

Fig. 4 Primitive unit cell of a C3v hexagonal lattice (dashed parallelogram): (a) the symmetry axes (triangles), fundamental domain (shaded region) and the positions (black dots) of the crystallographic orbit for a general Wyckoff position; (b) the same primitive unit cell, with the fractional lattice vectors d′ and d′′ indicated for the two different points, O′ and O′′ , that belong to the same crystallographic orbit.


177

Table 2 Special Wyckoff positions in the C3v hexagonal primitive unit cell. Wyckoff

Site

Multiplicity

Location

position

symmetry

(t)

(Cartesian basis)

Wa

C3v

1

( 0, 0 )

Wb

C3

2

Wc

C1h

3

1 2

(

a −1,1

) (

3 , a 0,1

)

3 ,

Triangle line segments

Let us examine the points associated with Wb Wyckoff position, as illustrated in Fig. 4b. These two points are part of the same crystallographic orbit, and according to 0, their site symmetry groups are conjugate with the C3 point group, where C3 = { E , C3 , C3−1} . Applying the transformation operator

(

(

with rO = ( 0, 0 ) being the origin and d′ = − a 2, a 2 3

( E d′ )

to (38),

)) a secondary axis O′ ,

yields

( E d′)( R a ) ε ( r ) = ( E d′) ε ( r ) ( E d′)( R a )( E d′) ( E d′) ε ( r ) = ( E d′) ε ( r ) ( R d′ − Rd′) ε ( r ) = ε ( r ) , −1

O

O

O

O

O′

(41)

O′

where we have used (29), ( E d′ ) ε ( rO ) = ε ( rO′ ) and R ∈ C3 . The site symmetry group of the point O ′ will contain all the elements

( R d′ − Rd′) ⊂ G ,

where

R ∈ C3 . To determine the conjugate subgroup at the second crystallographic orbit location O ′′ , we apply the mirror reflection operator about the principal axis

(σ ′ 0 ) to (41): y

(σ ′ 0 ) ( R d′ − Rd′) ε ( r (σ ′ 0) ( R d′ − Rd′) (σ ′ 0 ) (σ ′ 0) ε ( r (σ ′ Rσ ′ (σ ′ d′ − σ ′ Rσ ′ σ ′ d′) ) ε ( r

) = (σ y′ 0 ) ε ( rO′ ) ′ O ′ ) = (σ y 0 ) ε ( rO ′ ) O ′′ ) = ε ( rO ′′ ) , O′

y

−1

y

y

y

−1

y

y

−1

y

y

y

where we have again used (29), site

symmetry

group

(σ ′ Rσ ′ (σ ′ d′ − σ ′ Rσ ′ −1

y

y

y

y

y

of

the

)

y

(σ ′ 0 ) ε ( r

O′

y

point

O ′′

) = ε ( rO′′ )

(42)

and R ∈ C3 . The

will contain all elements

σ y′ d′ ) ⊂ G , where R ∈ C3 . Therefore, the second

−1

178


point O ′′ of the crystallographic orbit has a site symmetry group GO′′ that is

(

)

conjugate to site symmetry group of the first point GO′ by the operator σ y′ 0 .

3.3 Eigenmode symmetry transformations at secondary axes The local symmetries deriving from the photonic crystal space group were discussed above in the context of the dielectric profile, but they may also describe the photonic crystal eigenmodes. Earlier, in (23), we established the transformation properties of photonic crystal eigenmodes and classified them by the v irreducible matrix representations D ( ) ( R ) of their k-group, whose transformation operators act with respect to the principle axis. We now establish the irreducible matrix representations D ( v ) ( R d − Rd ) for the transformation operators acting with respect to any symmetry axis given by the space group G. For particular v v cases, we derive the relation between D ( ) ( R d − Rd ) and D ( ) ( R ) , focusing on the global transformation properties of the photonic crystal eigenmodes. If we apply a general operator ( R d − Rd ) ∈ G to the scalar Bloch mode (16), we obtain an equation similar to (18):

( R d − Rd ) u ( r ) e

ik ⋅ ( r )

kn

= ukn ( R −1r + d − R −1d ) e = uk′n ( r ) e

(

i R k ir

)

(

ik i d − R −1d

) i( R k ir ) e

(43)

.

Again, we focus on the TM case, but these results are equally applicable to the TE case. In general, application of any transformation operator of the photonic crystal space group G will create a new Bloch mode uk′n ( r ) e

(

i R k ir

)

with a Bloch

wave vector of Rk . Applying the periodic translation operator T ( R ) = ( E R ) to (43), we obtain

( E R ) u′ (r ) e (

i R k ir

kn

)

= ukn ( R −1r + R −1R + d − R −1d ) e =e

(

i R k iR

i ( R k ir ) ) ′ ukn ( r ) e ,

(

ik i d − R −1d

) i ( R k i(r + R )) e

(44)

where the periodicity of uk′n ( r ) = ukn ( R −1r + R −1R + d − R −1d ) follows from (20), since R −1R is a lattice vector (Evarestov and Smirnov 1993). The above equation

( defines eiRk ⋅R as the irreducible matrix representation of ( E R ) in the uk′n ( r ) e

i R k ir

)


179

basis. Equation (22) notes that wave vector satisfying R k = k ′ = k + G are equivalent, hence ukn ( r ) eik ⋅r and uk′n ( r ) e

(

i R k ir

)

are associated with the same irreducible

matrix representation of ( E R ) , because eiRk ⋅R = ei (k + G )⋅R = eik ⋅R .

The irreducible matrix representation eik ⋅R and the frequency (i.e., k and n) identify the eigenmode. When k and n are the same, then ukn ( r ) eik ⋅r and uk′n ( r ) e

(

i R k ir

)

must be either an equivalent eigenmode of the Maxwell operator, or

a linear combination of the degenerate eigenmodes also associated with the same irreducible matrix representation eik ⋅R . Therefore, as for the k-group, the operators ( R d − Rd ) ∈ G that satisfy

( R d − Rd ) k = k ′ = k + G

(45)

form the space k-group, Gk , of the photonic crystal eigenmode. Since all symmetry axes satisfy (45), there will be eigenmode space k-groups associated with them. If we revisit (43) and assume that R ′ = d − R −1d is a lattice vector, then ukn ( R −1r + d − R −1d ) = ukn ( R −1r + R ′ ) = ukn ( R −1r ) ,

(46)

based on the periodicity of ukn ( R −1r ) given by (21). Substituting (46) into (43), we find

( R d − Rd ) u ( r ) e kn

ik i ( r )

=e

(

ik i d − R −1d

)

ukn ( R −1r ) e

(

i R k ir

)

.

(47)

If the transformation operator R is part of the photonic crystal eigenmode’s k-group, then using (8) and (23) we find

Ruk( vni) ( r ) eik ⋅(r ) = uk( vni) ( R −1r ) ei ( R k i r ) = ∑ j D (jiv ) ( R ) uk( vnj) ( r ) ei (k ir ) , l

(48)

where the scalar Bloch mode of (16) has been explicitly introduced. Finally, we can substitute the results of (48) into (47) to obtain

180


( R d − Rd ) u ( r ) e (v) kni

ik i ( r )

=e

(

ik i d − R −1d

∑

l j

D (jiv ) ( R ) uk( vnj) ( r ) ei ( k i r )

= ∑ j D (jiv ) ( R d − Rd ) uk( vnj) ( r ) e ( l

D(

)

i k ir )

.

(49)

Equation (49) defines the elements of the irreducible matrix representation ( R d − Rd ) of the transformation operator ( R d − Rd ) , where uk(vni) ( r ) eik ⋅(r ) is

v)

the basis. This means that D(

v)

( R d − Rd ) = e

(

ik i d − R −1d

)

D(

v)

( R) ,

(50)

and thus the two irreducible matrix representations differ only by a phase factor. To exemplify irreducible matrix representations of site symmetry groups at non-principal axes, consider a non-degenerate eigenmode with C3v point group symmetry, such as the fundamental electric field component of the TM mode at k = xˆ 4π ( 3a ) ; it is associated with the A2 irreducible representation (Sakoda 2005). The fundamental symmetries are illustrated within the unit cell of Fig. 4. The irreducible matrix representations associated with the eigenmode are onev v dimensional ( D ( ) ( R ) = χ ( ) ( R ) ), and may thus be obtained from Table 1. The special Wyckoff position Wb identifies the non-principle axes, located within the unit cell at

1 2

(

)

a −1,1

(

3 and a 0,1

)

3 , with C3 = { E , C3 , C3−1} site sym-

metries. Table 3 enumerates the characters associated with the C3 group. Table 3 The C3 character table. Conjugacy classes

v

E

C3

C3−1

A E

1

1

1

1

−1 2 + i 3 2

−1 2 − i 3 2

E∗

1

−1 2 − i 3 2

−1 2 + i 3 2

Equation (50) may be used to calculate the irreducible matrix representations of the transformation operators ( R d − Rd ) associated with Wb from those associated with Wa (i.e., to determine Table 3 from Table 1) only if R ′ = d − R −1d (for all R ∈ C3 ) is a linear combination of the hexagonal lattice vectors a1 = a xˆ and

(

)

(

a 2 = 12 a xˆ + yˆ 3 . For d1 = 12 a −1,1

)

3 , we find that, using the usual Cartesian


181

matrix representations for the rotation operators (Sakoda 2005), d1 − E −1d1 = 0 , d1 − C3d1 = −a1

and

(

d1 − C3−1d1 = a 2 − a1 . Similarly, for

d 2 = a 0,1

)

3 ,

d 2 − E −1d 2 = 0 , d1 − C3d1 = a 2 − a1 and d1 − C3−1d1 = a 2 . Thus, the irreducible matrix representations (the characters, in this illustration) for the non-principal axis operators ( R d − Rd ) may be determined from those of the principle axis

operators. Substituting the above results into (50), expressed as χ ( v ) ( R di − Rdi ) = e

(

ik i di − R −1di

)

χ ( A ) ( R ) (for all R ∈ C3 and i = 1, 2 ), identifies χ ( v ) ( R d1 − Rd1 ) as 2

the E irreducible representation of C3 and χ (

v)

(R d

2

− Rd 2 ) as the E ∗ irreducible

representation of C3 ; see Table 3 for their respective character sets. Equation (50) has thus been explicitly validated in this instance. As we will next see, knowledge of the irreducible matrix representations of the photonic crystal space k-group will enable us to determine the location and type of polarization singularities in the electromagnetic field.

4

Expressing polarization singularities with group theory

4.1 The local polarization state The state of polarization is, in general, a spatially-dependent quantity that may be represented by an ellipse extracted at every point by mapping the modulation of the field vector over a temporal period. We express the spatially dependent complex electric field as E ( r ) = P ( r ) + iQ ( r ) = ⎡⎣a ( r ) + ib ( r ) ⎤⎦ e

iτ ( r )

,

(51)

where the orthogonality of a ( r ) and b ( r ) is established by choosing

τ ( r ) = 12 tan −1 ⎡⎣ 2P ( r ) ⋅ Q ( r )

( P ( r ) − Q ( r ) )⎤⎦ + n π , 2

2

r

(52)

with nr a spatially dependent integer ensuring a ≥ b at each point in the field (Born and Wolf 2003). The vectors a ( r ) and b ( r ) are the semi-major and semiminor axes, respectively, of the polarization ellipses traced out by E ( r, t ) = a ( r ) cos ⎡⎣τ ( r ) − ωt ⎤⎦ + b ( r ) sin ⎡⎣τ ( r ) − ωt ⎤⎦

in the time domain.

(53)

182


The electric field polarization response of the fundamental mode ( n = 1 ) in the hexagonal lattice at the Κ point, k = xˆ ( 4π 3a ) is shown in Fig. 5a. The extended unit cell has been shown for clarity, with the primitive unit cell dashed in red. The k-group associated with this eigenmode is C3v = { E , 2C3 ,3σ y } , and the electric field is associated with the A2 irreducible representation. The field has been analytically determined in the vanishing dielectric contrast limit (Wheeldon et al. 2007) The blue ellipses were generated for an equally-spaced array of points by mapping (53) over one temporal period, normalizing the major axis length for clarity. The large grey circles are the putative locations of the hexagonally arrayed cylinders. The local polarization directions, shown in Fig. 5b, are determined from P ( r ) × Q ( r ) , which are perpendicular to the plane defined by the polarization ellipses. The left-hand polarization is + zˆ (red, counter-clockwise), while the right-hand polarization −zˆ (blue, clockwise). (a)

(b)

Fig. 5 Polarization response for the electric field associated with the A2 irreducible representation of C3v . (a) Calculated local polarization states: polarization ellipses (blue), dielectric columns (closed grey circles), linear polarization (black lines), circular polarization (red circles) disclinations (green circles); the primitive unit cell is delineated by the dashed red lines and the fundamental domain is shown in gold; (b) P ( r ) × Q ( r ) .

The polarization state is circumscribed by three distinct types of singularities: pure circular polarization (indeterminate ellipse rotation angle); pure linear polarization (indeterminate polarization handedness); and disclinations (indeterminate polarization direction with a concomitant vanishing field) (Wheeldon and Schriemer 2009). A singularity is a location r0 where at least one of the fundamental field metrics becomes undefined. C points are locations, shown as red circles in Fig. 5a, where the local state of polarization is circular. Equation (51) then requires P ( r0 ) ⋅ Q ( r0 ) = P ( r0 ) − Q ( r0 ) = 0 ; in consequence, (52) asserts that τ ( r0 ) is 2

2

undefined. These locations correspond, in Fig. 5b, to extrema in P ( r ) × Q ( r ) .


183

L lines are continuous arrays of points where the local state of polarization is linear. On an L line, the vector normal to the ellipse becomes indeterminate, P ( r0 ) × Q ( r0 ) = 0 . This may occur when either P ( r0 ) or Q ( r0 ) vanish, or are (anti)parallel. In Fig. 5a, the L lines are denoted by the straight line segments in black. These lines correspond, in Fig. 5b, to the demarcation between regions of right-handed (negative) and left-handed (positive) polarization. Disclinations are locations where the local field magnitude vanishes, the state of polarization being indeterminate since (52) implies τ ( r0 ) is undefined due to P ( r0 ) = Q ( r0 ) = 0 . In Fig. 5a, these locations are indicated by the green circles.

4.2 Symmetry transformation and the local field The photonic crystal eigenmode k-group, Cnv , is fixed by the choice of ⎡⎣ε ( r ) , k ⎦⎤ , where the eigenmodes are the partner functions of the irreducible matrix representations D ( v ) ( R ) of the transformation operators R. Restricting ourselves to nondegenerate modes, (2) and (23) imply that the partner functions transform as

R F ( v ) ( r ) = χ R( v ) F ( v ) ( r )

(54)

for all R ∈ Cnv , where F ( v ) ( r ) is either a magnetic field H k , n ( r ) or its associated electric field Ek , n ( r ) . This equation quantifies the global symmetry transformation properties of any Bloch mode about the principal axis, and is invariant to all translations of the principal axis by a lattice vector R (i.e., all points of the crystallographic orbit where the origin is the generating point). The photonic crystal space k-group, Gk , is given by the set of all symmetry operations

( R d − Rd ) ,

where d is any distance such that R ′ = d − R −1d is a

lattice vector. For any such point a distance d away from the principal axis, (50) asserts that global transformations of the partner functions be expressed as

( R d − Rd ) F ( r ) = e (v)

(

ik i d − R −1d

)

χ ( v ) ( R ) F ( v ) ( r ) = χ ( v ) ( R d − Rd ) F ( v ) ( r ) ,

(55)

where χ ( v ) ( R ) is the character of the associated eigenmode k-group. The character of a select site symmetry group is thus phase shifted with respect to the character associated with the corresponding point group. Equations (33) through (36) established the symmetry equivalent points that defined the crystallographic orbits within the dielectric profile, ε ( r ) , where the

184


value of the dielectric profile was shown to be the same at each symmetry equivalent point. In like manner, we can define crystallographic orbits within the partner functions, which we take to be the electric field for illustration. A crystallographic orbit Qk of the space k-group Gk is generated by applying all its

symmetry operators ( R d − Rd )

−1

to a generating point r1 , yielding

( R d − Rd )

−1

r1 = ri ,

(56)

for all ri ∈ Qk . Since (55) must be valid at every point r1 within the electric field, the complementary local electric field relation may be given as

( R d − Rd ) E ( r ) = e (v)

(

ik i d − R −1d

)

1

χ ( v ) ( R ) E( v ) ( r1 ) .

(57)

Applying the operator ( R d − Rd ) to the vector and scalar parts of E( v ) ( r1 ) , as per (32) gives

( ( R d − Rd ) E )( ( R d − Rd ) (v)

−1

)

r1 = e

(

ik i d − R −1d

)

χ ( v ) ( R ) E( v ) ( r1 ) .

(58)

Upon substituting (56) into (58), we find

( ( R d − Rd ) E )

(v)

ik i d − R d ( ri ) = e ( ) χ (v ) ( R ) E(v ) ( r1 ) , −1

(59)

for all ri ∈ Qk . This defines the relationship between the local electric field

vectors, E( v ) ( ri ) , at their symmetry equivalent points ri ∈ Qk . Since every point in

the electric field can be associated with a crystallographic orbit, this allows us to quantify the local transformation properties of the entire electric field. Since the operators ( R d − Rd ) ∈ Gk transform the partner function, (55) asserts a corresponding transformation of the polarization response as

( R d − Rd ) P ( r ) × ( R d − Rd ) Q ( r ) = e

(

i 2 k i d − R −1d

)

( χ ( R )) P (r ) × Q (r ) (v)

2

because P( r) and Q (r ) must have the same character, e

(

−1

ik i d − R d

)

χ ( v ) ( R ) . The transformation of the mode

(60)

χ ( v ) ( R d − Rd ) =

( R d − Rd ) E ( r ) , at the generat1

ing point r1 , drives a corresponding transformation in the local state of polarization,

P ( r1 ) × Q ( r1 ) , as


185

( R d − Rd ) P ( r ) × ( R d − Rd ) Q ( r ) , = ( ( R d − Rd ) P ) ( r ) × ( ( R d − Rd ) Q ) ( r ) 1

1

i

(61)

i

where (32) and (56) have been employed. From (60), the complementary local relation for the states of polarization is obtained

( R d − Rd ) P ( r ) × ( R d − Rd ) Q ( r ) = e 1

(

i 2 k i d − R −1d

1

)

( χ ( R ) ) P ( r ) × Q ( r ) . (62) 2

(v)

1

1

Equating (61) and (62), noting that χ ( v ) ( R ) = ±1 , expresses the relationship between states of local polarization within a crystallographic orbit as

( ( R d − Rd ) P ) ( r ) × ( ( R d − Rd ) Q ) ( r ) = e i

(

i 2 k i d − R −1d

i

)

P ( r1 ) × Q ( r1 ) ,

(63)

for all ri ∈ Qk . To further progress our understanding of the symmetry relationships between states of local polarization within a crystallographic orbit, we now focus on the left hand side of (63). Since R ′ = d − R −1d is a lattice vector, the effect of

( R d − Rd ) on the components of the field is purely through the point operator R, see (25) and (32). Now if any two vectors P and Q are transformed in space by R, the cross product of the vectors RP and RQ creates another vector RP × RQ . The transformation law expresses the relationship between these two cross products as (Jackson 1999)

(

)

RP × RQ = det R R ( P × Q ) .

(64)

From (60), for any point operator R applied to the electric field mode, P ( r ) × Q ( r ) will remain completely invariant, since χ ( v ) ( R ) = ±1 for nondegenerate modes. Since R ( P × Q ) = P × Q , (64) may be written as

(

)

RP × RQ = det R ( P × Q ) .

(65)

Applying (65) to (63) allows the transformations within a crystallographic orbit to be expressed in terms of the generating point as

( det R ) P ( r ) × Q (r ) = e i

i

(

i 2 k i d − R −1d

)

P ( r1 ) × Q ( r1 ) ,

(66)

186


since det R = 1 for proper transformations (rotation operators, Cn ), and det R = −1 for improper transformations (reflection operators, σ y ). The polarization response magnitudes (i.e., ellipticities) are thus seen, from (66), to be the same for all members of the crystallographic orbit, P ( ri ) × Q ( ri ) = P ( r2 ) × Q ( r2 ) . The relative polarization direction, however, depends strictly upon e

(

−1

−i 2k i d − R d

)

( )

det R .

We have shown the polarization response of the fundamental mode ( n = 1 ) of the electric field in the hexagonal lattice at the Κ point, k = xˆ ( 4π 3a ) in Fig. 5a, and its fundamental symmetries in Fig. 4a. For any operation ( R d − Rd ) applied with respect to the principal axis ( d = 0 ), rotations preserve the handedness while reflections reverse it. There are, however, two axes located at the centers of the equilateral triangles, possessing C3 site symmetry, that are not equivalent to the principal axis. Table 3 enumerates the characters of their transformation operators.

( ) Since we have there shown that R ′ = d − R −1d is a lattice vector, e =1, and again rotations preserve the handedness, while reflections reverse it. The optical response shown in Fig. 5 clearly possesses the full translational invariance of the primitive unit cell. In Fig. 4a, selecting an arbitrary position r1 close to the origin, as a general Wyckoff position, we see that there are six elements comprising the crystallographic orbit ( t = nC3 v nGr1 = 6 / 1 = 6 ) in the i 2 k i d − R −1d

primitive unit cell. The crystallographic orbit with generating point r1 within the primitive unit cell has two elements deriving from the subgroup of rotation operators 2C3 , where ellipticity is again preserved, and polarization handedness is maintained (e.g., P ( ri ) × Q ( ri ) = P ( r1 ) × Q ( r1 ) where i = 2,3 ); see Fig. 5b. The

rotation angle of the polarization ellipse is advanced by 2π 3 , see Fig. 5a, which can be understood by the local transformation properties of the electric field given by (59). Likewise, it has three elements, deriving from the reflection operator σ y′ applied to the first three elements, where the orientation and ellipticity of the polarization ellipses are mirror-symmetric, with Fig. 5b demonstrating their opposite polarization directions (e.g., P ( ri ) × Q ( ri ) = −P ( r1 ) × Q ( r1 ) , where i = 4,5, 6 ). The full optical response of the entire extended system can thus be expressed by the fundamental domain, denoted by the shaded one-third equilateral triangle in Figs. 4a and 5a, where one element from each crystallographic orbit of all local field vectors exists.


187

4.3 Singularities at special Wyckoff positions We have thus far established only the transformation properties of the local polarization state at general Wyckoff positions. However, for local fields at special Wyckoff positions, we may show that the site symmetry reveals the local state of polarization to be a polarization singularity. Using (59) with i = 1 , the local field symmetry transformation proceeds as

( R d − Rd ) E ( r ) = e (v)

(

ik i d − R −1d

1

This is an eigenvalue equation, where e

(

)

χ ( v ) ( R ) E( v ) ( r1 ) .

ik i d − R −1d

)

(67)

χ ( v ) ( R ) is the eigenvalue and

E( v ) ( r1 ) the eigenvector. To relate the singular nature of a local state of polarization to a particular site symmetry, we will express the local field, via (51), as

⎛ Px ( r1 ) + iQx ( r1 ) ⎞ ⎛ D ( r1 ) + iB ( r1 ) ⎞ E( v ) ( r1 ) = ⎡⎣ P ( r1 ) + iQ ( r1 ) ⎤⎦ = ⎜⎜ ⎟⎟ = Ω ( r1 ) ⎜ ⎟ ; (68) P iQ r + r C ( r1 ) ( ) ( ) y 1 ⎠ ⎝ ⎠ ⎝ y 1 components D ( r1 ) = Px ( r1 ) Py ( r1 ) + Qx ( r1 ) Qy ( r1 ) ,

(69)

B ( r1 ) = − Px ( r1 ) Qy ( r1 ) + Qx ( r1 ) Py ( r1 ) ,

(70)

C ( r1 ) = Py2 ( r1 ) + Qy2 ( r1 )

(71)

and

are purely real, while Ω ( r1 ) = ( Py ( r1 ) − iQy ( r1 ) )

−1

(72)

is a complex phase factor. For any r1 , substitution of (67) into (68) will yield a self-consistency condition contingent on the local symmetry. We now consider, in general, local fields with either reflection or rotation symmetry (or both), with the goal of identifying the site symmetry subgroups of C3v , and the respective natures of their local polarization states.

188

(σ


Elements of the crystallographic orbit related by reflections, such as v

)

d − Rd E( v ) ( r2 ) = e

opposite e

(

−1

i 2k i d − R d

polarization

)

(

ik i d − R −1d

)

χ ( v ) ( R ) E( v ) ( r1 ) from (59), have an equal but P ( r2 ) × Q ( r2 ) = − P ( r1 ) × Q ( r1 ) ,

handedness,

= 1 . Hence, for

(σ

v

)

d − Rd E( v ) ( r1 ) = e

(

−1

ik i d − R d

)

when

χ ( v ) ( R ) E( v ) ( r1 ) , we

require P ( r1 ) × Q ( r1 ) = 0 , whence the sense of rotation is undefined, and thus mirror planes must constitute L lines. It now remains to determine the linear polarization rotation angle, vis-à-vis the respective mirror planes. Let us

(σ 0 )

investigate the self-consistency condition, employing

y

for illustration,

where σ y ∈ C3v . The argument is general and proceeds identically for all the other mirror reflection planes passing through the principal axis. The site symmetry of the general case of a field vector located on this mirror plane is clearly C1h = { E , σ y } , where the symmetry axis is located at the principal axis. Suppressing r1 , for convenience, we obtain ⎡1 0 ⎤ ⎛ D + iB ⎞ ⎛ D + iB ⎞ ik i ( d − R−1d ) ( v ) ⎛ D + iB ⎞ χσ ⎜ ⎟, ⎢0 −1⎥ ⎜ C ⎟ = ⎜ −C ⎟ = e y ⎣ ⎦⎝ ⎠ ⎝ ⎠ ⎝ C ⎠

(73)

As it stands, (73) does not provide any insight into the required form of the eigenvector, E( v ) ( r1 ) (i.e., local field) upon a mirror symmetry plane. The global transformation properties illustrated by (55) reveal that field vectors located on

(

)

our mirror plane have characters χ ( v ) σ y d − Rd = e

(

ik i d − R −1d

)

χ ( v ) (σ y ) = −1 , the

character of the A2 irreducible representation about the principal axis. Substituting this value into (73) then generates the constraint D = B = 0 , where

C is given by (71), whence E ( r1 ) = Ω ( r1 ) ( 0,C ( r1 ) ) , either in the +y-direction or

in the –y-direction, depending on phase. The polarization direction is thus orthogonal to the mirror plane. Similar expressions may be found for any other mirror plane that passes through the principal axis. All points on such planes are identified by the Wyckoff position Wc , which then identify locations of linear polarization as shown in Fig. 5a and b, respectively. We next determine if there are any locations where (67) is valid for rotation operations. However, because polarization handedness is maintained upon rotation, no specific insight is gained from the general local transformation properties. Employing Cn in (68), the self-consistency condition yields


189

⎡cos ( 2nπ ) − sin ( 2nπ ) ⎤ ⎛ D + iB ⎞ ik i ( d − R−1d ) ( v ) ⎛ D + iB ⎞ χ Cn ⎜ ⎢ ⎥⎜ ⎟=e ⎟, 2π 2π ⎝ C ⎠ ⎣⎢ sin ( n ) cos ( n ) ⎦⎥ ⎝ C ⎠

(74)

where only the point transformations of ( R d − Rd ) have been applied to (68) because lattice vector translations merely shift the origin to equivalent principal axes. Inspection reveals that there are no local field solutions where (74) identifies singularities for n = 1 or n = 2 . However, for n = 3 , we find two distinct types of singularities, dependent on the site symmetry to which the C3 operator belongs. The expected form of the state of local polarization can be determined for the rotation operator C3 , whence (74) becomes ⎛ D + iB ⎞ 1 ⎛ − ( D + iB ) − C 3 ⎞ ik i ( d − R−1d ) ( A2 ) ⎜ ⎟=e χ ( C3 ) ⎜ ⎟. 2 ⎜⎝ ( D + iB ) 3 − C ⎟⎠ ⎝ C ⎠

(75)

In contrast to (73), we can determine the form of the eigenvector E( v ) ( r1 ) and its eigenvalue e

(

ik i d − R −1d

)

(

ik i d − R −1d

χ ( A ) ( C 3 ) from (75). When the eigenvalues e 2

)

χ ( A ) ( C3 ) 2

match the global transformation characters of (55), then the local field must satisfy (75). If they do not match, then the local field must vanish. Taking the ratio of the two vector component equations, to eliminate e

(

ik i d − R −1d

)

χ ( A ) ( C3 ) , 2

cross-multiplying and equating the consequent real and imaginary parts yields B 2 = C 2 + D 2 and BD = 0 , respectively. The only possible solutions to this constraint system, for non-zero fields, are D = 0 , B = ±C or B = 0 , D = ±iC . From all possibilities, there are only two unique eigenvectors for (75): the eigenvector in the form of left-hand circularly polarized light given by

E( v ) ( r1 ) = C ( r1 ) Ω ( r1 )( i,1, 0 ) , where the eigenvalue is e

(

ik i d − R −1d

)

(

(76)

)

χ ( A ) ( C3 ) = −1 + i 3 2 ; or the eigenvector in 2

the form of right-hand circularly polarized light given by

E( v ) ( r1 ) = C ( r1 ) Ω ( r1 )( −i,1, 0 ) , where the eigenvalue is e

(

ik i d − R −1d

)

(

(77)

)

χ ( A ) ( C3 ) = −1 − i 3 2 . The centers of C3 2

rotational symmetry must therefore evidence local states of circular polarization (i.e., C points).

190


The cyclic group C3 = { E , C3 , C3−1} identifies points of pure threefold rotational symmetry in the hexagonal lattice at the centers of equilateral triangles located at

(

d1 = 12 a −1,1

3

)

(

and d 2 = a 0,1

3

)

in the unit cell, as shown in Fig. 4a.

These two points are part of the same crystallographic orbit, previously identified by the Wyckoff position Wb , and their characters are shown in Table 3

(

as χ ( E ) ( C3 d1 − C3d1 ) = −1 + i 3 respectively.

χ

(E ) ∗

(C

3

Matching

the

)

2 and χ

global

(E ) ∗

(C

3

characters

(

)

d 2 − C3d 2 ) = −1 − i 3 2 ,

χ ( E ) ( C3 d1 − C3d1 )

and

d 2 − C3d 2 ) to the two different eigenvalues determined by (75), we can

predict that, at d1 , the state of local polarization will be left-hand circularly polarized, as in (76) and that at d 2 the state of local polarization will be righthand circularly polarized, as in (77). This is indeed what we see in Fig. 5a, the red circles showing the points of circular polarization. Moreover, the observed polarization handedness, as noted in Fig. 5b, conforms to that anticipated by the symmetry arguments (i.e., left-handed P × Q positive, right-handed P × Q negative). The remaining C3 points within the hexagonal unit cell are found at the intersections of the mirror planes, where they possess both C3 and C1h

symmetries, that is, the point group symmetry of the mode, C3v = { E , 2C3 ,3σ y } .

These locations are associated with the Wyckoff position Wa . These positions are the points of the crystallographic orbit where the principal axis is the generating A point and χ ( 2 ) ( C3 ) = 1 . This character does not match the two possible eigenvalues as determined by (75), therefore the field must vanish and form a disclination. Such disclinations are seen in Fig. 5a at the anticipated locations, and are indicated by green circles.

5

Conclusions

A definitive study on the sub-wavelength features of the optical modes of photonic crystals expressed through the polarization singularities of the electromagnetic field was successfully made using fundamental principles of group theory. A notable result was the application of site symmetry to express how the eigenmode should transform under all permissible local symmetries. Previous work had focused on transformation operators of the photonic crystal eigenmode k-group ( R ∈ Cnv ), with respect to the principal axis of the dielectric structure. We have extended this to the derivation of the irreducible matrix representations of the


191

transformation operators ( R a ) of the photonic crystal eigenmode space k-group

( ( R a ) ∈ G ), which may be applied with respect to any symmetry axis. As a result, the symmetry transformation properties of the local state of polarization could be derived. The complex state of local polarization of the electromagnetic field was systematically analyzed, locating the polarization singularities within the field and deriving the symmetry transformation properties of the local polarization states. The polarization symmetry insights were derived based on the site symmetry of the optical system. From the global perspective of the electric field, E ( r ) ,

we showed that the state of polarization, characterized by the vector field P ( r ) × Q ( r ) , does not change even when the electric field is transformed in space by a symmetry operator of its point k-group. We described the crystallographic orbits, which are sets of symmetry-equivalent points in the field. Relative polarization relationships between points of the crystallographic orbit were established based on the symmetry transformation properties of the Bloch mode. Finally, for points within the electromagnetic field congruent to positions of high symmetry, we demonstrated that the local states of polarization must be singular. Such points in the field appeared were noted to be the special Wyckoff positions. For the hexagonal lattice, it was determined that the linear polarization state (L lines) must occur on mirror reflection planes ( C1h symmetry); that circular polarization states (C points) arise at secondary axes of rotational symmetry ( C3 symmetry); and that disclinations appear at the principal axes, where mirror reflection planes and rotational symmetries intersect ( C3v symmetry). In each case it was found that the polarization singularities were at locations predicted by their site symmetry maps. As a consequence of this body of work, we now have the tools to express the general mapping of the local states of polarization from the underlying system symmetries. This follows from our observation that symmetry does not merely constrain global modal transformation properties, but that it equally defines the nature of the mode’s local properties. The resulting prescriptions regarding the fine structure of the electromagnetic field will enable more measured investigations of the nature of light at subwavelength length scales, and are anticipated to play a critical role in maturing current and future nanotechnologies. One avenue of immediate interest is codifying the symmetry relationships between optical vortex arrays (Wheeldon et al. 2007) and the polarization singularities. In certain fields, such as Laguerre-Gaussian laser modes, it has been demonstrated that the light can transfer orbital angular momentum to dielectric particles, causing them to rotate around a vortex structure (Allen et al. 1992). While preliminary investigations have revealed that the optical vortices within two-dimensional photonic crystals are not suited to the rotation of isotropic dielectric particles (Wheeldon 2009), a more comprehensive investigation needs to be conducted. Since it is known that differing states of polarization can rotate such

192


anisotropic particles (Nieminen et al. 2004), a group theoretic understanding of the optical forces is necessary to fundamentally advance technologies of nano-particle manipulation.

References Allen, L., Beijersbergen, M.W., Spreeuw, R.J.C., Woerdman, J.P.: Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes. Phys. Rev. A 45(11), 8185– 8189 (1992) Berry, M.V., Dennis, M.R.: Singularities polarization singularities in isotropic random vector waves. Proc. R. Soc. Lond. A 457, 141–155 (2001) Born, M., Wolf, E.: Principles of optics: electromagnetic theory of propagation, interference and diffraction of light, 7th edn. Cambridge University Press, Cambridge (2003) Evarestov, R.A., Smirnov, V.P.: Site symmetry in crystals theory and applications, Springer series in solid-state sciences 108. Springer, New York (1993) Hahn, T.: International tables for crystallography, Volume A: space-group symmetry, 4th edn. Kluwer, London (1995) Hajnal, J.V.: Singularities in the transverse fields of electromagnetic waves, I: Theory. Proc. R. Soc. Lond. A 414, 433–446 (1987) Jackson, J.D.: Classical electrodynamics, 3rd edn. Wiley, Hoboken, NJ (1999) John, S.: Strong localization of photons in certain disordered dielectric superlattices. Phys. Rev. Lett. 58(23), 2486–2489 (1987) Nieminen, T.A., Parkin, S.J.W., Heckenberg, N.R., Rubinsztein-Dunlop, H.: Optical torque and symmetry. Proc. SPIE 5514, 254–263 (2004) Nye, J.F.: Natural focusing and the fine structure of light: caustics and wave dislocations. IOP, Bristol (1999) Sakoda, K.: Optical properties of photonic crystals, 2nd edn. Springer, Berlin (2005) Tinkham, M.: Group theory and quantum mechanics. Dover Mineola, New York (1992) Wheeldon, J.F.: Group theoretic expressions of optical singularities in photonic crystals. Ph.D thesis, University of Ottawa, Ottawa (2009) Wheeldon, J.F., Schriemer, H.P.: Wyckoff positions and the expression of polarization singularities in photonic crystals. Opt. Express 17, 2111–2121 (2009) Wheeldon, J.F., Hall, T.J., Schriemer, H.P.: Symmetry constraints and the existence of Bloch mode vortices in linear photonic crystals. Opt. Express 15, 3531–3542 (2007) Yablonovitch, E.: Inhibited spontaneous emission in solid-state physics and electronics. Phys. Rev. Lett. 58(20), 2059–2062 (1987)

On the Photonic Dispersion of Periodic Superlattices Made of Left-Handed Materials Solange B. Cavalcanti Instituto de F´ısica, Universidade Federal de Alagoas, Maceió – AL, 57072-970, Brazil, and Inmetro, Campus de Xerém, Duque de Caxias – RJ, 25250-020, Brazil [email protected]

Ernesto Reyes-Gómez Instituto de F´ısica, Universidad de Antioquia, AA 1226, Medell´ın, Colombia [email protected]

Alexys Bruno-Alfonso Faculdade de Ciências, Universidade Estadual Paulista – UNESP, Bauru – SP, 17033-360, Brazil [email protected]

Carlos A. A. de Carvalho Instituto de F´ısica, Universidade Federal do Rio de Janeiro – UFRJ, Rio de Janeiro – RJ, 21945-9702, Brazil, and Inmetro, Campus de Xerém, Duque de Caxias – RJ, 25250-020, Brazil [email protected]

Luiz E. Oliveira Instituto de F´ısica, Universidade Estadual de Campinas – UNICAMP, CP 6165, Campinas – SP, 13083-970, Brazil, and Inmetro, Campus de Xerém, Duque de Caxias – RJ, 25250-020, Brazil [email protected]

Abstract Studies of the band gap properties of one-dimensional superlattices with alternate layers of air and left-handed materials are carried out within the framework of Maxwell’s equations. By left-handed material, we mean a material with dispersive negative electric and magnetic responses. Modeling them by Drude-type responses or by fabricated ones, we characterize the n(ω ) = 0 gap, i.e., the zeroth order gap, which has been predicted and detected. The band structure and analytic equations for the band edges have been obtained in the long wavelength limit in case of periodic, Fibonacci, and Thue–Morse superlattices. Our studies reveal the nature T.J. Hall and S.V. Gaponenko (eds.), Extreme Photonics & Applications, © Springer Science + Business Media B.V. 2010

193

194

S.B. CAVALCANTI ET AL.

of the width of the zeroth order band gap, whose edge equations are defined by null averages of the response functions. Oblique incidence is also investigated, yielding remarkable results. Keywords: One-dimensional stacks, left-handed materials, plasmon polaritons

1 Introduction The propagation of classical waves in periodic media has been investigated as early as 1887 by Lord Rayleigh (1887) in his studies on laminated one-dimensional (1D) structures. With the advent of nanostructured materials which exhibit the so-called photonic band gap (PBG), i.e., a range of forbidden propagation frequencies analogous to electronic bandgaps in periodic crystalline systems, the subject has gained a new thrust (Yariv and Yeh 1984). This renewed interest has led many research groups to devote attention to waveguide arrays, optically induced lattices, and photonic crystals (PCs) because of their potential to affect the properties of light. In addition, super-refractive phenomena, such as the super-prism effect (Kosaka et al. 1998), and tunable band structures (Busch and John 1999; Andre and Lukin 2002) have been used in a variety of optical applications involving PCs (Konoplev et al. 2005a,b). Likewise, 1D PBG heterostructures (Mishra and Satpathy 2003; Longhi and Janner 2004; Murzina et al. 2005; Kleckner et al. 2005) have also been extensively studied, and proposed as basic ingredients in various devices. Another subject that has attracted the attention of researchers worldwide over the last decade is the study of systems with a negative index of refraction (Veselago 1968). It was once considered a mere mathematical curiosity, even though Veselago’s seminal paper predicted unusual properties such as backward wave propagation, reverse refraction, reverse Doppler shift, and reverse Cherenkov radiation in the propagation of electromagnetic radiation through media with negative refraction. Recently, however, theoretical and experimental developments resulted in the construction of micro-structured materials that exhibit negative refraction. Indeed, the equally unusual property of focusing predicted for such materials has in fact been observed in photonic crystals (Parimi et al. 2003; Cubuku et al. 2003). A remarkable example of materials with negative refraction is given by the so-called metamaterials. They exhibit negative dispersive electric and magnetic responses, and are also known as left-handed materials (LHMs). This is because the electromagnetic waves that propagate in LHMs have their electric E and magnetic B fields related to the direction of the propagation vector by means of the left-hand rule. Thus, the phase velocity vector is opposite to the Poyinting vector. LHMs have not only opened a new era for optical devices, but also given considerable thrust to plasmonics. This is an area which studies the coupling of plasmons with light fields, i.e., electron density waves excited by resonant interactions between optical fields and mobile electrons at a metal surface. In metal-dielectric

ON THE PHOTONIC DISPERSION OF PERIODIC SUPERLATTICES

195

interfaces, resonant surface plasmons have much shorter wavelengths than that of the radiation, which enables them to propagate along nanoscale wires (Barnes et al. 2003; Maier and Atwater 2005; Ozbay 2006). This feature could lead to increases in the resolution of microscopes, in the efficiency of LEDs, and in the sensitivity of chemical and biological devices, among other applications. Lately, materials with ε near zero have also received considerable attention, as they permit squeezing and tunnelling of radiation through them (Edwards et al. 2008). In fact, the interaction between plasmon excitations and radiation in the frequency region around null electric and magnetic responses, a requisite for the existence of longitudinal waves, results in the excitation of modes that couple plasmons and optical fields. Those are known as plasmon polaritons and have been described in hexagonal lattices. By applying a magnetic field, the frequency range where the coupled mode appear will change with the cyclotron frequency, so that there is the possibility of creating tunable plasmon polaritons (Duque et al. 2009). In the present study, we are concerned with the photonic band properties of light propagating through 1D superlattices composed of alternating slabs (cf. Fig. 1), of widths a and b, of two materials, one with a positive and constant refractive index (nA > 0), and the other with a negative dispersive refractive index (i.e., nB (ω ) may have negative values). Theoretical and experimental results for normal incident light in such systems (Fredkin and Ron 2002; Wu et al. 2003a,b; Bria et al. 2004; Wang et al. 2004) have evidenced the existence of a non-Bragg photonic bandgap also known as a zeroth order gap (Li et al. 2003; Jiang et al. 2003; Shadrivov et al. 2004; Daninthe et al. 2006). It opens at a frequency that satisfies the n(ω ) = 0 equation. That will happen even if one changes the lattice parameter at will, as long as the refractive indices average is kept null. In the sequel, we present the results of our investigations. In Section 2, normal incidence dispersion relations are calculated by transfer matrix methods for periodic systems, borrowed from solid state theory. Section 3 is devoted to the study of the band structure properties of Fibonacci and Thue–Morse superlattices. In Section 4, we investigate the influence of an additional parallel component of the wave vector

Fig. 1 (Color online) Schematic view of a multilayer structure with slabs A and B in periodic arrangement. The TE-like and TM-like electromagnetic modes are also indicated.

196


in periodic superlattices for both TE and TM polarizations. Finally, in Section 5, we present our conclusions.

2 Periodic systems Let us begin by considering incident light, either s polarized (or TE, E-field parallel to the interface, see Fig. 1) or p polarized (or TM, H-field parallel to the interface), on the 1D superlattice depicted in Fig. 1 (d = a + b). We choose the origin located at the center of a first slab (with dielectric constant εA and magnetic permeability μA ) of width a, whereas b is the slab width of the LHM, whose dielectric constant εB and magnetic permeability μB , for a lossless medium, are best described by (Li et al. 2003; Jiang et al. 2003; Shadrivov et al. 2004; Daninthe et al. 2006; Pacheco et al. 2002; Eleftheriades et al. 2002; Grbic and Eleftheriades 2002; Liu et al. 2002):

ωe2 ωm2 ; μ ( ω ) = μ − , (1) B ω2 ω2 respectively. In Fig. 2, we illustrate the behavior of both responses by plotting the ratio between the dispersion laws of materials A and B according to Eq. (1). Let us now define an average index such as, εB (ω ) = ε −

n(ω ) =

1 d

n(z, ω )dz =

anA + bnB(ω ) , d

(2)

where nA and nB are the refractive indices of layers A and B, respectively (n(z) = ε (z) μ (z) is the z-position dependent refractive index). For the particular case of zero average, one finds that anA − b|nB(ω0 )| =0 d

(3)

meaning that light propagating with a particular frequency ω0 satisfying Eq. (3) feels in average, a null index of refraction. We shall be interested in studying the properties of both the transverse-electric (TE) and transverse-magnetic (TM) polarizations of a monochromatic electromagnetic field of frequency ω propagating through the 1D periodic system. For the TE case (the electric field perpendicular to the incidence plane), one may write E(r,t) = E(z) exp [i (qx − ω t)] ey ,

(4)

Similarly, one may write the magnetic field for the TM-polarization case (the electric field parallel to the incidence plane) as H(r,t) = H(z) exp [i (qx − ω t)] ey .

(5)


197

a -

b -

Fig. 2 (Color online) The ratio between electric permitivities, magnetic permeabilities and indices of refraction of materials A and B for (a) the dispersion law as proposed by Li et al. (2003) and (b) the Drude-like response as used by several investigators (Li et al. 2003; Jiang et al. 2003; Shadrivov et al. 2004; Daninthe et al. 2006; Pacheco et al. 2002; Eleftheriades et al. 2002; Grbic and Eleftheriades 2002; Liu et al. 2002).

In both expressions, we have assumed that the superlattice was grown along the z axis, q is the wavevector component along the x direction, and ey denotes the cartesian unitary vector along the y direction. The electric and magnetic amplitudes satisfy the differential equations d ω 2 1 d q2 E(z) = −ε (z) E(z) (6) − 2 dz μ (z) dz c n (z) and

respectively.

d ω 2 1 d q2 H(z) = −μ (z) H(z), − 2 dz ε (z) dz c n (z)

(7)

198


Equations (6) and (7) may be solved by means of the transfer-matrix technique (Cavalcanti et al. 2006, 2007). According to this procedure, the TE-modes dispersion relation may be obtained from the solution of the transcendental equation cos(kd) = PS + KR,

(8)

where k is the Bloch wave vector along the z direction (Cavalcanti et al. 2007). Note that, because of (1), the refraction index in layer B (nB ) is a function of the frequency, and may be a real positive, real negative or a pure imaginary number. Also, note that the q wavevector component along the x direction may be obtained as a function of the angle of incidence θ ≡ θA by q = nA ωc sin θ . If nB is a real number and n2B − n2A sin2 θ ≥ 0, then a Q μ b b a A B − , P = cos QB sin QB cos QA sin QA 2 2 QB μ A 2 2 a Q μ a b b B A − , S = cos QB sin QB cos QA sin QA 2 2 QA μ B 2 2 a μA b a μB b K=+ + , cos QB sin QB sin QA cos QA QA 2 2 QB 2 2 a Q a QB b b A − , R=− sin QB cos QB cos QA sin QA μB 2 2 μA 2 2 (9)

and

ω QA = nA | cos θ | , (10) c

ω QB = n2B − n2A sin2 θ . (11) c Note that functions P, S, K, and R depend on the frequency ω and incidence angle. On the other hand, if nB is real and n2B − n2A sin2 θ < 0, then a Q μ b b a A B − , P = cosh QB sinh QB cos QA sin QA 2 2 QB μ A 2 2 a Q μ a b b B A S = cosh QB + , sinh QB cos QA sin QA 2 2 QA μ B 2 2 a μA b a μB b + , K = cosh QB sinh QB sin QA cos QA QA 2 2 QB 2 2 a Q a QB b b A R= − , sinh QB cosh QB cos QA sin QA μB 2 2 μA 2 2 (12)

where QA is given by (10) and QB =

ω c

n2A sin2 θ − n2B.

(13)


199

Moreover, if n2B < 0, then the functions P, S, K, and R are given by (12), QA is given by (10), and

ω QB = n2BI − n2A sin2 θ , (14) c where nBI is the imaginary part of nB . For the case of TM polarization, the transcendental equations (8) and (12), as well as the definitions of QA and QB for the different cases, are also valid provided that one replaces μA by εA and μB by εB . In this section and the next we limit ourselves to the case of normal incidence. In Section 4 we shall analyse the case of oblique incidence. For normal incidence, we find that a gap opens up around the frequency that cancels the average effect of the refractive index that light sees. The optical wavelengths in media A and B have the same modulus but opposite sign, so that all the phase acquired in medium A is lost in medium B. In this way there is no light propagation. In contrast with Bragg gaps, the n(ω ) = 0 gap is insensitive to the size of the unit cell (Li et al. 2003), as it is depicted in Fig. 3 where the curves around the n(ω ) = 0 gap are plotted. It is clear from these results that the gap is a function of the ratio ab as we show next. Upon using the macroscopic response functions ε and μ , one must have in mind that the size of the unit cell of the superlattice must be small compared with the 2.0

a a/b = 1

1.5

ν0

frequency (GHz)

1.0

0.5 -0.10 -0.05 0.00 1.5

0.05

0.10

b

a/b = 2

ν0

1.0

0.5 -0.06 -0.03 0.00

0.03

0.06

k (in units of 2π/d) Fig. 3 Photonic dispersion displaying the invariance of the n(ω ) = 0 non-Bragg gap. Solid, dashed, dotted, and dashed-dotted curves correspond to a = 12 mm, 14 mm, 16 mm, and 18 mm, respectively (results after Cavalcanti et al. (2007)).

200


wavelength of the incident radiation, i.e., λi = |n2πi |ωc >> a, b. Assuming that a and b are in the range of 1−20 mm, one may therefore show (Cavalcanti et al. 2007) that, at the k = 0 BZ center, the ν− frequency, corresponding to the even solution associated with the top of the lower band around the n(ω ) = 0 gap, illustrated in Fig. 3, is obtained through the solution of

εA b =− , εB (ω ) a

(15)

which is equivalent to ε (ω ) = 0, whereas the ν+ frequency, corresponding to the odd solution associated with the bottom of the higher band around the n(ω ) = 0 gap in Fig. 3, may be derived from

μA b =− μB (ω ) a

(16)

which is equivalent to μ (ω ) = 0. Notice that, as it happened with the ω0 frequency associated with the n(ω ) = 0 gap, both ν− and ν+ are only dependent on the a/b ratio characterizing the robustness of the n(ω ) = 0 gap. In that respect, Fig. 4 displays the gap profile as a function of the relative layer width, i.e., the a/b–dependence of ν0 , and of the ν+ (ν− ) frequencies corresponding to the top (bottom) of the first (second) photonic band displayed in Fig. 3. So, in this section, we have characterized the n(ω ) = 0 gap for normal incident light. We have provided analytical expressions for the band edges of this gap, which are shown to agree with the averages of both response functions and depend solely on the a/b ratio.

gap profile (GHz)

2.0

ν0

1.5

ν+

1.0

ν-

0.5 0

1

2

3

4

a/b Fig. 4 Frequency profile of the n = 0 non-Bragg gap with the top (ν− ) or bottom (ν+ ) frequency of the first or second photonic bands of the previous figure, and the ν0 frequency at which n(ω ) = 0 (results after Cavalcanti et al. (2007)).


201

3 Quasiperiodic sequences Let us now, for normal incidence, extend these investigations on the structure of the n(ω ) = 0 gap to a photonic 1D periodic heterostructure where the unit cell is a Fibonacci or a Thue–Morse sequence of layers A and B. Within the transfer-matrix approach illustrated in the previous section, one may write the photonic dispersion as (Bruno-Alfonso et al. 2008a,b) cos(kL) = R(ω ),

(17)

where R(ω ) is the semitrace of the corresponding transfer matrix M(ω ). Moreover, the allowed (forbidden) frequencies are those satisfying |R(ω )| ≤ 1 (|R(ω )| > 1). The unit cells of the Thue–Morse and Fibonacci lattices consist of layers A and B, arranged according to chains of specific lengths. Those chains are labelled by an integer m (they are denoted as Sm ) and obey simple recurrence relations. In both cases, it is straightforward to obtain the number of layers A and B within Sm , which are denoted as NA,m and NB,m , respectively. The length of the unit cell Sm is Lm = NA,m a + NB,m b and the optical path along this chain is δm = NA,m a nA + NB,m b nB . Hence, the condition for constructive interference of reflected waves is given by δm = Nωπ c where N = 0 for the zeroth order gap, and N = 1, 2, 3, ... for Bragg gaps. The average refractive index is defined by n(ω )m = Lδmm and the n(ω ) = 0 gap occurs around a frequency satisfying the condition n(ω )m = 0, i.e, b |nB | NA,m = . a nA NB,m

(18)

For the Thue–Morse chain of mth order, the zeroth order gap occurs around a freB| quency for which the ratio of the optical paths in media A and B is given by b|n anA = 1. This result is independent of the Thue–Morse order m, and coincides with the wellknown case of a periodic lattice based on the unit cell AB, i.e., S1 . In the limit of high Fibonacci order, the zeroth order gap occurs (Bruno-Alfonso et al. 2008a,b) around a frequency for which the ratio of the optical paths are in the golden ratio, B| i.e, b|n anA = τ . The average permittivity and the average permeability are given by ε m = (NA,m a εA + NB,m b εB )/Lm and μ m = NA,m a μA + NB,m b μB /Lm , respectively. The edge frequency corresponding to zero ε is given by (Bruno-Alfonso et al. 2008a,b)

and that for zero μ obeys

b|εB | NA,m = , a εA NB,m

(19)

b|μB | NA,m = . a μA NB,m

(20)

In the Thue–Morse case, one has NA,m /NB,m = 1. For Fibonacci lattices, NA,m /NB,m = Fm /Fm−1 = τm−1 , in which the Fibonacci number Fm is given by

202


b

4 3.5

Frequency GHz

Frequency GHz

a

3 2.5 2 1.5 1

0

2

4 6 Thue Morse order

4 3.5 3 2.5 2 1.5 1

8

2

4 6 8 Fibonacci order

10

Fig. 5 (Color online) Photonic spectra in Thue–Morse (a) and Fibonacci (b) photonic superlattices (a = 4 mm and b = 2 mm) as functions of the chain order. Photonic bands are depicted as verticalline segments of different widths (results after Bruno-Alfonso et al. (2008b)).

a

b 3.0

3.0

= 0

2.8

ν+

Frequency (GHz)

2.6 2.4

n>

= 0 2.0

ν−

1.8

= 0

1.6

8

1.4

2

4

6

8

10

12

14

Fibonacci order

Fig. 6 (Color online) The n = 0 gap profiles of Thue–Morse (a) and Fibonacci (b) photonic superlattices as functions of the chain order, with a/b = 2. Crosses correspond to the long-wavelength limit. Results are after Bruno-Alfonso et al. (2008ab).

Fm = Fm−2 + Fm−1 , with F1 = F2 = 1 and the limiting quotient of sucessive numbers in the case of large m is the τ golden mean (Bruno-Alfonso et al. 2008a,b). In Figs. 5 and 6, the photonic spectra and gap profiles of Thue–Morse and Fibonacci superlattices are plotted as functions of the chain order, illustrating the limiting behavior of the mth order chains. Figure 7 shows the gap profiles of a Fibonacci structure as functions of the ratio a/b. It is important to note that the n = 0 frequency is within the considered gap. This frequency is denoted as ν0m , and satisfies the relation nB /nA = −τm−1 a/b. The ratio nB /nA , as displayed in Fig. 2, is the geometric mean of the ratios εB /εA and μB /μA . Therefore, it is clear that ν0m is between νεm and νμm . In this section, we have found similar results as the previous ones, with the overall conclusion that, as long as one finds a way of combining a balanced average of positive and negative responses along suitable chosen optical paths, one is doomed


203

a =

= =

b =

= = =

Fig. 7 (Color online) The n = 0 gap profile of the Fibonacci photonic lattice as a function of the ratio a/b, with a = 4 mm. The Fibonacci order m is 2 and 11 in panels (a) and (b), respectively. The solid lines (dots) correspond to the conditions ε m = 0 and μ m = 0 (numerical results for the gap edges). Results are after Bruno-Alfonso et al. (2008a).

to find a robust gap. In simple periodic structures this is achieved by combining equal optical paths. In Fibonacci sequences one must combine the optical paths in the irrational golden ratio, i.e., the optical paths in media A and B must be incommensurate,

4 Oblique incidence In this section we study the photonic band properties when light is obliquely incident. Figure 8 shows a TE dispersion curve, and one notices that the second photonic band for θ = 0, just above the zeroth order gap, is a pure photonic mode and, for

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Fig. 8 (Color online) TE dispersion ν = ν (k) in photonic periodic superlattices. Calculations were performed for εA = 1, μA = 1, a = b = 12 mm, and ωe /2π = ωm /2π = 3 GHz for the Drude model (with ε = 1.21, μ = 1, see Eq. (1)) in the second slab. Solid and dashed lines correspond to an incidence angle θ = π /12 and θ = 0, respectively.

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Fig. 9 (Color online) As in Fig. 8, but for ωe /2π = 2 GHz and ωm /2π = 2.5 GHz.

θ = π /12, there is a coupling between the radiation field and the plasmon mode (note, in Fig. 8, that the dotted line indicates, for θ = 0, the pure magnetic plas√ mon mode at νm = ωm /2π μ ) which leads to a pair of coupled plasmon-polariton modes. Let us now turn to Fig. 9, where the dispersions are depicted for TE (a) and TM modes (b), for layer widths a = b = 12 mm. Comparing the results for normal and oblique incidence, it is clear that, for θ = π /12, resonant plasmon-polariton waves are excited by the coupling of the electric (or magnetic) plasmon modes with the incident eletromagnetic field. Figs. 8 and 9 illustrate that, for θ = 0, coupled plasmon-polariton modes show up. One might notice that, in the TE case (Fig. 9 a), the plasmon-polariton waves


a 10

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Fig. 10 (Color online) TE and TM dispersions in a photonic periodic superlattice with a = b = 12 mm. Numerical calculations were carried out for εA = 1 and μA = 1, and in the second slab we use the Drude model with ωe /2π √= 1.5 GHz and ωm /2π = 2.5 GHz. Note that, in (b), the electric plasmon frequency νe = ωe /2π ε is in the gap region, leading to a flat plasmon-polariton band in the case of TM polarization.

correspond to coupling of light with magnetic plasmons whereas, in the TM case (Fig. 9 b), the plasmon-polariton modes result from the coupling of light with electric plasmons and are driven by the electric field. Note that, in Fig. 8, as we have chosen ωe /2π = ωm /2π = 3 GHz for the Drude model (with ε = 1.21, μ = 1, see Eq. (1)) in the second slab, the lower coupled plasmon-polariton mode is asymptotic to the pure θ = 0 magnetic plasmon mode at νm . In Fig. 9, however, where ωe /2π = 2 GHz and ωm /2π = 2.5 GHz are different, it is apparent that the plasmonpolariton mode is driven by the magnetic field, in the case of TE polarization, since this mode is asymptotic to the θ = 0 magnetic plasmon mode at νm , whereas the plasmon-polariton mode in the TM-polarization case is driven by the electric field and exhibits a coupled mode asymptotic to the θ = 0 electric plasmon mode at νe . Finally, Fig. 10 illustrates that, by choosing the resonant electric plasmon frequency within the photonic zeroth order bandgap, the coupled electric plasmon-polariton mode (Fig. 10 (b)) essentially gives way to a pure electric plasmon mode.

5 Conclusions Summing up, we have studied the band gap properties of 1D superlattices with alternate layers of air and a LHM within the framework of Maxwell’s equations and the transfer-matrix approach. By modeling the LHM by Drude-type responses or by fabricated ones, we characterize the n(ω ) = 0 gap, i.e., the zeroth order gap, which has been predicted and detected. The band structure and analytic equations for the band edges have been obtained in the long wavelength limit in case

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of periodic, Fibonacci, and Thue–Morse heterostructures. The present work clearly reveal the nature of the width of the zeroth order band gap, whose edge equations are defined by null averages of the response functions. The case of oblique incidence is also investigated, yielding remarkable results. The photonic dispersion indicates that, in the frequency region around ε (ω ) = 0 and μ (ω ) = 0, coupled plasmonpolariton modes show up for both TE and TM cases. The coupling of light with plasmons is weakened by choosing the plasma frequency inside the zeroth order gap. As light propagation is forbidden in that particular gap-frequency region, the coupled mode becomes essentially a pure plasmon mode. Finally, we note that we have studied an ideal system in which losses have been neglected. Work to include the effects of losses is in progress. Acknowledgements The authors would like to thank the Brazilian Agencies CNPq, FAPESP, FAPERJ, as well as the Colombian CODI - Univ. of Antioquia and COLCIENCIAS, for partial financial support.

References Andre, A., Lukin, M.D.: Manipulating light pulses via dynamically controlled photonic band gap. Phys. Rev. Lett. 89(14), 143602 (2002). Barnes, W.L., Dereux, A., Ebbesen, T.W.: Surface plasmon subwavelength optics. Nature 424(6950), 824 (2003). Bria, D., Djafari-Rouhani, B., Akjouj, A., Dobrzynski, L., Vigneron, J.P., El Boudouti, E.H., Nougaoui, A.: Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials. Phys. Rev. E 69(6), 066613 (2004). Bruno-Alfonso, A., Reyes-Gómez, E., Cavalcanti, S.B., Oliveira, L.E.: Band edge states of the n = 0 gap of Fibonacci photonic lattices. Phys. Rev. A 78(3), 035801 (2008a). Bruno-Alfonso, A., Reyes-Gómez, E., Cavalcanti, S.B., Oliveira, L.E.: Band-edge states of the zeroth-order gap in quasi-periodic photonic superlattices. In Tománek, P., Senderáková, D., Hrabovský, M. (eds.) Photonics, Devices, and Systems IV. Proc. of SPIE Vol. 7138, 71381A (2008b). Busch, K., John, S.: Liquid-crystal photonic-band-gap materials: The tunable electromagnetic vacuum. Phys. Rev. Lett. 83(5), 967 (1999). Cavalcanti, S.B., Dios-Leyva, M., Reyes-Gómez, E., Oliveira, L.E.: Band structure and band-gap control in photonic superlattices. Phys. Rev. B 74(15), 153102 (2006). Cavalcanti, S.B., Dios-Leyva, M., Reyes-Gómez, E., Oliveira, L.E.: Photonic band structure and symmetry properties of electromagnetic modes in photonic crystals. Phys. Rev. E 75(2), 026607 (2007), and references therein. Cubuku, E., Aydin, K., Ozbay, E., Foteinopolou, S., Soukoulis, C.M.: Subwavelength resolution in a two-dimensional photonic-crystal-based superlens. Phys. Rev. Lett. 91(20), 207401 (2003). Daninthe, H., Foteinopoulou, S., Soukoulis, C.M.: Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials. Phot. Nanostruct. Fund. Appl. 4(3), 123 (2006). Duque, C.A., Porras-Montenegro, N., Cavalcanti, S.B., Oliveira, L.E.: Photonic band structure evolution of a honeycomb lattice in the presence of an external magnetic field. J. Appl. Phys. 105(3), 034303 (2009). Edwards, B., Alù, A., Young, M.E., Silveirinha, M., Engheta, N.: Multifrequency optical invisibility cloak with layered plasmonic shells. Phys. Rev. Lett. 100(11), 033903 (2008).


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Eleftheriades, G.V., Iyer, A.K., Kremer, P.C.: Planar negative refractive index media using periodically L-C loaded transmission lines. IEEE Trans. Microwave Theory Tech. 50(12), 2702 (2002). Fredkin, D.R., Ron, A.: Effectively left-handed (negative index) composite material. Appl. Phys. Lett. 81(10), 1753 (2002). Grbic, A., Eleftheriades, G.V.: Experimental verification of backward-wave radiation from a negative refractive index metamaterial. J. Appl. Phys. 92(10), 5930 (2002). Jiang, H., Chen, H., Li, H., Zhang, Y., Zhu, S.: Omnidirectional gap and defect mode of onedimensional photonic crystals containing negative-index materials. Appl. Phys. Lett. 83(26), 5386 (2003). Kleckner, T.D., Modotto, D., Locatelli, A., Mondia, J.P., Linden, S., Morandotti, R., De Angelis, C., Stanley, C.R., van Driel, H.M., Aitchison, J.S.: Design, fabrication, and characterization of deep-etched waveguide gratings. J. Light. Tech. 23(12) , 3832 (2005). Konoplev, I.V., McGrane, P., Cross, A.W., Ronald, K., Phelps, A.D.R.: Wave interference and band gap control in multiconductor one-dimensional Bragg structures. J. of Appl. Phys. 97(7), 073101 (2005a). Konoplev, I.V., McGrane, P., Phelps, A.D.R., Cross, A.W., Ronald, K.: Observation of photonic band-gap control in one-dimensional Bragg structures. Appl. Phys. Lett. 87(12), 121104 (2005b), and references therein. Kosaka, H., Kawashima, T., Tomita, A., Notomi, N., Tamamura, T., Sato, T., Kawakami, S.: Superprism phenomena in photonic crystals. Phys. Rev. B 58(16), 10096 (1998). Li, J., Zhou, L., Chan, C.T., Sheng, P.: Photonic band gap from a stack of positive and negative index materials. Phys. Rev. Lett. 90(8), 083901 (2003). Liu, L., Caloz, C., Chang, C.C., Itoh, T.: Forward coupling phenomena between artificial lefthanded transmission lines. J. Appl. Phys. 92(9), 5560 (2002). Longhi, S., Janner, D.: Diffraction and localization in low-dimensional photonic bandgaps. Optics Lett. 29(22), 2653 (2004). Lord Rayleigh: On the maintenance of vibrations by forces of double frequency and on the propagation of waves through a medium endowed with a periodic structure. Phil. Mag. XXIV, 145 (1887). Maier, S.A., Atwater, H.A.: Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures. J. Appl. Phys. 98(1), 011101 (2005). Mishra, S., Satpathy, S.: One-dimensional photonic crystal: The Kronig-Penney model. Phys. Rev. B 68(4), 045121 (2003). Murzina, T.V., Sychev, F.Yu., Kim, E.M., Rau, E.I., Obydena, S.S., Aktsipetrov, O.A., Bader, M.A., Marowsky, G.: One-dimensional photonic crystals based on porous n-type silicon. J. Appl. Phys. 98(12), 123702 (2005). Ozbay, E.: Plasmonics: Merging photonics and electronics at nanoscale dimensions. Science 311(5758), 189 (2006). Pacheco, J.Jr., Grzegorczyk, T.M., Wu, B.-I., Zhang, Y., Kong, J.A.: Power propagation in homogeneous isotropic frequency-dispersive left-handed media. Phys. Rev. Lett. 89(25), 257402 (2002). Parimi, P.V., Lu, W.T., Vodo, P., Sridhar, S.: Photonic crystals - Imaging by flat lens using negative refraction. Nature 426(6965), 404 (2003). Shadrivov, I.V., Zharova, N.A., Kivshar, Y.S.: Defect modes and transmission properties of lefthanded bandgap structures. Phys. Rev. E 70(4), 046615 (2004). Veselago, V.G.: The electrodynamics of substances with simultaneously negative values of ε and μ . Sov. Phys.-Usp. 10(4), 509 (1968). Wang, L., Chen, H., Zhu, S.-Y.: Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials. Phys. Rev. B 70(24), 245102 (2004). Wu, L., He, S., Chen, L.: On unusual narrow transmission bands for a multi-layered periodic structure containing left-handed materials. Opt. Exp. 11(11), 1283 (2003a). Wu, L., He, S., Shen, L.: Band structure for a one-dimensional photonic crystal containing lefthanded materials. Phys. Rev. B 67(23), 235103 (2003b). Yariv, A., and Yeh, P.: Optical waves in crystals. John Wiley & Sons, New York (1984).

Slow Light Propagation and Disorder-Induced Localization in Photonic Crystal Waveguides Mark Patterson, Stephen Hughes Department of Physics, Queen’s University, Kingston, ON K7L 3N6, Canada [email protected]

Sylvain Combrié, Nguyen-Vi-Quynh Tran, and Alfredo De Rossi Thales Research and Technology, Route Départementale 128, 91767 Palaiseau CEDEX, France [email protected]

Renaud Gabet, Yves Jaouën Telecom ParisTech, 46 Rue Barrault, 75634 Paris CEDEX 13, France

Abstract We investigate the phenomenon of slow-light propagation in planar photonic crystal waveguides and present a theoretical formalism and matching experiments to describe disorder-induced coherent scattering. The theory uses a coupled-mode approach to track multiple forward and backward reflections and a rigorous Green function technique to introduce the effect of disorder. Simulations based on this theory and experimental measurements for high-quality GaAs photonic crystal membranes are compared through transmission measurements and frequency-delay reflectometry. The excellent qualitative agreement between theory and experiment provides clear physical insight into naturally occurring light localization and multiple coherent-scattering phenomena in slow-light waveguides. Further, we briefly connect with related phenomena of absorption-induced losses in slow-light metamaterial waveguides. Keywords: Photonic crystal waveguides, slow light, disorder-induced scattering, light localization, nanophotonics

1 Introduction Photonic crystals (PCs) are periodic dielectric structures that can strongly alter the propagation of light due to the interaction of coherent reflections from the constituent periodic features. One property of PCs is that they can exhibit a photonic bandgap: a range of frequencies where light cannot propagate due to destructive T.J. Hall and S.V. Gaponenko (eds.), Extreme Photonics & Applications, c Springer Science + Business Media B.V. 2010

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interference between the reflection paths. This process is analogous to the formation of an electronic bandgap in the periodic lattice potential of a semiconductor. By exploiting photonic bandgaps to confine light in complex structures, a number of interesting phenomena, such as light trapping on sub-wavelength spatial dimensions in high-quality cavities (Akahane et al. 2003), or engineered waveguide band dispersions with a vanishing group velocity (Notomi et al. 2001; Vlasov et al. 2005; Baba 2008), can be observed. Both of these effects give rise to novel regimes of enhanced light-matter interaction and have broad applications in nano- and quantumtechnologies. A particularly promising platform for PC technologies are semiconductor-based planar PCs, due to their ease of fabrication using standard etching and lithography techniques. A planar PC can be constructed, for example, by etching an array of circular holes in a membrane of high-refractive-index semiconductor. A waveguide can be created by omitting a role of holes as shown in Fig. 1. The waveguide dispersion is shown in Fig. 2 with the solid curve indicating the mode of interest. The shaded region in the left of the plot indicated the continuum of radiation modes which do not satisfy the criteria for total internal reflection; the waveguide modes above the

Frequency [THz]

Fig. 1 Schematic of a section of a planar PC waveguide showing a pulse and its slowly-varying envelope.

230 220 210 200 190 0

0.25 Wavevector, k [2π /a]

0.5

Fig. 2 The band structure for the waveguide in Fig. 1 showing the dispersion relations for the two lowest-order waveguide modes. The lowest-order (solid) mode is used for the calculations. The grey region at the left represents the continuum of radiation modes above the light line.

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light line are poorly confined and will quickly leak out of the slab. The propagation mode has the peculiar property that the group velocity vg , equal to the slope of the dispersion curve dω /dk, tends to zero at the band edge, leading to slow-light propagation. Below the light line, the waveguide mode is confined by the photonic bandgap to the sides and total internal reflection above and below, and will, ideally, propagate indefinitely without loss. In practice, scattering losses are observed that are attributed to unavoidable structural imperfections, collectively termed “disorder.” Indeed, it is now well established that slow-light PC waveguides suffer from significant losses attributed to scattering at disordered surfaces and other device imperfections (Kuramochi et al. 2005; Parini et al. 2008). Rigorous modeling of this generally undesired extrinsic scattering phenomena is essential for understanding the underlying physics of measurements and for eventually producing functional devices. However, since the spatial scale of a disordered hole interface is only around 3 nm or less (for approximately 200 nm holes), and because the disorder varies rapidly across the highindex-contrast interface (Skorobogatiy et al. 2005), the theoretical description of light scattering in these intriguing periodic media presents enormous challenges. Previously, Hughes et al. (2005) introduced a PC waveguide model for incoherent extrinsic scattering in the slow-light regime. With vg the group velocity, they predicted that the ensemble average backscattering loss, for weak perturbative disorder, scales as 1/v2g and dominates over scattering into radiation loss modes (1/vg scaling). Similar scalings were also implied by Povinelli et al. (2004) and by Gerace and Andreani (Gerace and Andreani 2004), though the precise details depend on the nature of the propagating modes and how they sample the disorder regions. These general approximate loss-scaling relations have now been confirmed experimentally by a number of groups, e.g. (Kuramochi et al. 2005; O’Faolain et al. 2007; Engelen et al. 2008), except that they naturally break down at extremely low group velocities where using the ideal band structure is no longer a good approximation (Pedersen et al. 2008). Recent experimental measurements have reported interesting features such as narrow-band resonances near the band edge (Topolancik et al. 2007; Parini et al. 2008) that are not explained at all by the incoherent scattering theory, and simple vg scaling rules do not make sense. Also the previous loss formulas employ an expectation-value calculation and, as such, they should be compared with an ensemble average of loss measured on a set of nominally identical structures and omit zero-mean features visible in individual samples. It is vital to address behaviors such as multiple coherent reflections between disorder sites since such effects are now being clearly observed in experiments. In this work, we present a theory of coherent optical scattering loss and model individual, fully 3D, disordered waveguides instead of ensemble averages. Specifically, we use a coupled-mode formalism to track bidirectional propagation in the waveguide and introduce the disorder with a rigorous Green function formalism. The theoretical results are also presented along side measurements on stateof-the-art GaAs PC waveguides which are probed with transmission measurements (Combrié et al. 2006) and optical low-coherence reflectometry (OLCR) (Combrié et al. 2007). OLCR allows the measurement of the back-reflected signal

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as a function of propagation time inside the PC waveguide. The experimental setup is basically a Michelson interferometer illuminated with a broadband, partially coherent source. The waveguide is placed in one arm and a translating mirror in the other. The backreflected signal returning from the sample with an accumulated time delay interferes constructively only with the signal returning from the reference arm with a similar time delay, because the source has limited temporal coherence. Thus, the amplitude of the interference fringes is proportional to the signal backreflected within a localized spatial region inside the sample, which depends on the position of the reference arm. Properties such as group velocity and complex frequency-dependent reflectance can be extracted from the interference pattern. Here, we also present a powerful and recently developed analytic technique: time–frequency reflectance maps (TFRMs) (Parini et al. 2008), that can be used to visualize the frequency-dependent impulse response and reveal a number of interesting features such as disorder-induced scattering, facet reflections, and long-time trapping of the light in the structure. Our formalism also provides fresh insights into the long standing question of light localization in PC waveguides. It was proposed by John (1984) and Anderson (1985), that strong localization may be observable in PCs. Strong localization, or Anderson localization, was originally proposed in the context of electron propagation in a disordered atomic lattice, however it is a general wave phenomenon that applies equally to light. Strong localization occurs when the mean free path l, of a propagating Bloch mode is reduced to the order of the Bloch wave vector k, totally disrupting propagation; formally kl < 1 (John 1987). Wiersma et al. (1997) reported optical Anderson localization in a random powder in 1997 however it can be argued that this is not pure Anderson localization since it is not a disordered periodic media with Bloch modes that become localized. Vlasov et al. (1999) observed the thickness dependance of transmission through a disordered face-centred-cubic lattice of silica spheres near the band edge and argued that their observed exponential decrease in transmission was due to the interplay of Bragg scattering and incoherent scattering. There have been a number experiments (Schwartz et al. 2007; Lahini et al. 2008) that showed strong localization in periodic structures transverse to the direction of propagation. This is an easier experimental feat since the transverse component of the wave vector can be significantly smaller than the wave vector magnitude. In this discussion, we are particularly interested in periodic (quasi-) one dimensional structures. Mookherjea et al. (2008) have reported on localization effects in a coupled resonator strip waveguide. They showed that band edge states where the group velocity ideally goes to zero are particularly susceptible to disorder which causes them to be localized on the order of a few periods. Recently, Topolancik et al. reported the observation of resonances at the band edge of an artificially roughened PC waveguide and argued these were due to localization (Topolancik et al. 2007). In contrast, here we report measurements on structures with only unavoidable disorder and provide deeper insight into localization phenomena and the correct criteria for strong localization. Our non-perturbative theory using a fully solvable 3D model with no fit parameters, presented along side several key experiments, brings to the


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fore an unparalleled and undisputed view into the rich and much misunderstood role of disorder-induced scattering in PC waveguides. The basic foundations of this work have recently been published in Physical Review Letters (Patterson et al. 2009).

2 Theory Since the waveguide is a quasi one-dimensional periodic structure, we can exploit Bloch’s theorm to write the ideal waveguide mode electric field as the product of a period function and a forward-propagating plane wave phase Ew (r; k) = eikx ek (r) + c.c., where k is the wave vector which implicitly depends on the the angular frequency ω , the Bloch-mode electric field ek (r) is normalized by cell dr εi (r) |ek (r)|2 = 1 and c.c. denotes the complex conjugate. The waveguide dispersion ω (k) and electric field profiles are determined through finite-difference time-domain (FDTD) (Taflove and Hagness 2005) calculations and frequency-domain methods (Johnson and Joannopoulos 2001). The dispersion curve for the PC waveguide mode is shown in Fig. 2 with the mode of interest marked with a solid line. We begin with an ideal structure (no disorder), described through the spatiallydependent dielectric constant εi (r), as schematically illustrated in Fig. 1. If the electric field of the ideal waveguide mode Ei (r; ω ), is incident on a disordered waveguide described by ε (r) = εi (r) + Δ ε (r), the exact electric field has a well known analytical form: E(r; ω ) = Ei (r; ω ) +

← → dr G (r, r ; ω ) · [E(r ; ω ) Δ ε (r )],

(1)

← → where G (r, r ; ω ) is the Green function for the ideal structure. The Green function is obtained from a polarization-dipole solution to Maxwell’s electrodynamics equations and is a sum over all the modes at a given frequency. It is inefficient to calculate the exact Green function for complicated structures. Instead, the Green function can be decomposed into a superposition of an exact bound mode contribution which dominates in the slow-light regime and an approximation of the ← → summed background radiation modes G rad (r, r ; ω ). The bound mode contribution is calculated as ← → iaω H(x − x ) ek (r) ⊗ e∗k (r ) eik(x−x ) G bound(r, r ; ω ) = 2vg (2) + H(x − x) e−k (r) ⊗ e∗−k (r ) e−ik(x−x ) where a is the periodicity of the PC, H(x) is the Heaviside step function, and ⊗ denotes a tensor product. Note that the vg in the denominator ensures that the bound mode contribution dominates in the slow-light regime.

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Since we are interested in the propagation in the waveguide modes, we take a coupled-mode approach and introduce “slowly-varying envelopes,” ψf (x) and ψb (x) for the forward and backward waves respectively, to approximate the solution including scattering as E(r; ω ) = E0 [ek (r) eikx ψf (x) + e∗k (r) e−ikx ψb (x)] + c.c. + · · · ,

(3)

where E0 is an amplitude, and ‘· · · ’ includes contributions from radiation modes which quickly leak from the slab. Combining these expansions with Equation 1, we then derive a pair of coupled-mode equations for the evolution of the envelopes: dψf (x) = i cff (x) ψf (x) + i cfb (x) e−i2kx ψb (x) dx +i cfr (x) ψf (x), dψb (x) = i cbb (x) ψb (x) + i cbf (x) ei2kx ψf (x) −vg dx +i cbr (x) ψb (x). vg

(4)

(5)

The coupling coefficients can be physically interpreted as cff = cbb driving scattering from a mode into itself, cbf = c∗fb driving scattering into the counterpropagating mode, and cfr and cbr driving scattering from the waveguide mode into radiation modes above the light line. The presence the group velocity in the equations accounts for an enhancement in scattering as the light slows down and the local field strength increases. In the limit of single scattering events, backscattering scales as 1/v2g and radiation scattering scales as 1/vg. The coefficients are evaluated from the detailed configuration of the disorder and properties of the modes as

ωa dy dz e∗k (r) · ek (r) Δ ε (r), 2 ωa dy dz ek (r) · ek (r) Δ ε (r), cbf (x) = 2 ← → ωa dy dz dr Δ ε (r) e˜ ∗n,k (r) · G rad (r, r ; ω ) cnr (x) = 2 · e˜ n,k (r ) Δ ε (r ), n = f, b, cff (x) =

(6) (7)

(8)

where e˜ f,k (r) = e˜ ∗b,k (r) = ek (r) eikx . The interesting frequency response of the system is dominated by scattering between waveguide modes. Radiative scattering merely leaks energy from the waveguide and is a much smaller effect in the slow-light regime as shown in previous studies. For numerical efficiency, and for the reason given above, we calculate effective radiation coefficients ceff nr using our incoherent theory, and then solve the coupled mode equations numerically for simulated configurations of disorder.


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3 Experimental Device Except for the transmission spectra of Fig. 4, the experimental device is a W1.1 PC waveguide fabricated √ from GaAs as schematically shown in Fig. 1. The width of the waveguide is 1.1 3 a, the periodicity is a = 400 nm, the thickness is h = 265 nm, the hole radius is R = 0.27 a, and the length is 250 μm. In the literature, most planar PCs are fabricated from silicon due to its dominance in established semiconductor applications. The GaAs fabrication process showcased here has been shown to be competitive with state-of-the-art silicon processes in recent cavity experiments (Combrié et al. 2008) and has the potential to offer enhanced nonlinear optical effects and light amplification. The samples were analyzed using the high resolution scanning electron microscopy and the image processing technique of Skorobogatiy et al. (2005). The disorder statistics were found to be well described by small perturbations of the radius around the hole perimeter Δ R(φα ), that follow the distribution Δ R(φα )Δ R(φα ) =

σ 2 eR|φα −φα |/l p δα ,α , where α indexes the holes, and φα is the angular coordinate of the point relative to the hole centre. The root-mean-square roughness σ , and correlation length l p , are estimated to be 3 and 40 nm respectively from the measurements and these values are used in our calculations. A disordered hole used in the calculation is shown schematically in Fig. 3. The disorder used in Equations 6–8 is then given as 2 2 Δ ε (r) = (ε2 − ε1 ) H(h/2 − |z|) ∑ Δ R(φα (x, y)) δ (x − xα ) + (y − yα ) − R α

(9)

where ε1 (ε2 ) is the dielectric constant of the membrane (holes) and hole α is centred at (xα , yα ).

o lp

Nominal Radius

Fig. 3 Schematic of a disordered hole showing the RMS roughness σ and the in-plane correlation length l p .

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4 Transmission spectra

10−1 10−2 10−3 10−4 100 10−1 10−2 10−3

150 100 50 0

ng

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Experimental Transmission

An experimental transmission spectrum is shown in the top plot of Fig. 4 for a 1.5 mm W1 waveguide of a different design. At high frequencies, in the fast-light (or nominal-light) regime, losses are low and the transmission spectrum is relatively flat. Approaching the band edge, the transmission rolls-off approximately with 1/v2g , however there are numerous sharp resonances where the transmission varies by orders of magnitude. Two theoretical models for this waveguide are shown in the bottom plot. The previous incoherent loss calculation (Hughes et al. 2005) (dashed, black) captures the approximate 1/v2g roll-off but does not explain the resonances. Recall that the incoherent calculation gives the expectation value for the transmission when averaged over many nominally identical waveguides but here we are comparing with just a single waveguide. In contrast, the coherent loss calculation presented in this paper (solid, grey) reproduces the resonances since it, in addition to calculating the transmission for a single disordered waveguide, accounts for multiple scattering events which are necessary to build up Fabry-Pe´rot–like resonance between disorder sites. For reference the group index near the band edge is show in the inset (dashed, black). While we obtain the general trends of the experiments, over more than three orders of magnitude, and without any fit parameters, we have not included the disorder-induced broadening of the slow-light-regime band structure. Pendersen et al. (2008) have shown that the presence of small amounts of absorption will soften the band edge and prevent the group velocity from reaching zero. From perturbative

191 191.5 191.5 192 192.5 193 193.5 194 194.5 195 Frequency [THz]

Fig. 4 Top: Experimental transmission spectra for a 1.5 mm W1 waveguide showing resonances near the band edge. Bottom: Theoretical transmission spectra calculated using the incoherent (dashed, black) and coherent (solid, grey) scattering theories. Inset: Ideal group index ng = c/vg (dashed, black) compared with an estimate of the effective value due to disorder (solid, grey). The experimental spectrum is shifted by 1 THz to account for uncertainty in the slab thickness.


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calculations with identical disorder statistics, we calculate that disorder-induced local frequency shifts will cause a similar softening of the group velocity as shown in the grey curve in the inset of Fig. 4. We estimate that the group velocity will be noticeably altered from the ideal value for vg c/35 and will have a minimum of around c/80. For different structures, this minimum vg will vary. Although these findings are not important for the reflectance maps analyzed below, our calculations are broadly consistent with our own experiments, those reported by Engelen et al. (2008), and theoretical analysis of PC coupled-cavity structures (Fussell et al. 2008).

5 Time–Frequency Reflectance Maps Time–frequency reflectance maps (TFRMs) are intensity plots of the reflected signal as a function of time (horizontal axis) when the structure is excited with a narrowband pulse centred at some frequency of interest (vertical axis). The map is generated using the complex reflectance of the waveguide r(ω ) which, for physical samples, is deduced from a single set of OLCR data or, for simulated structure, is calculated directly (Parini et al. 2008). Formally, the TFRM is given by M(t, ω0 ) = IFFT[r(ω ) G(ω ; ω0 )],

(10)

where G(ω ; ω0 ) is the narrow frequency spectrum of the probe centred at ω0 . In the figures, we will plot the reflected intensity |M(t, ω0 )|2 . A simulated TFRM is shown in Fig. 5 for a waveguide with perfectly transmissive facets. The dashed black lines indicate the time the pulse is injected and the

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Fig. 5 A TFRM for a 250 μm simulated disordered waveguide with perfectly transmissive facets showing the strength of the back reflection in dB. The left black dashed line indicates when the pules was injected and the black dashed curve is the expected round trip time in an ideal structure. The solid white line shows the group index ng = c/vg on the top scale.

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Fig. 6 Comparison of simulated (top) and experimental (bottom) TFRMs for a 250 μm waveguide with a partially reflective front facet and a strongly reflective back facet. The simulation uses the same disorder configuration as in Fig. 5.

round-trip time. The solid white line indicates the ideal group index ng = c/vg for comparison. Away from the band edge, at higher frequencies, the back reflections are small and are confined between the two time limits, indicating that only single scattering events are occurring. Approaching the band edge, for vg < c/20, scattering becomes significant with strong back reflections and multiple scattering events clearly evidenced by the hot spots and the continued reflections after the time for one round trip. This agrees well with Engelen et al. (2008) who observed total disruption of propagation for vg c/30. Experimental samples have more complicated TFRMs due to reflections from the sample facets. Fig. 6 compares simulated (top) and experimental (bottom) TFRMs for the same device geometry. The multiple facet reflections1 are clearly visible and the time for a round trip lengthens as the group velocity decrease. At low group velocity, the pulse is washed out by strong multiple scattering events, making transmission of signals difficult. Since the simulations do not incorporate details of the experimental setup beyond the sample, they tend to give richer features than the experimental maps. Nevertheless, there is an excellent correspondence between the measured and simulated maps, and this agreement has been found for a number of different sample lengths and facet reflections. 1 We use 50% and 100% reflectances for the front and back facets respectively to match the sample properties.


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6 Localization Both our measured and simulated transmission spectra (Fig. 4) exhibits sharp resonances near the band edge, similar to those reported in Topolancik et al. (2007). These features can also be resolved in high resolution TFRMs. To rigorously address the question of whether these features are indicative of localization, a localization length l can be defined as l −1 = −ln T /L (11) where T is the transmitted power and L is the sample length (Vlasov et al. 1999). For the experimental structure at k = 0.45 × 2π /a where vg = c/45, the localization length is calculated to be ∼100 μm, far from the criteria for strong localization. Experimental imaging of light leaking from the waveguide plane confirms that these features are distributed over a large number of waveguide periods. Thus these features are better described as Fabry–Pérot-like resonances between scattering sites and not localization, in agreement with the interpretation of Vlasov et al. (1999). This conclusion is further supported by examining the position-dependent distribution of energy in the waveguide under c.w. illumination. Figure 7 shows the forward wave intensity in the waveguide at two points near the band edge. The points were chosen so that the grey curve corresponds to a local transmission maximum and the black to a local transmission minimum. From comparing the two curves, it is clear that the scattering is occurring in similar sites but that differences in accumulated phase cause constructive forward or backward interference. That this is not in the regime of strong localization in no way alters the fact that backscattering in slow light modes leads to highly disordered propagation and low transmission, which has profound implications for both fundamental physics and slow light applications.

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7 Connection to slow-light effects in metamaterial waveguides The phenomenon that slow-light waveguides are subject to strong scattering or other disruptions of propagation is not exclusive to PCs and can be observed in other materials. For example, Tsakmakidis et al. (2007) proposed a metamaterial waveguide that can slow down and actually stop light over broadband frequencies. However, they neglected to include absorption in their metamaterial model, which is needed to ensure causality; in this regard, Reza et al. (2008b) recently reanalyzed a proposal for a metamaterial waveguide in the presence of material loss and showed that propagation is severely disrupted: there was no region of stopped light (zero energy velocity) and the propagation losses are impractically high. The structure was a core layer of metameterial with a negative refractive index sandwiched between cladding layers of normal dielectric. The structure supports a bound slab mode with the peculiar property that the energy fluxes in the core and cladding regions propagate in opposite directions; when the core and cladding fluxes are almost perfectly opposed, slow-light propagation is achieved. The unrealistic (without material absorption) dispersion of the metamaterial waveguide mode is shown in Fig. 8 by the black curve. The band consist of a lossless component (the bottom hill) with a zero energy velocity point at the peak. The portion of the band above the peak is an evanescent mode that is absorbed without propagating energy. Metamaterials, in contrast to semiconductor PCs, suffer from intrinsic ohmic losses in the metallic components used to create the negative refractive index. Reza et al. shows that in the presence of even arbitrarily small absorptive loss, the ideal band splits into two lossy bands (grey) and the zero energy velocity point is completely lost.

Fig. 8 Dispersion of the metamaterial waveguide mode of interest without (black) and with (grey) disorder. (Taken, with permission, from A. Reza (2008a).)


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Although the mechanism for the disruption of propagation is different – small absorptive losses instead of scattering sites – the general effect of unavoidable nonperturbative light scattering for slow-light modes is similar. In the slow-light regime, the local field strength and the interaction time of the light with the disorder are enhanced, exacerbating any disruption of the field propagation. Consequently, the minimum achievable group velocity will be strongly limited by the presence of unavoidable absorbers and scatterers (even if on the atomic scale) in the structure.

8 Conclusion We have presented a coherent loss formalism and matching experiments that allows us to calculate and directly compare the properties of naturally disordered PC waveguides by considering the full 3D structure. In particular, we capture the behavior of a slow-light regime with narrow-band resonances superimposed on the average 1/v2g roll-off of the incoherent theory. Our simulated waveguides show excellent qualitative agreement with measurements on GaAs waveguides, as demonstrated by comparing transmission spectra and TFRMs. We also determine the localization length near the band edge of a PC waveguide and show that while propagation is highly disordered, strong localization is not occurring. Finally, we have connected with slow-light metamaterial waveguides and shown that disruption of propagation is common to all slow-light structures. Acknowledgements We thank Jeff Young, John Sipe, Lora Ramunno, Arvin Reza and Marc Dignam for useful discussions. This work was supported by the National Sciences and Engineering Research Council of Canada and the Canadian Foundation for Innovation.

References Akahane, Y., Asano, T., Song, B.S., Noda, S.: High-Q photonic nanocavity in a two-dimensional photonic crystal. Nature 425(6961), 944–947 (2003) Anderson, P.W.: The question of classical localization theory of white paint? Philos. Mag. B 52(3), 505–509 (1985) Baba, T.: . Slow light in photonic crystals. Nat. Photonics 2(8), 465–473 (2008) Combrié, S., Weidner, E., DeRossi, A., Bansropun, S., Cassette, S., Talneau, A., Benisty, H.: Detailed analysis by Fabry–Pérot method of slab photonic crystal line-defect waveguides and cavities in aluminium-free material system. Opt. Expr. 14(16), 7353–7361 (2006) Combrié, S., Tran, N.V.Q., Weidner, E., Rossi, A.D., Cassette, S., Hamel, P., Jaouen, Y., Gabet, R., Talneau, A.: Investigation of group delay, loss, and disorder in a photonic crystal waveguide by low-coherence reflectometry. App. Phys. Lett. 90(23), 231104 (2007) Combrié, S., Rossi, A.D., Tran, Q.V., Benisty, H.: GaAs photonic crystal cavity with ultrahigh Q: microwatt nonlinearity at 1.55 mu m. Opt. Lett. 33(16), 1908–1910 (2008) Engelen, R.J.P., Mori, D., Baba, T., Kuipers, L.: Two regimes of slow-light losses revealed by adiabatic reduction of group velocity. Phys. Rev. Lett. 101(10), 103901 (2008)

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Fussell, D.P., Hughes, S., Dignam, M.M.: Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides. Phys. Rev. B 78(14), 144201 (2008) Gerace, D., Andreani, L.C.: Disorder-induced losses in photonic crystal waveguides with line defects. Opt. Lett. 29(16), 1897–1899 (2004) Hughes, S., Ramunno, L., Young, J.F., Sipe, J.E.: Extrinsic optical scattering loss in photonic crystal waveguides: Role of fabrication disorder and photon group velocity. Phys. Rev. Lett. 94(3), 033903 (2005) John, S.: Electromagnetic absorption in a disordered medium near a photon mobility edge. Phys. Rev. Lett. 53(22), 2169–2172 (1984) John, S.: Strong localization of photons in certain disordered dielectric superlattices. Phys. Rev. Lett. 58(23), 2486–2489 (1987) Johnson, S.G., Joannopoulos, J.D.: Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis. Opt. Expr. 8(3), 173–190 (2001) Kuramochi, E., Notomi, M., Hughes, S., Shinya, A., Watanabe, T., Ramunno, L.: Disorderinduced scattering loss of line-defect waveguides in photonic crystal slabs. Phys. Rev. B 72(16), 161318(R) (2005) Lahini, Y., Avidan, A., Pozzi, F., Sorel, M., Morandotti, R., Christodoulides, D.N., Silberberg, Y.: Anderson localization and nonlinearity in one-dimensional disordered photonic lattices. Phys. Rev. Lett. 100(1), 013906 (2008) Mookherjea, S., Park, J.S., Yang, S.H., Bandaru, P.R.: Localization in silicon nanophotonic slowlight waveguides. Nat. Photonics 2(2), 90–93 (2008) Notomi, M., Yamada, K., Shinya, A., Takahashi, J., Takahashi, C., Yokohama, I.: Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs. Phys. Rev. Lett. 87(25), 253902 (2001) O’Faolain, L., White, T.P., O’Brien, D., Yuan, X., Settle, M.D., Krauss, T.F.: Dependence of extrinsic loss on group velocity in photonic crystal waveguides. Opt. Expr. 15(20), 13129–13138 (2007) Parini, A., Hamel, P., Rossi, A.D., Combrié, S., Tran, N.V.Q., Gottesman, Y., Gabet, R., Talneau, A., Jaouën, Y., Vadalà, G.: Time-Wavelength Reflectance Maps of Photonic Crystal Waveguides: A new view on disorder-induced scattering. J. Lightwave Technol. 26(21–24), 3794–3802 (2008) Patterson, M., Hughes, S., Combrié, S., Tran, N.V.Q., Rossi, A.D., Gabet, R., Jaouën, Y.: Disorderinduced coherent scattering in slow-light photonic crystal waveguides. Phys. Rev. Lett. 102, 253903 (2009) Pedersen, J.G., Xiao, S., Mortensen, N.A.: Limits of slow light in photonic crystals. Phys. Rev. B 78(15), 153101 (2008) Povinelli, M.L., Johnson, S.G., Lidorikis, E., Joannopoulos, J.D., Soljacic, M.: Effect of a photonic band gap on scattering from waveguide disorder. App. Phys. Lett. 84(18), 3639–3641 (2004) Reza, A.: The optical properties of metamaterial waveguide structures. Master’s thesis, Dept. of Physics, Queen’s University, Kingston, Ontario, Canada (2008a) Reza, A., Dignam, M.M., Hughes, S.: Can light be stopped in realistic metamaterials? Nature 455(7216) (2008b) doi:10.1038/nature07359 Schwartz, T., Bartal, G., Fishman, S., Segev, M.: Transport and Anderson localization in disordered two-dimensional photonic lattices. Nature 446(7131), 52–55 (2007) Skorobogatiy, M., Bégin, G., Talneau, A.: Statistical analysis of geometrical imperfections from the images of 2D photonic crystals. Opt. Expr. 13(7), 2487–2502 (2005) Taflove, A., Hagness, S.C.: Computational electrodynamics: the finite-difference time-domain method, 3rd ed. Artech House Publishers, Norwood, MA (2005) Topolancik, J., Ilic, B., Vollmer, F.: Experimental observation of strong photon localization in disordered photonic crystal waveguides. Phys. Rev. Lett. 99(25), 253901 (2007) Tsakmakidis, K.L., Boardman, A.D., Hess, O.: ‘Trapped rainbow’ storage of light in metamaterials. Nature 450(7168), 397–401 (2007) Vlasov, Y.A., Kaliteevski, M.A., Nikolaev, V.V.: Different regimes of light localization in a disordered photonic crystal. Phys. Rev. B 60(3), 1555–1562 (1999)


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Silicon Photonic Waveguide Structures and Devices: From Fundamentals to Implementations in Spectroscopy and Biological Sensing Pavel Cheben, Adam Densmore, Jens H. Schmid, Dan-Xia Xu, André Delâge, Mirosław Florjańczyk, Siegfried Janz, Boris Lamontagne, Jean Lapointe, Edith Post, Martin Vachon, Philip Waldron Institute for Microstructural Sciences, National Research Council Canada, 1200 Montreal Road, Ottawa, ON, K1A 0R6, Canada [email protected]

Abstract In this chapter we discuss recent advances in silicon photonics research at the National Research Council Canada. We review our work on first implementations of subwavelength grating structures in silicon waveguides, including efficient fiber-chip coupling structures, and anti-reflective and high-reflectivity structures formed at the SOI waveguide facets. Silicon planar waveguide spectrometer chips are introduced, namely a high-resolution arrayed waveguide grating spectrometer and the first planar waveguide Fourier-transform spectrometer with a largely increased light gathering capability. Finally, we review our work in developing silicon-wire biological sensors with excellent surface sensitivity, ultracompact sensor designs, and new waveguide geometries that allow these sensors to be densely arrayed for compatibility with conventional microarray spotters. These sensors provide a practical route to the development of label-free microarray biochips for multi-analyte monitoring.

Keywords: Silicon photonics, subwavelength grating, silicon wire waveguide, arrayed waveguide grating, waveguide spectrometer, Fourier-transform spectrometer, biological sensor, label-free sensing

1

Silicon photonics: Promises and challenges

The fundamental discovery in 1958 by Kilby that integrated circuits can be made in a single-crystal semiconductor triggered an unprecedented growth in electronic device integration density, with silicon being the dominant platform for the microelectronics industry. Since the 1980s, silicon photonics has been the subject of T.J. Hall and S.V. Gaponenko (eds.), Extreme Photonics & Applications, © British Crown 2010

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intense research activity, with the goal of developing photonic components that are factory-compatible with silicon microelectronic integrated circuits (Pavesi and Lockwood 2004; Reed and Knights 2004; Cheben et al. 2008c). Silicon is an excellent material for confining and manipulating infra-red light at the submicrometer scale, and possesses the added advantage of leveraging the enormous manufacturing infrastructure developed by the silicon microelectronics industry. Silicon optoelectronic integrated devices can be miniaturized and mass-produced at affordable cost for many applications and markets, including telecommunications, optical interconnects, and biological and chemical sensing. Recent developments in diverse areas such as light sources, modulators, switches, detectors, photonic crystals, resonators, sensors, and various sub-systems, attest that Si photonics is an extremely active research field (Reed and Knights 2004; Soref 2008a). Among the optical properties of silicon, the high refractive index (n ~ 3.5 at 1.5 μm wavelength) is particularly attractive. When silicon is used as the waveguide core surrounded by, for example silica cladding (n ~ 1.5), an index step of Δn ~ 2 is obtained. Such high index contrast waveguides can be made with cross sections as small as ~250 × 250 nm and bending radii of a few micrometers. Thus, very compact waveguide devices can be made in Si, amenable to high integration density. Also to be noticed is a fairly broad transparency range of Si-waveguides. Silicon photonic devices can operate at any wavelength from 1.2 μm to about 3.4 μm mid-wave infrared, where the SiO2 becomes highly absorbing, with a potential extension into long-wave infrared being envisioned (Soref 2008b; Soref et al. 2006). Silicon photonic devices are typically fabricated in the silicon-on-insulator (SOI) platform. SOI consists of a top single-crystal silicon layer separated from the silicon substrate by an insulating SiO2 layer. In photonics applications, the top Si layer acts as the waveguide core and the insulating oxide as the bottom cladding to provide the vertical confinement for light. Lateral confinement of light is achieved by forming waveguide channels using microfabrication processes. The potential for fabricating monolithic photonics devices with high-integration density is an important advantage of SOI waveguides. For example, using silicon wire waveguides with a bend radius of 5 μm, one can fold millimeter-long waveguides into dense spirals occupying an area with diameter of less then 100 μm. Such spiral elements can be advantageously used as biological sensors, as we will describe in Section 4. They can be densely arrayed in two dimensions for compatibility with microarray spotters, and at the same time provide long interaction length and improved molecular capture efficiency. For sensing applications, an additional benefit is the highest intrinsic surface sensitivity of silicon wire waveguides compared to any other commonly used waveguide materials. Such Si-based waveguide structures show great promise as fundamental sensing elements for label-free microarrayed biochips (see Section 4). Using high index contrast SOI waveguides, complex photonic circuits can be built, such as arrayed waveguide grating (AWG) spectrometer chip with footprint of only 8 × 8 mm2, discussed in Section 3. A high spectral resolution of 0.1 nm is

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achieved by implementing silicon-wire technology in the focal plane of the spectrometer. State-of-the-art AWG devices based on glass waveguide technology and with an equivalent wavelength resolution would occupy a full 4-in. diameter wafer. The benefits and advantages of implementing photonic circuits in silicon waveguides are accompanied by some technological challenges. For example, a comparatively large waveguide propagation loss, polarization sensitivity, and fiber-chip coupling loss, have been the main obstacles to practical applications of silicon waveguides. Various approaches have been pursued to address these challenges. The waveguide sidewall roughness and the resulting loss can be reduced by using an oxidation step to smoothen the sidewall, and low-loss Si waveguides have also been fabricated using deep-UV lithography (Dumon et al. 2004). Requirements on polarization sensitivity typically vary for different applications. For example, in wavelength dispersive devices such as multiplexers or spectrometers (Cheben 2007; Cheben et al. 2008a), polarization independent properties are often required. This is because the input light polarization is generally not known and can vary in time and this would produce polarization-dependent shifts in measured spectral lines, thus degrading spectrometer performance. An important source of polarization sensitivity in silicon waveguides is waveguide birefringence. The birefringence arises from the asymmetry in the waveguide cross-section in the vertical and horizontal directions. This so-called geometrical birefringence can be eliminated by using stress engineering for a wide range of waveguide geometries (Xu et al. 2004, 2005). In this technique, a stressed cladding film is deposited over the silicon waveguide core. The film exerts a force on the silicon core that can effectively eliminate the waveguide birefringence via the photoelastic effect. Using such stress engineering, polarization-insensitive SOI devices, such as arrayed waveguide gratings, have been demonstrated (Xu et al. 2004). When it is difficult to achieve polarization independent device characteristic, such as in devices based on waveguides of sub-micrometer dimensions (so-called photonic wires) a polarization diversity approach can be used (Bogaerts et al. 2007). In the latter, the light is split into two orthogonal polarizations, and each of these is processed independently by a dedicated device optimized for the specific polarization. In yet other applications, a well defined input polarization state can be assumed and the device performance can be optimized for that single polarization state. For example, the evanescent field waveguide biosensor discussed in Section 4 is designed to operate with TM polarized light, since this polarization provides maximal device sensitivity. A major problem in practical implementations of microphotonic devices, including the biosensors and spectrometers presented in this chapter, is low efficiency of optical coupling between the microphotonic waveguides and the macroscopic input/output ports, such as lenses and optical fibers. For example, the mode area of the silicon wire waveguide can be ~103 times smaller than that of standard optical fiber, making fiber-chip coupling highly inefficient. Various spot size converters have been proposed to reduce the coupling loss (Thourhout et al.

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2008). The highest coupling efficiency mode converters typically involve 3D overgrowth to increase the modal field diameter and are used in conjunction with edge-deposited anti-reflective coatings (Smith et al. 2006). However, these are comparatively complex processes and a single-step fabrication is desirable. This is achieved in an inverse taper fiber-chip coupler (Shoji et al. 2002), where the waveguide width is reduced to ~100 nm near the chip coupling facet, causing the mode to expand and to eventually match that of the fiber. However, it is a challenge to reproducibly fabricate such narrow waveguide tips. In order to increase the taper tip width and thus improve the fabrication robustness, subwavelength grating (SWG) fiber-chip couplers have been proposed (Cheben et al. 2006). The principle of the sub-wavelength grating fiber-chip coupler is based on a gradual modification of the waveguide mode effective index by the subwavelength nanostructure, so that efficient index matching that of the optical fiber is obtained at the chip coupling facet, as we discuss in Section 2. There are also fundamental challenges of the SOI material system related to the fixed value of the refractive indices of the constituent materials (Si and SiO2). To circumvent this limitation, we have proposed the use of the sub-wavelength effect to engineer materials with intermediate effective indices simply by lithographic patterning, as we discuss in Section 2. The aim of this chapter is to provide an overview of several related aspects of our silicon photonics research. In Section 2, we review our work on first implementations of subwavelength grating structures in silicon waveguides. In particular, we describe two applications of sub-wavelength nanostructures in silicon photonics, namely as efficient fiber-chip coupling structures, and anti-reflective and highreflectivity structures formed at the SOI waveguide facets. In Section 3, silicon planar waveguide spectrometer chips are presented. A high-resolution arrayed waveguide grating spectrometer is discussed, including its use to interrogate a fibre Bragg grating sensor with a high wavelength accuracy of 1 pm. The first planar waveguide Fourier-transform spectrometer is also introduced. The Fourier-transform spectrometer chip has an advantage of a largely increased light gathering capability compared to conventional spectrometers, with the additional benefit of a fully static design, i.e. absence of any moving parts, unlike in conventional scanning Fourier-transform interferometers. Finally, in Section 4 we review our work developing label-free biological sensors using silicon photonic wire waveguides. We demonstrate that the high index contrast of these waveguides provides excellent surface sensitivity and compact sensor designs. We also describe new waveguide geometries that allow these sensors to be densely arrayed in two dimensions for compatibility with conventional microarray spotters, while simultaneously providing long interaction length and improved molecular capture efficiency. These sensor designs are shown to provide a practical route to the development of label-free, micro-array biochips for multi-analyte monitoring.


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2.1 Subwavelength grating (SWG) principle The concept of using periodic dielectric structures with a periodicity shorter than the wavelength of light λ has been known for many years (Mait and Prather 2001; Kikuta et al. 2003). Such structures, commonly referred to as subwavelength gratings (SWGs), have been used to create new effective media with useful and interesting optical properties. An example of an established application for SWGs is antireflective optical layers on bulk dielectric surfaces. A one-dimensional SWG is illustrated in Fig. 1. The structure is comprised of alternating slabs of dielectric materials with refractive indices n1 and n2, with a period Λ. For an SWG, Λ < λ/n1,2 and hence diffraction is suppressed. Light propagating through such a structure macroscopically behaves as in a homogeneous effective medium with an averaged dielectric permittivity. For the case of light incident parallel to the dielectric slabs with a wavevector kx (Fig. 1), the effective index of the SWG depends on the polarization of the light. According to the effective medium theory (Rytov 1956), the effective refractive index is to the first approximation given by: n|| = (fn12 + (1-f)n22)1/2

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for a wave with the electric field parallel and perpendicular to the slabs, respectively, where f = a/Λ is the volume fraction of the material with index n1. The effective index is thus a weighted average of the refractive indices of the constituent materials in either case; however, due to the fact that the averaging is polarization dependent, the SWG exhibits form birefringence. The optical properties of the structure shown in Fig. 1 for light incident perpendicular to the dielectric slabs (wavevector kz) are well-known from the study of photonic crystals (Joannopoulos et al. 2008). Usually, photonic crystals are used in a regime where the periodicity is comparable to the wavelength of light, in which case bandgaps form in the dispersion relation; however, in the long wavelength limit, which corresponds to an SWG, photonic crystals exhibit linear dispersion, consistent with the concept of an effective homogeneous medium. Generally, silicon-on-insulator waveguide devices are designed using just two materials with a high refractive index contrast, namely silicon (n = 3.48 at λ = 1.55 μm) and silicon dioxide (n = 1.45). The use of SWGs makes designs possible with effective dielectric materials covering a continuous range of intermediate refractive indices, which can be fabricated by standard nanofabrication techniques. In the following two sections, we will discuss some examples of the use of SWG structures in SOI waveguides, namely SWG waveguide fiber-chip couplers, antireflective waveguide facets, and waveguide mirrors.

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Fig. 1 One-dimensional SWG comprised of two materials with refractive indices n1 and n2.

2.2 Subwavelength grating waveguides for fiber-chip coupling A SWG photonic wire waveguide is an example of a structure where the light propagation direction is collinear to the grating vector. We have recently demonstrated an application of SWG waveguides for fiber-to-waveguide coupling, that mitigates losses due to the mode size mismatch of optical fibers and submicrometer SOI waveguides (Cheben et al. 2006; Schmid et al. 2008a). The principle of this SWG fiber-chip coupler is based on a gradual modification of the waveguide mode effective index by the SWG effect. The idea is illustrated in a schematic side view of a coupler structure in Fig. 2a. The waveguide mode effective index is altered by chirping the SWG fill factor f(z) = a(z)/Λ, where a(z) is the length of the waveguide core segment. The volume fraction of the Si waveguide core is modified such that at one end of the coupler the effective index is matched to the SOI waveguide while at the other end, near the chip facet, it matches that of the optical fiber. Using finite difference time-domain simulations, we have calculated efficiencies as large as 76% (1.35 dB loss) for coupling from a standard optical fiber (SMF-28, mode field diameter 10.4 μm) to a 0.3 μm wide SOI waveguide through a 50-μm-long SWG coupler (Cheben et al. 2006), including reflection loss.

Fig. 2 (a) Schematic side view of an SOI photonic wire SWG waveguide coupler as explained in the text. (b) Scanning electron microscope image of a fabricated structure (top view).


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The SWG coupler was also demonstrated experimentally for 0.26 × 0.45 μm silicon photonic wire waveguides (Schmid et al. 2008a). A close-up view of a fabricated coupler structure connecting to the waveguide is shown in Fig. 2b. Coupling losses of 6.5 and 4 dB for transverse electric (TE) and transverse magnetic (TM) polarization, respectively, were obtained using these SWG couplers. The loss penalty compared to the simulation was attributed to substrate leakage, which is expected to be reduced in subsequent designs. The corresponding values of coupling loss for waveguides without SWG couplers were 18 dB (TE polarization) and 11 dB (TM).

2.3 Antireflective and highly reflective waveguide facets For the modification of waveguide facet reflectivity, we have directly adapted the known SWG anti-reflective effect on bulk optical surfaces to waveguide facets (Schmid et al. 2007a, 2008a). An example of a fabricated SWG facet with triangular teeth is shown in Fig. 3a. For this structure, the wave vector and the grating vector are orthogonal. As a waveguide mode propagates in the SWG region, the effective waveguide core index varies continuously from the value of silicon (n = 3.48) to the value of air (n = 1) over the length of the SWG section, denoted D in Fig. 3a. Therefore the mode effectively propagates through a gradedindex section. In contrast, for a conventional flat waveguide facet the abrupt refractive index change at the facet-air interface yields a Fresnel power reflectivity of ~30% for silicon waveguides. As the modulation depth (D) of the SWG is increased, an efficient anti-reflective effect is observed experimentally. This is shown in Fig. 3b, for the case of a transverse electric (TE) mode. The experimental results are in good agreement with the behavior expected from effective medium theory. An efficient anti-reflective effect is also observed for the transverse magnetic (TM) mode; however, due to the form birefringence of the structure the facet reflectivity value is different for TM. We obtained a minimum reflectivity of 2.0% for TE and 2.4% for TM polarization for a modulation depth of D = 0.72 μm, which is the maximum grating depth used in our experiments. An anti-reflective effect can also be achieved with a rectangular SWG (Schmid et al. 2008a), in which case the SWG structure is akin to a single layer λ/4 antireflective coating. More surprisingly perhaps, highly reflective facets can be achieved with rectangular grating patterns (Schmid et al. 2008b). This effect is analogous to that first described for bulk optical surfaces (Goeman et al. 1998). It is observed for gratings with a period Λ which is smaller than the wavelength of light in air (λ) but larger than the wavelength inside silicon (λ/nSi). This means that diffraction of light inside the silicon is allowed in principle but suppressed by the judicious choice of grating parameters. We have found by simulations and experiments, that

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Fig. 3 (a) Scanning electron microscope image of a ridge waveguide facet patterned with a triangular SWG. (b) Measured reflectivity of SWG facet as a function of grating modulation depth compared to effective medium theory.

the high-reflectivity effect is strongly dependent on grating size. For example, the maximum reflectivity of a patterned facet on a 1.5 μm thick waveguide is 77%, but power reflectivities larger than 90% can be achieved for silicon waveguides with thicknesses of 4 μm or more.

3

Silicon planar waveguide spectrometers

3.1 Arrayed waveguide grating spectrometer The arrayed waveguide grating (AWG) (Cheben 2007; Cheben et al. 2008a) is a planar waveguide device that acts as an optical analogue of a microwave phase array antenna. By controlling the phase relationship between radiating sources forming the array output, the radiated beam direction can be changed without mechanical movement. The AWG (Fig. 4) has been extensively used in optical telecommunications, where it performs functions such as wavelength multiplexing, signal routing, wavelength filtering, and optical cross-connects (Cheben 2007). New applications of these spectrometer devices are emerging, for example in infrared spectroscopy, signal processing, optical interconnects, metrology, chemical and biological sensing, medical instrumentation, and satellite-based sensing (Cheben et al. 2008a). Their operation range can potentially be extended from the nearinfrared to the long-wavelength infrared region. The basis for the AWG operation is the waveguide phase array (Fig. 4). Each successive waveguide in the array is longer than the adjacent waveguide by a constant length increment. The resulting phase delay between array waveguides is similar to that between light scattered by adjacent grooves in a conventional diffraction grating. Therefore, the light emerging from the waveguide array into the output slab waveguide combiner region is spectrally dispersed. In the output


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Fig. 4 Schematics of an arrayed waveguide grating (AWG) spectrometer.

combiner, different wavelengths propagate at different angles θ relative to the combiner axis (Fig. 4), thus arriving with a spatial separation at the AWG focal plane. The spectrum is typically obtained using a photodetector array attached to the chip output facet (Fig. 5a). AWGs can be implemented with many wavelength channels and a high spectral resolution using high index contrast silicon-on-insulator (SOI) waveguides. The large refractive index contrast ensures a very small device footprint, and the first miniature SOI AWG of sub-centimeter size was developed at our laboratory (Pearson et al. 2000). The device had eight channels spectrally separated by 1.6 nm, and an overall chip size was 5 × 5 mm2. Using this miniature AWG, we demonstrated various techniques to achieve polarization-independent operation (Xu et al. 2004; Cheben et al. 2001, 2003). Recently, we have developed a 50-channel AWG spectrometer with high spectral resolution (Cheben et al. 2007) (Fig. 5a). It is commonly known that, for a given grating and imaging optics, the spectral resolution of a bulk optic grating spectrometer is set by the input and output slit widths. By narrowing the slit width, the resolution is improved. In an AWG spectrometer, the input and output waveguide mode size plays an analogous role to the slit width. This mode size is limited to several micrometers in glass waveguide AWGs. The mode size and coupling between closely spaced waveguides also limits the maximum number of channels. In silicon however, well confined waveguides can be fabricated with sub-micrometer dimensions. We have used such sub-micrometer waveguides in our AWG design to increase both the wavelength resolution and number of spectral channels.

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Ik+1 I k+1 Ik

Ik

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Fig. 5 (a) High-resolution SOI arrayed waveguide grating (AWG). The spectrum is measured by a photodetector array positioned next to the chip output facet; (b) light intensities and their ratio for two adjacent AWG channels as a function of resonance wavelength shift in TFBG sensor interrogation experiment.

In our high-resolution AWG spectrometer, the input waveguides and arrayed grating waveguides use a conventional ridge geometry with a nominal waveguide width of 1.5 μm and the same (1.5 μm) thickness. However, near the Rowland circles, the ridge waveguides are adiabatically transformed down to 0.6 μm wide channel waveguides etched down to the bottom cladding oxide (Schmid et al. 2007b). This results in high aspect ratio 0.6 × 1.5 μm waveguide apertures at the positions where the light is injected into the input combiner (at the input Rowland circle) and intercepted by different output channels (output Rowland circle), respectively. The ridge waveguides and the adiabatic mode converters (Schmid et al. 2007b) are fabricated using a self-aligned two-step electron-beam patterning and plasma etching process (Cheben et al. 2007). The measured wavelength channel spacing of 0.2 nm allows spectral lines separated by 0.1 nm to be resolved at 1.5 μm wavelength. This is the highest resolution reported to date for an SOI AWG. Furthermore, state-of-the-art glass AWGs with an equivalent wavelength resolution occupy a full 100 mm wafer, compared to the 8 × 8 mm2 footprint of this SOI AWG microspectrometer.

3.2 Interrogating a fibre Bragg grating sensor with SOI AWG spectrometer We have used the high-resolution AWG spectrometer (Fig. 5a) to measure the relative wavelength spacing between the cladding and Bragg mode resonances of a fiber Bragg grating refractometric sensor (Cheben et al. 2008b). The titled fibre Bragg grating (Chan et al. 2007) was used to sense the refractive index of the


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medium surrounding the fiber. Since the cladding modes of tilted fibre Bragg grating have an evanescent tail that propagates in the surrounding medium, their propagation constants depend on the refractive index of the medium. On the other hand, the effective index of the core mode does not depend on the refractive index of the surrounding medium, since the core is optically isolated from the medium by the cladding. However, the temperature sensitivities of both the core and cladding mode indices are nearly equal. Therefore, by measuring the wavelength separation between the Bragg and the cladding resonance, the refractive index of the medium is monitored, while the influence of temperature fluctuations is minimized. The wavelength shifts of the Bragg and cladding mode resonances were determined from the optical power I k and I k+1 in two adjacent AWG output channels (k and k + 1) with the Bragg grating resonance minima located between the two, i.e. λk < λres < λk+1. Figure 5b shows the variation in output power of two such AWG channels as the wavelength of the Bragg resonance shifts with sensor temperature. As the Bragg resonance minimum moves, the intensity decreases in channel λk+1, while it increases in channel λk. By monitoring the power ratio Ik+1/Ik, we were able to measure the resonance wavelength shift with an accuracy of 1 pm. This value is comparable to that achieved with complex and dedicated laboratory instruments such as optical spectral analyzers and wavelength swept laser interrogation systems.1

3.3 Fourier-transform waveguide spectrometer Development of miniature spectrometers that are robust and environmentally reliable is of great interest for space-borne sensing and spectral imaging. It is important that miniaturization of the instrument does not compromise its performance and the instrument gathers as much light as possible. In optical instrumentation, it is known that interferometric spectrometers have the advantage of increased optical throughput (étendue) compared to dispersive spectrometers at comparable resolution (Jacquinot 1954). The étendue advantage is one of the main reasons of the widespread use of scanning Fourier-transform interferometers in infrared spectroscopy. However, in space applications, the moving mirrors required for scanning in Fourier-transform spectrometers impose problems of size, mass, and reliability. Fortunately, an interferometric technique exists that does not require moving parts. This technique, called spatial heterodyne spectroscopy, relies on the formation of stationary interference pattern by two wavefronts interfering at a wavelength-dependent angle (Harlander et al. 1992; Powell and Cheben 2006). The input spectrum is calculated by Fourier transformation of the measured spatial interferogram.

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High resolution swept laser interrogator, Model Si720, from Micron Optics, Inc., Atlanta, Ga.

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Fig. 6 (a) The schematics of the waveguide spectrometer formed by arrayed Mach-Zehnder interferometers; (b) spatial fringe at the arrayed Mach-Zehnder outputs for monochromatic input at the Littrow wavelength λL, wavelengths λL+Δλ and λL+2Δλ, and polychromatic input.

We have adopted the principle of the spatial heterodyning to the planar waveguide geometry to design the first Fourier transform planar waveguide spectrometer. The spectrometer uses two interleaved reflective AWGs arranged in a Michelson configuration (Cheben et al. 2005). To further increase the spectrometer light throughput, we have extended the Fourier-transform waveguide spectrometer concept into configurations with multiple input apertures. One such device is the Mach-Zehnder interferometer array (Florjańczyk et al. 2007) schematically shown in Fig. 6a. The Mach-Zehnder interferometer is an established device both in bulk optics and waveguide implementations. Its transmittance is a periodic function of the optical path difference between the two interferometer arms. The Fourier-transform spectrometer comprises an array of independent Mach-Zehnders with different phase delays ΔLi (Fig. 6a). It has a multi-aperture input formed by N waveguides each feeding into an individual Mach-Zehnder. An obvious advantage of this device is that the optical throughput is largely increased by using multiple inputs simultaneously.


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The operating principle of the device can be understood as follows. The path difference ΔLi in the Mach-Zehnder array changes by a constant increment across the array. For a given monochromatic input, different transmittance function of each Mach-Zehnder results in a different power value at the corresponding output (Fig. 6b). A monochromatic input results in a periodic (sinusoidal) spatial distribution of power across the output ports Piout = Pout (xi), which is the Fourier transform of the monochromatic input spectrum. Since the spatial power distribution Pout(xi) and the input spectrum are a Fourier transform pair, a polychromatic input produces a power distribution (Fig. 6b) from which the input spectrum can be calculated using discrete Fourier transformation.

a

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Fig. 7 (a) Simulated power distribution at the outputs of an Mach-Zehnder interferometer array for water vapor absorption spectrum at 15 km altitude (inset); (b) FT spectrometer chip fabricated at the National Research Council Canada.

An example of a spatial interferogram and a calculated spectrum is illustrated in Fig. 7a. The input spectrum corresponds to the absorption of water vapor using solar occultation satellite sensing at 15 km altitude in the near infrared spectral region at ~1.36 μm wavelength (Solheim 2005). The first prototype of the MachZehnder interferometer array was recently fabricated (Fig. 7b). In addition to the large optical throughput, an important advantage of this device is that deviations from the ideal design appear as systematic errors in the interferograms. Once the waveguide device has been fabricated and characterized, the errors can simply be corrected for by software calibration. The calibration ability is an important advantage of this device compared to an AWG. In an AWG there is no direct physical access to the arrayed waveguide output aperture, which makes measuring and correcting the phase errors a formidable task. Unlike in the AWG, the Mach-Zehnder interferometer array provides physical access to each of the interferometer outputs where both phase and amplitude errors can be readily measured as part of the spectrometer calibration procedure.

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Silicon photonic biological sensors

4.1 Optical sensors for label-free detection of biological molecules Over the past decade there has been significant effort directed towards the development of new, label-free optical diagnostic technologies based on planar optical waveguides for genomics, proteomics and pathogen detection (Densmore et al. 2008; Armani et al. 2007; Sepulveda et al. 2006). These technologies rely on the affinity binding reaction that occurs between appropriate molecular pairs, such as complementary DNA or RNA molecules, or antibody – antigen systems. These sensor devices are capable of detecting extremely dilute concentrations of analyte molecules and can monitor binding reactions in real time, allowing important kinetic information to be obtained. In the most common configuration, receptor molecules are immobilized on the planar waveguide surface using a specific binding chemistry. As the target molecules are captured from the analyte solution a localized change of the near-surface refractive index is induced, resulting in a phase shift of the light propagating through the waveguide. By monitoring this phase shift, the binding reaction can be observed in real time without the use of fluorescent labels. The two most commonly reported devices for this application are the surface plasmon resonance sensor and the waveguide evanescent field sensor. The surface plasmon resonance sensor utilizes the surface plasmon mode of a thin metal film deposited on a dielectric substrate. As molecules bind to the receptors immobilized on the metal film the effective index of the plasmon is altered, changing the resonant coupling condition between the excitation beam and the surface plasmon. By monitoring the resonance peak position in the angular reflection spectrum, the binding reaction can be monitored. Although surface plasmon resonance is currently the most widely employed and commercially successful technique, it is fundamentally limited by the high loss and short propagation length of the surface plasmon mode, in the range of tens of micrometers. This short plasmon propagation length leads to a relatively broad angular resonance spectrum imposing a limit in the attainable sensitivity. An alternative and emerging technique is the evanescent field sensor, where the binding reaction is sensed by the evanescent tail of an optical mode in a dielectric or semiconductor waveguide, as shown in Fig. 8. Unlike the surface plasmon resonance structures, these waveguides can be several centimeters long and still maintain negligible optical loss. The extended interaction length results in increased sensitivity compared to surface plasmon resonance sensors (Lukosz 1991). In addition, these devices may be straightforwardly combined with other optical, electrical and microfluidic components for the development of highly functional lab-on-a-chip devices.

SILICON PHOTONIC WAVEGUIDE STRUCTURES AND DEVICES Target molecules

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Thin oxide (optional) Evanescent optical field

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Fig. 8 Typical evanescent field waveguide sensor configuration.

4.2 Evanescent field silicon photonic wire waveguide biosensors In this section, we overview our work developing EF sensors using silicon photonic wire waveguides. We show that the high index contrast of these submicrometer waveguides creates a strong highly localized evanescent field at the waveguide surface for high surface sensitivity. We also discuss how the high lateral index contrast of these waveguides makes new sensing structures possible (e.g., densely folded spirals). Such structures are particularly interesting for the development of a fully integrated sensor microarray device for the simultaneous monitoring of multiple analytes. The phase shift Δφ, induced by molecular binding to the waveguide is a function of the induced effective index change of the waveguide mode ΔΝeff and the sensor length L: Δφ = 2π ΔΝeff L/λ. The effective index change is determined by the overlap of the waveguide mode electric field intensity with the captured molecules. It has been shown that the strength of this interaction grows with increasing refractive index contrast of the waveguide layers (Parriaux and Veldhuis 1998) and can therefore be maximized using the SOI platform (Densmore et al. 2006). ΔΝeff is a strong function of the waveguide core thickness as shown in Fig. 9a, where the optimum Si-core thickness (at a wavelength of 1,550 nm) is 0.22 μm for the more sensitive TM polarization. Figure 9b shows the calculated TM polarized electric field distribution for the 0.26 × 0.45 μm2 single mode photonic wire waveguide used in our work. In this regime, the waveguide dimensions are smaller than the wavelength of guided light, which greatly reduces the confinement of the optical mode in the core region. This results in the substantial part of the optical field traveling outside the waveguide core, yet the light being strongly confined in close proximity to the waveguide surface (Fig. 9b). This is beneficial for maximum interaction with the molecular layer formed on the waveguide surface.

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a)

b)

Fig. 9 (a) Surface sensitivity of an SOI slab waveguide as a function of waveguide core thickness at a wavelength of 1,550 nm. The optimum core thickness for the TM polarization is 0.22 μm. (b) Electric field distribution for the TM polarization for a 0.26 × 0.45 μm2 silicon photonic wire waveguide with an aqueous upper cladding.

Near this optimum silicon thickness, the single-mode guiding condition can be met using a photonic wire waveguide where the silicon layer is completely etched down to the buried oxide (lower cladding) layer. Long sensing path lengths are desirable as the number of molecules captured by the waveguide will increase with waveguide length, thereby amplifying the phase response. However, long straight waveguides are not well suited for the development of densely packed, two-dimensional sensor arrays. The latter rely on the use of microarray spotting tools to immobilize different receptor molecules on each sensor. In order to achieve compatibility with two-dimensional arrays using long waveguide lengths, we have developed sensors exploiting the high lateral index contrast of the photonic wire waveguide. This large lateral-index-contrast allows spiral waveguide geometries to be formed with radii of a few micrometers and having several millimeters of waveguide length folded within a small 150μm-diameter circular area, as shown in Fig. 10. The circular sensor geometry is well suited for two-dimensional sensor arrays and the spiral diameter is comparable to the drop size dispensed by common microarray spotters. Furthermore, the spiral waveguide design is also beneficial for use with microfluidic channels that are used to deliver small volumes of analyte solution to the sensor array and commonly have widths of several hundred micrometers. In the folded sensor configuration, a higher percentage of the channel floor is occupied per unit length of the channel by the folded sensing area. In this way, improved molecular capture efficiency is achieved compared to conventional straight waveguide geometries. To track the induced phase shift, the spiral waveguides were implemented into two sensor configurations: a Mach-Zehnder interferometer (Densmore et al. 2006) and a ring resonator (Xu et al. 2008). The Mach-Zehnder transforms the phase


a)

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100 μm Fig. 10 Top views of the fabricated sensors employing photonic wire spiral waveguide geometries: (a) Mach-Zehnder interferometer; (b) ring resonator sensor.

shift in the sensing arm of the interferometer into a readily observable intensity change at a fixed signal wavelength, while the ring resonator produces a measurable shift in the resonance wavelength. Figure 10 shows the top views of the MachZehnder interferometer (a) and the ring resonator (b) sensors. A 2-μm-thick SU-8 polymer layer was used to isolate the non-sensing regions of the device from the analyte solution, and it was removed over the sensing ring resonator cavity and the sensing arm of the Mach-Zehnder. To evaluate the individual sensor performance, we used the biotin – streptavidin binding system, which has well understood model behavior and very strong binding affinity. Biotin was immobilized on a 2 nm thick SiO2 layer formed with oxygen plasma on the silicon waveguide surface, using a common silanization process for glass surfaces. Streptavidin solution was then passed through a microfluidic channel aligned and attached to the sensing area of the device. A 1,550 nm laser signal was coupled into the sensors and the transmitted optical power was monitored for the MachZehnder sensor at a fixed wavelength. For the ring resonator sensor, the resonance wavelength peak position was tracked using a tunable laser. The binding curve (Fig. 11) shows the resonant wavelength shift of the ring resonator as streptavidin was captured from the solution. This experiment was repeated for different analyte concentrations (0.1, 1 and 10 μg/ml) and the binding rate is observed to scale proportionately. Based on the observed signal-to-noise ratios, we estimated a low mass detection limit of ∼5 fg of streptavidin, or a minimum surface coverage of less than 3 pg/mm2. These values are competitive with state-of-the-art surface plasmon resonance instruments and are expected to be further improved with upgrades of our measurement station for better control of flow rate and temperature. Our future work will focus on implementing the proposed sensor designs into an array format for the development of a fully integrated microarray biochip for multi-analyte sensing.

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Fig. 11 Ring resonator peak wavelength shift due to streptavidin binding to the waveguide surface for varying analyte concentrations.

5

Conclusions

In this chapter we have reviewed several interrelated aspects of our silicon photonics research. We discussed implementations of subwavelength grating structures in silicon waveguides. These subwavelength structures show great promise to overcome some important challenges, which have delayed practical implementation of this high index contrast technology platform, particularly largely inefficient fiberchip coupling and limited values of refractive indices (nSi and nSiO2) in the SOI platform. Using the sub-wavelength grating effect, it is possible to realize effective dielectric materials with a continuous range of intermediate refractive indices, which can be fabricated by standard nanofabrication techniques. We have also reviewed our work in the development of comparatively complex SOI planar waveguide circuits, including waveguide spectrometers and biosensors. The highresolution arrayed waveguide grating spectrometer and the Fourier-transform multiaperture spectrometer were introduced. The latter is the first planar waveguide implementation of a Fourier-transform spectrometer, and has advantages of a large light throughput and a static design with no moving parts. Finally, we have discussed recent developments in our laboratory in label-free molecular sensing using silicon photonic wire waveguides. The high index contrast of SOI waveguides provides excellent surface sensitivity and compact sensor designs, including new waveguide geometries that allow these sensors to be densely arrayed in two dimensions for compatibility with microarray spotters. At the same time, long interaction length and improved molecular capture efficiency are provided. Availability of


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similar sensor devices is an important prerequisite for practical implementations of label-free microarrayed biochips for multi-analyte monitoring. Acknowledgments We would like to thank Prof. Jacques Albert, Dr. Albane Laronche, Dr. Greg Lopinski, Dr. Ross McKinnon, Dr. Pedro Barrios, Trevor Mischki, Przemek Bock, Craig Storey, Dr. David Lockwood, Prof. Richard A. Soref, Prof. María L. Calvo, Prof. Brian Solheim, Dr. Alan Scott, Prof. Tatiana Alieva, and Dr. Jose Rodrigo for insightful discussions, and the Institute for Microstructural Sciences, National Research Council Canada, for continuous support. The support from the Canadian Space Agency, York University, COM DEV Ltd., and the NRC Genomics and Health Initiative, is also gratefully acknowledged.

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Index 1 100 kHz, 14 2 2.9–3.3 µm, 14 A Absorption coefficient, 122, 127, 128, 129 Absorption length, 124, 125, 127, 129 Acyclovir (ACV), 96 Angular momentum, 191 Anthrax, 68 Antibody, 238 Antigen, 235 Antireflective coating, 231 Antireflective waveguide facet, 229 Arrayed Waveguide Grating (AWG), 226 B Band gap, 125, 132, 133 Benzophenone, 96 Binding reaction, 238 Biochip, 228 Biological chromophore, 32 Biotin, 241 Bloch mode, 161, 163, 179, 183, 191 Bragg gaps, 199, 201 Bravais lattice, 165, 167 Breakdown threshold fluence, 129 Brillouin zone, 168, 169, 170 Bulk modulus, 137 Buried oxide, 240 C Carotenoids, 37 energy levels, 40 hot-S0, 40 Carrier envelope phase (CEP), 71 Cavity diameter, 140 CEP stability, 12 Character, 170, 180, 181, 183, 186, 188, 190, 191 Chirped-mirror recompression, 14 Coherence atomic, 75 maximal, 76 molecular, 78, 85

Coherent Anti-Stokes Raman Scattering (CARS), 57, 85 hybrid, 85 microscopy, 87 Conduction band, 128, 132 Coupled-mode, 209, 211, 214 Critical density, 128, 138 Cross section, 134 Crystallographic orbit, 173, 174, 183, 185, 186, 188, 190 Cut-off wavelength, 147 D Damage threshold, 122, 123 Degenerate four wave mixing, 45 Density of photon states, 108 Density of States (DOS), 108 local (LDOS), 114 Dielectrics, 121, 123, 125, 128, 142, 145, 146 Difference frequency generation, 13 Diffraction, 232 Disclination, 183, 190, 191 Disorder, 211, 215 Dispersion relation, 162, 195, 198 Dispersive response, 193 Displacement field, 133 DNA molecules, 238 Drude response, 193, 197 E Effective medium theory, 231 Efficiency of the shortening, 10 Efficient self-compression, 10 Electric current, 133 Electric plasmon mode, 205 Electromagnetically Induced Transparency (EIT), 75 Electron-beam patterning, 234 Electrons generation, 129 Electrospray-ionisation mass-spectrometry (ESI-MS), 32 Emission, 108 spontaneous, 108 Equations of State (EOS), 134 tuning, 143 étendue, 235 Evanescent field sensor, 258

247

248 F Fabry-Pérot, 216, 219 Few-cycle, 3 Few-cycle Mid-IR, 11 Fiber-chip coupling, 230 Fibonacci sequence, 203 Fibre Bragg grating, 234 Field enhancement, 109 enhancement factor, 111 envelope, 6 Filament, 7 Filamentation, 1, 2, 3, 6 Finite difference time-domain, 230 Fluorescent label, 238 Focal volume, 123, 125, 127, 130, 142, 144, 145 Fourier-transform CARS, 67 Fourier-transform spectrometer, 228 Fresnel power reflectivity, 231 Full revival, 23 G Gas density gradients, 8 Gas-filled hollow fibers, 11 GaAs, 209, 211, 221 Genomics, 238 Geometrical birefringence, 227 Golden mean, 202 Green function, 209, 211, 213 Group velocity, 6, 77, 210 Grüneisen coefficient, 132, 134, 135, 140, 142

INDEX K Ketoprofen, 96, 102 L Label-free biological sensor, 228 Laser Induced Forward Transfer, 144 Laser tight focusing, 125, 126 Latent heat of sublimation, 138, 140, 142 Localization, 25 Localization length, 219, 221 Long wavelength limit, 193, 205, 229 M Mach-Zehnder interferometer array, 236 Magnetic plasmon mode, 204, 205 Matter waves, 44 Maxwell operator, 167, 179 Maxwell’s equations, 125, 133, 134 Metal nanoparticles, 109 Metamaterial, 194, 209, 220 Microarray spotter, 242 Microfluidic, 241 Mid-IR, 11, 12 Mode size, 233 Modulation molecular, 78, 79, 82 Molecular capture efficiency, 242 Molecular Rydberg wave packets, 24 Multi-analyte monitoring, 225 Multiphoton ionization, 133 Multiplexer, 227

H Herpes Simplex Virus (HSV), 95 High index contrast waveguide, 236 Higher-order mode, 151 Hollow-fiber, 3 Hugoniot, 134, 135, 137, 140

N Nanocrystals, 118 Nanofabrication, 242 Nanojet formation, 144 Negative refraction, 194 Nonlinear propagation, 148 Numerical simulation, 6

I Incommensurate, 203 Infrared spectroscopy, 235 Interferogram, 237 Interferometric detection, 69 Intersystem crossing, 31 Ionization current, 133 model, 134 threshold, 125, 129, 139 Isentropic expansion, 131

O Oblique incidence, 194, 199, 203 OPCPA amplifier, 13 Optical low-coherence reflectometry (OLCR), 211 Optical force, 192 Optical path, 59, 87, 201 Optical telecommunications, 232 Optoelectronic integrated devices, 226 Orbital angular momentum composition, 23

INDEX P Partial revival, 21 Partner function, 164, 169, 170, 183, 184 Pathogen detection, 238 Phase carrier-envelope, 79 profile, 26 shift, 241 Phase-preserving, 4 Photocage, 95, 98, 99, 102, 103 Photochemical release, 95 Photochemistry, 28 Photodecarboxylation, 97 Photolabile protecting group. See Photocage Photonic band gap, 194 Photonic crystal, 57, 88, 109, 147, 161 Photonic Crystal Fiber (PCF), 68, 147 Photorelease. See Photochemical release Plasma etching, 234 Plasmon polaritons, 194 Plasmons surface, 110, 111 Polarization ellipse, 181, 183, 186 sensitivity, 228 Principal axis, 165, 166, 171, 175, 177, 183, 190 Protein, 118, 143 Proteomics, 238 Pulse characterization, 63 shaping, 59 Q Q-factor, 116 Quantum Control Spectroscopy (QCS), 38 depletion action trace, 42 electronic coherence, 50 enhancement of vibrational coherence, 43 mode filtering, 44 pump-depletion-probe, 38 Quantum dots, 117, 118 Quotidian Equations of State (QEOS), 134 R Ramsey detection method, 22 Rarefaction wave, 131 Refractive index, 125, 127, 128 Refractometric sensor, 234 Repetition rate, 14

249 Resonances, 109, 211, 235 Ring resonator, 240 RNA molecules, 238 Rowland circle, 234 Rude-like response, 197 Rydberg electron wave packets, 20 S Satellite sensing, 237 Satellite-based sensing, 232 Scattering, 108 Raman, 108 Rayleigh, 108 Self-compression, 3, 8 Self-focusing, 125, 126 Self-phase modulation, 155 Self-steepening, 157 SESAME EOS, 135, 136, 137, 138 Shaped, UV femtosecond laser pulses, 32 Shaper-Assisted Collinear SPIDER, 64 SHG-FROG, 14 Shock wave, 122, 130, 131, 135, 142, 145 Silanization, 241 Silica, 226 Silicon, 229 Silicon photonics, 242 Silicon wire waveguide, 227 Silicon-On-Insulator (SOI), 226 Single-cycle, 77 Singularity, 182, 187 Sinusoidal phase function, 45 SMF-28, 230 Soliton fission, 155 Spatial heterodyne spectroscopy, 235 Spatio-temporal properties, 7 Spectral Phase Interferometry for Direct Electric-Field Reconstruction (SPIDER), 64 Spectrometer, 228 Stimulated Raman scattering, 157 Streptavidin, 241 Stress engineering, 227 SU-8, 241 Subfemtosecond, 77 Subwavelength grating (SWG), 229 Sub-wavelength nanostructure, 228 Suitable pressure gradients, 10 Supercontinuum, 151 generation, 60 Surface plasmon resonance, 241 Surface sensitivity, 225, 239

250 T Three-dimensional (3D) structures, 122 Thue-Morse chain, 201 Time-domain single-beam CARS, 67 Time-frequency reflectance maps, 212, 217 Time-resolved photoelectron spectroscopy, 19 Transfer-matrix, 198, 201, 205 Transformation operator, 168, 170, 171, 172, 174, 177, 178, 179, 180, 183, 186, 190, 191 Transform-limited pulse, 4 Transmission, 84, 209, 216 Transverse electric (TE), 231 Transverse magnetic (TM), 231 U Ultrashort, 1, 11 Ultrashort pulse, 7 V Valence band, 132, 133 Void formation, 123, 144 Vortex, 191

INDEX W Waveguide birefringence, 231 core, 230 mirror, 229 propagation loss, 227 sidewall roughness, 227 Wavelength dispersive devices, 227 multiplexing, 232 resolution, 238 Wyckoff position, 177, 180, 186, 187, 190, 191 X Xanthone, 99, 100 Y Young’s double slit experiment, 20 Z Zeroth order gap, 193, 195, 201, 206

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