Plasmons: Theory and Applications [1 ed.] 9781617614217, 9781617613067

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Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

PHYSICS RESEARCH AND TECHNOLOGY

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

PLASMONS: THEORY AND APPLICATIONS

No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services. Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

PHYSICS RESEARCH AND TECHNOLOGY

PLASMONS: THEORY AND APPLICATIONS

KRISTINA N. HELSEY

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

EDITOR

Nova Science Publishers, Inc. New York Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

Copyright © 2011 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works.

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book.

LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA

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Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

CONTENTS Preface Chapter 1

Chapter 2

Chapter 3

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Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

i Metal Nanostructures with Controlled Profiles Fabricated by Nanosphere Lithography for Localized Surface Plasmon Resonance Xiaodong Zhou, Kai Yu Liu and Nan Zhang

1

Strategies for the Enhancement of Surface Plasmon Resonance Immunoassays for High Sensitivity Biosensors John S. Mitchell and Yinqiu Wu

43

Plasmon Dispersion and Damping in Two-Dimensional Electron Gases on Metal Substrates Antonio Politano

75

Evanescent Coupling between Resonant Plasmonic Nanoparticles and the Design of Nanoparticle Systems T. J. Davis

111

SPR-Facilitated Interrogation of Enzymatic Reactions Involving Nucleic Acids Wendi M. David

143

Surface Plasmon Assisted Microscopy: Reverse Kretschmann Fluorescence Analysis of Kinetics of Hypertrophic Cardiomyopathy Heart J. Borejdo, P. Mettikolla, N. Calander, R. Luchowski, I. Gryczynski and Z. Gryczynski

161

Two-Dimensional Plasmon Polariton Nanooptics by Imaging in Far-Field Andrey L. Stepanov and Joachim R. Krenn

179

Surface Plasmon Resonance Spectroscopy for Biomimetic Membrane Assembly and Protein-Membrane Interactions Studies Joël Chopineau, Laure Beven Daniel Ladant and Claire Rossi

201

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vi Chapter 9

Plasmonic Phenomena in Nanoarrays of Metallic Particles Victor Coello, Rodolfo Cortes, Paulina Segovia, Cesar Garcia and Nora Elizondo

Chapter 10

Application of Localized Surface Plasmons to Study Morphological Changes in Metal Nanoparticles T. A. Vartanyan, N. B. Leonov and S.G. Przhibelskii

Chapter 11

Chapter 12

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Contents

Surface Plasmon Resonance -- Applications in Nanoparticle Detection from a Liquid Matrix Using the Maximum Entropy Method Jarkko J. Saarinen, Erik M. Vartiainen, and Kai-Erik Peiponen Enhanced Transmission of Light and Matter Through Subwavelength Nanoapertures by Far-Field Multiple-Beam Interference S. V. Kukhlevsky

217

237

247

267

Chapter 13

Graded Plasmonic Structures and Their Properties Jun Jun Xiao, Kousuke Yakubo and Kin Wah Yu

289

Chapter 14

Infrared Surface Plasmon Spectroscopy of Living Cells M. Golosovsky, V. Yashunsky, A. Zilberstein, T. Marciano, V. Lirtsman, D. Davidov and B. Aroeti

327

Chapter 15

Basic Properties of Plasmons J.T.Mendonca

345

Chapter 16

Exciton-Plasmon Interactions in Individual Carbon Nanotubes Igor V. Bondarev, Lilia M. Woods and Adrian Popescu

381

Chapter 17

Surface Phenomena caused by Optical Spin-Orbit Interaction: A Pedagogical Theory using Quantum Mechanical Concepts I. Banno

Index

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PREFACE A plasmon is a quantum of plasma oscillation. The plasmon is a quasiparticle resulting from the quantization of plasma oscillations just as photons and phonons are quantizations of light and heat. Plasmons are collective oscillations of the free electron gas density, for example, at optical frequencies. Plasmons can couple with a photon to create another quasiparticle called a plasma polariton. This book presents topical research data in the study of plasmons, including the profile control of the obtainable gold nanostructures for localized surface plasmon resonance application; cutting edge techniques to improve signal sensitivity and assay performance for surface plasmon resonance immunobiosensors; the application of surface plasmon resonance (SPR) technology for elucidating enzymatic reactions involving nucleic acids; and graded plasmonic structures and their properties. Chapter 1 – Localized surface plasmon resonance (LSPR) is coherent oscillation of conductive electrons confined in noble metallic nanoparticles excited by electromagnetic radiation. Metal nanoparticles of LSPR can either be periodic or non-periodic on substrate or in solution, and nanosphere lithography (NSL) is one of the cost-effective methods to fabricate many kinds of metal nanostructures for LSPR. NSL can be categorized into two major groups: dispersed NSL and closely pack NSL. In dispersed NSL, nanospheres are dispersed on substrate, and each nanosphere serves as an individual mask for metal evaporation and etching. Dispersed NSL provides an abundance of metal nanostructure shapes as the metal can be evaporated several times at different angles and thicknesses, and the metal can be dry etched at any angle or wet etched. Through dispersed NSL, nanocrescents and nanoholes are reported for LSPR sensing. In closely pack NSL, nanospheres are arranged hexagonally or quadrangularly in mono or multiple layers, and adjacent nanospheres shadow each other during gold evaporation to form the nanostructures. When metal is evaporated at a small angle to the normal of the substrate, nanotriangles on the substrate or nanocaps on silica nanospheres can be used for LSPR; when metal is evaporated at a large oblique angle, patchy nanoparticles formed on the top of the nanospheres can be used for LSPR. However, for both dispersed NSL and closely pack NSL, the experimentally reported metal nanostructure shapes for LSPR are only a small amount of the numerous obtainable shapes. This chapter investigates in detail on the profile control of the obtainable gold nanostructures for LSPR application. The formulas for calculating the profiles obtained with different nanosphere masks, metal evaporation and etching conditions are derived, and software which can tackle with complicated fabrication process is programmed. The software

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simulated profiles coincide well with the profiles of the fabricated metal nanostructures observed under scanning electron microscope (SEM) and atomic force microscope (AFM), which proves that the software is a useful tool for the process design of different LSPR nanostructures. Besides the well-known nanoholes with vertical sidewalls, nanocrescents, nanotriangles and nanocaps for LSPR sensing, this chapter reports several novel noble metal nanostructures fabricated by NSL for LSPR sensing, including 3D nanostructures obtained by evaporating metal obliquely on dispersed nanospheres, nanoholes with sidewalls in slope, nanostructures obtained by dispersed NSL and wet etching, and metal patches on closely pack nanospheres. Chapter 2 – Surface plasmon resonance (SPR) has achieved widespread use as a transduction technique for biosensors whereby binding interactions on a noble metal surface can be followed in real-time by examination of minute changes in the refractive index of the surface. For a fixed surface, the amount of bound mass on the sensor surface is reflected by changes in the index of refraction at the surface, which in turn changes the angle at which peak plasmon resonance occurs. This transduction technique lends itself to incorporation with immunoassays, where either antibodies or their antigen binding partners are immobilized on the sensor surface and changes in antibody / antigen binding are used to reflect free antigen concentrations. The sensitivity of such assays is critically dependent upon the format of the sensor optics, the structure of the underlying metal surface, the chemical functionalization of the sensing interface and the format of the binding events that compose the assay. As many of the important biomolecules targeted by immunoassays are present only in very low concentrations in real samples, there are stringent sensitivity requirements. Many researchers have examined ways of enhancing the signal response from SPR immunobiosensors both through adding mass to the surface and / or through cooperative plasmon enhancements between the noble metal surface and a noble metal colloid or nanoparticle. In this chapter, the authors examine a wide range of cutting-edge techniques used to improve signal sensitivity and assay performance for SPR immunobiosensors. This includes using proteins as high mass labels in sandwich and competitive immunoassay formats; noble metal nanoparticle enhancement through high mass and cooperative plasmon effects; use of other high mass labels; the effects of sensor surface construction on sensitivity and enhancement; and the use of plasmonic enhancement techniques for plasmonic immunoassays where the sensing surface is not fixed but takes the form of a free-floating nanoparticle. Furthermore, the authors examine how fundamental changes to the structure, shape, and distance from nanoparticle to sensor surface affect SPR signal strength and look briefly at how this plasmonic theory can be applied beyond conventional SPR immunosensors into fluorescence, acoustic and electrochemical biosensors. The authors also assess how well these immunosensors have been applied to measurements in complex biological matrices and where these sensor technologies may lead in the future. Chapter 3 – Herein the authors report on high-resolution electron energy loss spectroscopy (HREELS) measurements on surface plasmon dispersion in systems exhibiting quantum well states, i.e. Na/Cu(111), Ag/Cu(111), and Ag/Ni(111). Their results demonstrate that the dominant coefficient of surface plasmon dispersion for thin and layer-by-layer Ag films presenting quantum well states is quadratic even at small q||, in contrast with previous measurements on Ag semi-infinite media and Ag thin films deposited on Si(111). The authors suggest that this behavior is due to screening effects enhanced by the presence of quantum well states shifting the position of the centroid of the induced charge less inside the

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geometrical surface compared to Ag surfaces and Ag/Si(111). For ultrathin Ag films, i.e. two layers, the dispersion was found to be not positive, as theoretically predicted. Annealing of the Ag film caused an enhancement of the free-electron character of the quantum well states, thus inducing a negative linear term of the dispersion curve of the surface plasmon. Moreover, the authors report the first experimental evidence of chemical interface damping in thin films for K/Ag/Ni(111). As regards Na/Cu(111), the authors found a different dispersion curve compared to thick Na films, thus confirming the enhanced screening by Na quantum well states. Results reported here should shed light on the influence of quantum well states on dynamical screening phenomena in thin films. Chapter 4 – Localized surface plasmon resonances in metallic nanoparticles arise from the interaction between the free electrons in the metal and their associated electromagnetic fields. The resonant frequencies depend on the geometry of the nanoparticle, its electric permittivity and the electric permittivity of the surrounding medium. When two or more nanoparticles come in close proximity, the evanescent electric fields associated with the surface plasmons interact with the surface charges, changing the resonant frequencies. In this chapter a theory of the interaction of plasmonic nanoparticles is presented. The theory is based on the “electrostatic" eigenvalue method that describes the localized surface plasmon resonances in nanoparticles of any shape provided they are much smaller than the wavelength of light. The theory leads to simple algebraic expressions for the coupling between nanoparticles that can be used to deduce the resonant properties of the nanoparticle ensembles. In particular, the theory is used to deduce the relative energies of the hybrid states associated with nanoparticle ensembles, it is applied to the understanding of dark modes and plasmon-induced transparency in metamaterials, and it is shown how the theory can be used to design and analyse plasmonic circuits. Chapter 5 - Processing of nucleic acids during transcription, translation, replication, and repair must proceed smoothly for survival. Numerous proteins interact with DNA and RNA to mediate these processes, ranging from those that allow access to the genome and participate in transmission of coded information to those that directly repair damaged areas of the genome. Interactions between proteins and nucleic acids are often investigated using polyacrylamide gel electrophoresis (PAGE) analysis, which necessarily yields information concerning the end result of a particular interaction. More recent analytical techniques to monitor intermediate or real-time interactions include fluorescence resonance energy transfer (FRET) spectroscopy and surface plasmon resonance (SPR). SPR is label-free, an advantage for interrogating many biomolecular interaction events. Although SPR has been used extensively to determine small molecule/DNA and protein/DNA binding interactions, SPR analysis of enzymatic reactions involving nucleic acids has lagged behind. However, several recent studies of enzymatic reactions that extend nucleic acids or unwind different topological forms of DNA demonstrate significant advantages of SPR for discerning aspects of these interactions that are difficult to “see” using other techniques. This chapter focuses on the emerging application of SPR technology for elucidating enzymatic reactions involving nucleic acids. Chapter 6 - It is believed that the alteration of the kinetics of interaction between actin and myosin causes a serious heart disease called Familial Hypertrophic Cardiomyopathy (FHC) by making a heart pump blood inefficiently. To check this hypothesis in ex-vivo heart, the authors constructed Surface Plasmon Assisted Microscope (SPAM) and used it in a

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reverse Kretschmann (RK) configuration. In SPAM fluorescence is the result of near-field coupling of fluorophores excited in the vicinity of the metal coated surface of a coverslip with the surface plasmons propagating in the metal. Surface plasmons decouple on opposite side of the metal film and emit in directional manner as a far-field p-polarized radiation. In RKSPAM a sample is illuminated directly by the laser beam. During contraction of heart muscle a myosin cross-bridge imparts periodic force impulses to actin. The impulses were analyzed by RK-SPAM by Fluorescence Correlation Spectroscopy (FCS) of fluorescently labeled actin. The rate of changes of orientation were significantly faster in contracting cardiac myofibrils of transgenic (R58Q) mice than of wild type (WT). These results suggest a way to rapidly diagnose this disease. Chapter 7 - A review of the experimental realization of key high efficiency twodimensional optical elements, built up from metal nanostructures, such as nanoparticles and nanowires to manipulate plasmon polaritons propagating on metal surfaces is reported. Beamsplitters, Bragg mirrors and interferometers designed and produced by elelectron-beam lithography are investigated. The plasmon field profiles are imaged in the optical far-field by leakage radiation microscopy or by detecting the fluorescence of an organic film deposited on the metal structures. It is demonstrated that these optical far-field methods are effectively suited for direct observation and quantitative analysis of plasmon polariton wave propagation and interaction with nanostructures on thin metal films. Several examples of two-dimensional nanooptical devices fabricated and studied in recent years are presented. Chapter 8 - Surface Plasmon Resonance (SPR) spectroscopy is a powerful technique for monitoring in real time interactions at surface/liquid or air interfaces. This detection method is widely used in the area of biomimetic membrane sensors. Biomimetic membranes are adaptable lipid supported structures which can be used to study protein or ligand association with cellular membrane. Different lipidic environments and membrane architectures have been described: vesicles, hybrid bilayer membranes, supported lipid bilayers, tethered lipid bilayers and vesicle layers. The formation of these architectures and their complete characterization involved complementary techniques such as fluorescence recovery after photobleaching, atomic force microscopy, quartz crystal microbalance and surface plasmon resonance spectroscopy. SPR spectroscopy technique has been found as a method of choice for studying the interaction of protein or peptides and membranes. The membrane binding properties of extrinsic proteins such as myristoylated proteins was followed on gold supported biomimetic membranes. SPR spectroscopy allowed the determination of the kinetic binding parameters and equilibrium constant of these interactions. The reconstitution of integral membrane proteins in solid supported membranes represents a major goal for the researchers specialized in biomimetic membranes. The supported tethered lipid bilayers, in which the bilayer delimits two distinct compartments, is considered to be the preferred architecture to mimic an authentic cell membrane. SPR spectroscopy was used to monitor the insertion and/or reconstitution of proteins in such supported lipid environments. It was demonstrated to be very appropriate for the study of peptides or proteins insertion into or translocation across lipid bilayers. Chapter 9 - Current experimental and theoretical investigations of plasmonic phenomena are indented to be the basis for miniaturization of photonics circuits with length scales much smaller than currently achievable, inter-chip and intra-chip applications in computer systems, and bio/sensor-systems. In this chapter experiments and numerical developments conducted to the understanding of this area are outlined. The authors focus their attention in the

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interaction of Surface Plasmon Polaritons (SPP) with arrays of nano-particles. Numerical simulations and experimental results of different SPP elastic (in-plane) scattering orders, and the operation of simple plasmonic devices are presented. Furthermore, non-linear microscopy, with a tightly focused laser beam scanning over a sample surface with different densities of nano-particles is presented. Finally, a scanning near-field microwave microscope is presented as an alternative technique that is reliable enough to be used as a check of potential plasmonic components that are based on nano-particle arrays. In general, the stability with respect to geometrical parameters and dispersion were the main features investigated in all the presented plasmonic phenomena. Chapter 10 - All characteristics of surface plasmons localized in small metal particles are known to depend strongly on the particles size and shape. In the size regime 5 to 50 nm these dependencies approximately separate. The form of the nanoparticle is almost solely responsible for the spectral position of the plasmon resonance, while the particle size defines its strength. Although this separation is only valid in the quasistatic limit and disregards scattering and other radiation effects, it leads to many useful applications. In particular, this approximation was used to extract information about the homogeneous widths of localized surface plasmons hidden under huge inhomogeneous broadening of as grown ensembles of supported metal nanoparticles. Quasistatic approximation is also instrumental when one decides to go beyond the plane statement of the existence of correlations between the optical spectra of the particles ensembles and their morphology and to draw some quantitative conclusions. In their recent study, the results of which are presented in this chapter along with a short overview of the results of others, the authors employ the optical absorption spectroscopy in conjunction with the simple theory based on the quasistatic approximation to characterize several new mechanism of the morphological changes that the nanoparticles undergo in the course of the light and heat treatments. In particular, the authors report on the reversible changes of the particles shape in the cycles of heating and cooling as well as the possibility to speed up these changes via illumination. Chapter 11 - Light interaction with a medium depends on the wavelength of the light and the structure of the medium. In the UV-visible spectral range the interaction of the electromagnetic field with a homogenous medium, which is either insulator, semiconductor or conductor, typically occurs with the electron system of the medium, which may be in solid, liquid or gaseous state. The interaction can be fully described with the aid of light absorption and dispersion in the medium. The situation becomes much more complicated when the authors wish to study light interaction with a binary- or multiphase system. The binary system can be, for instance, a liquid matrix that contains either macroscopic solid particles (suspension) or nanoparticles (colloid). In such a case, in addition to the light absorption and dispersion one has to deal with light scattering, which can be characterized using the models of Mie and Rayleigh for spherical and small inclusions, respectively. In this chapter the authors deal with the optical properties of a binary system, which is a colloid. The study of optical properties of colloids has a rather long history. Indeed, for instance, Maxwell Garnett studied the colors of metallic solutions already more than a hundred years ago, and presented an elegant theory to predict the optical properties of effective media, a theory that is even nowadays widely used in the description of optical properties of effective media. The investigation of metal nanoparticles in liquid matrix has been a popular field for a long time, and different physical laws governing colloidal particles have been described in the literature The optical properties of a colloid depend on the size, shape and structure of the nanoparticles

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their complex refractive index and thermodynamic condition of the ambient liquid. These all have a role in the color of the colloid, which is subject to change. This color is due to surface plasmon resonance (SPR) of the incident light with the metal nanoparticle. Visible light, which is incident on a metal nanoparticle, couples with the plasma oscillation of the electrons that are confined in the nanoparticle. The color and also scattering of light is usually measured by recording light transmittance of the colloid. Much of the research work has been on fundamental optical properties of nanoparticles in liquid environments, but also on applications as nanoprobes and imaging in medicine and biology . Nanotoxicology of liquid food is also an issue that has been recently raised. Chapter 12 - Subwavelength aperture arrays in thin metal films can enable enhanced transmission of light and matter (atom) waves. The phenomenon relies on resonant excitation and interference of the plasmon-polariton or matter waves on the metal surface. The authors show a mechanism that could provide a great resonant and nonresonant transmission enhancement of the light or de Broglie's particle waves passed through the apertures not by the surface waves, but by the constructive interference of diffracted waves (beams generated by the apertures) at the detector placed in the far-field zone. According to the model, the light beams generated by multiple, subwavelength apertures can have similar phases and can add coherently. If the spacing of the apertures is smaller than the optical wavelength, then the phases of the multiple beams at the detector are nearly the same and beams add coherently (the light power and energy scales as the number of light-sources squared, regardless of periodicity). If the spacing is larger, then the addition is not so efficient, but still leads to enhancements and resonances (versus wavelength) in the total energy transmitted (radiated). The authors stress that the plasmon-polaritons do not affect the principle of the enhancement based on the constructive interference of diffracted waves (beams) generated by the subwavelength apertures at the detector placed in the far-field zone. Naturally, the plasmonpolaritons could provide additional enhancement by increasing the power and energy of each beam. The Wood anomalies in transmission spectra of gratings, a long standing problem in optics, follow naturally from the interference properties of their model. The point is the prediction of the Wood anomaly in a classical Young-type two-source system. The authors’ analysis is based on calculation of the energy flux (intensity) of a beam array by using Maxwell's equations for classical, non-quantum electromagnetic fields. Therefore the mechanism could be interpreted as a non-quantum analog of the super-radiance emission of a subwavelength ensemble of atoms (the light power and energy scales as the number of lightsources squared, regardless of periodicity) predicted by the well-known Dicke quantum model. In contrast to other models, the enhancement mechanism depends on neither the nature (non-quantum electromagnetic waves, quantum light or matter) of beams (continuous waves or pulses) nor material and shape of the multiple-beam source (arrays of one- and twodimensional subwavelength apertures, fibers, dipoles, and atoms). The quantum reformulation of their model is also presented. The Hamiltonian describing the phenomenon of interferenceinduced enhancement and suppression of both the intensity and energy of a quantum optical field is derived. The basic properties of the field energy determining by the Hamiltonian are analyzed. Normally, the interference of two or more waves causes enhancement or suppression of the light intensity, but not the light energy. The model shows that the phenomenon could be observed experimentally, for instance, by using a subwavelength array of the coherent quantum light-sources (one- and two-dimensional subwavelength apertures, fibers, dipoles, and atoms).

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Chapter 13 - Plasmonic structures have extraordinary optical propertiesin terms of engineering surface plasmon for optical waveguiding, switching, sensing, and enhanced optical spectroscopies such as surface enhanced Raman spectroscopy (SERS) and strong nonlinear optical effects. In particular, there is a type of plasmonic structures which have spatially varying characteristics---graded plasmonic structures---that enable extremely flexible and strong confinement of surface plasmon in real space, making it easier to control hot spots, surface signal propagation distance and direction. The authors show that a peculiar kind of confined Plasmon modes, called plasmonic gradons, can occur in such graded plasmonic structures. The plasmonic gradons are different from either disorder-induced Anderson-type localization in random media or Bragg-gap type confinement that is supported by periodic structures. The authors identify several localization-delocalization transitions between various plasmonic gradons which are of distinct flavors and are uniquely sustainable in these graded plasmonic structures. To this end, the authors also discuss the interplay between gradon confinement and a variety of oscillations, such as Bloch oscillation and breathing-like oscillation. To have a deep understanding of the plasmonic gradon physics, an analogous problem in graded elastic lattices with tunable on-site potentials are proposed. Chapter 14 - The authors report a spectroscopic technique that combines the FourierTransform Infrared Spectroscopy with the Surface Plasmon Resonance. This tool enables sensitive infrared spectroscopy of liquid and solid objects in the attenuated total reflectance mode. The FTIR-SPR technique is similar to FTIR-ATR technique but has higher sensitivity due to resonance amplification of the surface electric field. The label-free FTIR-SPR technique is especially advantageous for living cell studies since it combines spectroscopic information inherent to FTIR with structural information provided by the Surface Plasmon Resonance. The authors discuss the FTIR-SPR technique for label-free studies of cell sedimentation and spreading on substrate and for surface plasmon spectroscopy. Chapter 15 – The authors describe the basic properties of electron plasma waves or plasmons, giving particular emphasis to the wave and particle aspects of their behavior. The authors discuss the plasmon dispersion relations, including the cases of relativistic and quantum plasmas. Finite plasmas are also discussed, including Mie oscillations and TonksDattner resonances in cylindrical and spherical geometries, and Trivelpiece-Gould modes. The authors then consider resonant kinetic processes associated with Landau damping of electron plasma waves by both electrons and photons. Nonlinear processes such as solitons and harmonic generation are considered. In what concerns the particle-like properties of electron plasma oscillations, the authors discuss the plasmon effective mass, as well as the plasmon effective charge. The authors also discuss plasmon processes in a time varying plasma, such as time refraction and plasmon acceleration. The authors then consider plasmon kinetic instabilities, and in particular plasmon beam effects. Finally, they describe the properties of plasmon states with orbital angular momentum, and their relevance to Raman scattering. Chapter 16 - The authors use the macroscopic quantum electrodynamics approach suitable for absorbing and dispersing media to study the properties and role of collective surface excitations --- excitons and plasmons --- in single-wall and double-wall carbon nanotubes. The authors show that the interactions of excitonic states with surface electromagnetic modes in individual small-diameter ≲ 1 nm) single-walled carbon nanotubes can result in strong exciton-surface-plasmon coupling. Optical response of individual

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nanotubes exhibits Rabi splitting ∼.1 eV, both in the linear excitation regime and in the nonlinear excitation regime with the photoinduced biexcitonic states formation, as the exciton energy is tuned to the nearest interband surface plasmon resonance of the nanotube. An electrostatic field applied perpendicular to the nanotube axis can be used to control the exciton-plasmon coupling. For double-wall carbon nanotubes, the authors show that at tube separations similar to their equilibrium distances interband surface plasmons have a profound effect on the inter-tube Casimir force. Strong overlapping plasmon resonances from both tubes warrant their stronger attraction. Nanotube chiralities possessing such collective excitation features will result in forming the most favorable inner-outer tube combination in double-wall carbon nanotubes. These results pave the way for the development of new generation of tunable optoelectronic and nano-electromechanical device applications with carbon nanotubes. Chapter 17 - This chapter gives a pedagogical treatment of optical system with plane interfaces (multilayers), emphasizing analogy with one-dimensional quantum system. Familiar optical phenomena such as refraction, total reflection, Brewster's total transmission, surface plasmon polariton, and optical tunneling effect, etc. are described in a manner compatible with quantum mechanics. To make clear the analogy between optical and quantum systems, dual vector potential is used as the minimum degrees of freedom of electromagnetic field in optical regime (the regime with negligible magnetic response of the matter). The source of dual vector potential is separated into surface and volume magnetic currents; the surface source is responsible for the singularity expressed by Maxwell's boundary conditions. Furthermore, it is shown that the surface source originates in the optical spin-orbit interaction and that the coexistence of spin-orbit interaction (surface source) and volume source is essential for surface effects such as Brewster's total transmission and surface plasmon polariton.

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Chapter 1

METAL NANOSTRUCTURES WITH CONTROLLED PROFILES FABRICATED BY NANOSPHERE LITHOGRAPHY FOR LOCALIZED SURFACE PLASMON RESONANCE Xiaodong Zhou1, Kai Yu Liu2 and Nan Zhang1

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1

Institute of Materials Research and Engineering, A*STAR (Agency for Science, Technology and Research), 3 Research Link, Singapore 2 School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore

ABSTRACT Localized surface plasmon resonance (LSPR) is coherent oscillation of conductive electrons confined in noble metallic nanoparticles excited by electromagnetic radiation. Metal nanoparticles of LSPR can either be periodic or non-periodic on substrate or in solution, and nanosphere lithography (NSL) is one of the cost-effective methods to fabricate many kinds of metal nanostructures for LSPR. NSL can be categorized into two major groups: dispersed NSL and closely pack NSL. In dispersed NSL, nanospheres are dispersed on substrate, and each nanosphere serves as an individual mask for metal evaporation and etching. Dispersed NSL provides an abundance of metal nanostructure shapes as the metal can be evaporated several times at different angles and thicknesses, and the metal can be dry etched at any angle or wet etched. Through dispersed NSL, nanocrescents and nanoholes are reported for LSPR sensing. In closely pack NSL, nanospheres are arranged hexagonally or quadrangularly in mono or multiple layers, and adjacent nanospheres shadow each other during gold evaporation to form the nanostructures. When metal is evaporated at a small angle to the normal of the substrate, nanotriangles on the substrate or nanocaps on silica nanospheres can be used for LSPR; when metal is evaporated at a large oblique angle, patchy nanoparticles formed on the top of the nanospheres can be used for LSPR. However, for both dispersed NSL and closely pack NSL, the experimentally reported metal

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Xiaodong Zhou, Kai Yu Liu and Nan Zhang nanostructure shapes for LSPR are only a small amount of the numerous obtainable shapes. This chapter investigates in detail on the profile control of the obtainable gold nanostructures for LSPR application. The formulas for calculating the profiles obtained with different nanosphere masks, metal evaporation and etching conditions are derived, and software which can tackle with complicated fabrication process is programmed. The software simulated profiles coincide well with the profiles of the fabricated metal nanostructures observed under scanning electron microscope (SEM) and atomic force microscope (AFM), which proves that the software is a useful tool for the process design of different LSPR nanostructures. Besides the well-known nanoholes with vertical sidewalls, nanocrescents, nanotriangles and nanocaps for LSPR sensing, this chapter reports several novel noble metal nanostructures fabricated by NSL for LSPR sensing, including 3D nanostructures obtained by evaporating metal obliquely on dispersed nanospheres, nanoholes with sidewalls in slope, nanostructures obtained by dispersed NSL and wet etching, and metal patches on closely pack nanospheres.

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1. INTRODUCTION In recent two decades, localized surface plasmon resonance (LSPR), a coherent oscillation of conductive electrons in nanostructured noble metal with light excitation, is broadly used for reflective index variation based biosensors [1-7], surface-enhanced spectroscopic methods such as surface-enhanced Raman spectroscopy (SERS) [8, 9], and killing of cancerous cells through plasmonic heating [10, 11]. In general, LSPR exists for all kinds of metal nanostructures. In LSPR, each metal nanostructure acts as an individual emitting element, and the oscillation peaks of the LSPR spectrum is determined by the shape, size, interdistance, and materials of the nanostructure and its adjacent media. Although single metal nanoparticle LSPR has been heavily investigated and demonstrated with the detection of biological samples, a large amount of nanostructures with identical shape and even distribution on an area with > 10 μm critical length is desired for practical applications, such as for the further fabrication of the LSPR chip into a microfluidic device or a microfluidic sensing array. Besides the requirements of being identical and even distribution in a large area, the fabrication of the nanostructures is also expected to be cost-effective, able to generate sharp corners on the nanostructures to enhance the plasmonic signal, and can be fine tuned to control the oscillation peaks of the metal nanostructures. Nanosphere lithography (NSL), which uses a group of dispersed or closely packed nanospheres on a substrate as a mask for nanostructure fabrication, is an ideal candidate for obtaining various nanoparticles or nanoholes for LSPR application [12-29]. In reality, nanospheres are difficult to be dispersed or closely packed on substrate; they tend to aggregate on the substrate due to the strong capillary force among them. However, recent studies have facilitated the methods of dispersing or closely-pack the nanospheres on various kinds of substrates. In dispersed NSL [12-20], after the nanospheres are dispersed on the substrate, each nanosphere serves as a separate mask for metal evaporation or metal etching; while in closely packed NSL [21-29], the nanospheres are closely packed in mono or multiple layers, the shadows of several adjacent nanospheres contribute to the formation of a nanostructure. NSL includes the critical processes of nanosphere arrangement and metal

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evaporation, and optional subsequent processes of metal etching and nanosphere removal. A more detailed description of these processes is summarized in Table 1. Most importantly, any parameter in these processes will severely affect the shape of the nanostructures, thus the process control in NSL for obtaining the desired nanostructures becomes crucial. In this chapter, a profile simulation method is introduced to correlate the shape of the metal nanostructures to the process conditions and parameters in NSL fabrication. A complete set of profile calculation software that covers most of the current NSL fabrication techniques is programmed, it is applied to predict and verify the gold nanostructures fabricated in our experiments, and high coincidence is demonstrated between the simulated profile and the profile of the fabricated nanostructures examined under scanning electron microscope (SEM) and atomic force microscope (AFM). Moreover, simulated results provide more information in two cases, one is for the fine and delicate metal nanostructures which might not be discerned by SEM and AFM, or might have been damaged prior to observation; another is for the metal nanostructures underneath the nanospheres. Besides the profile control of the NSL fabricated nanostructures, this chapter also presents some of the nanostructures created in our laboratory for LSPR applications.

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2. NANOSTRUCTURES OBTAINED BY DISPERSED NSL The fabrication process of dispersed NSL is shown in Figure 1(a). Through this fabrication process, it can be seen that various kinds of metal nanostructures can be acquired for LSPR. After metal evaporation as shown in Figure 1(b), based on the nanosphere diameter r, the metal evaporation thickness t and the evaporation angle θ, the metal on the substrate can either be detached from the metal on the nanosphere under the condition of r sin   t cos  r to form a 2D nanostructure on the substrate, as demonstrated in Figure 1(c) left; or be attached to the metal on the nanosphere to form a 3D nanostructure as demonstrated in Figure 1(d) left for conformal metal deposition and Figure 1(e) left for nonconformal metal deposition [17, 18]. Without etching, these 2D or 3D nanostructures can be directly used for LSPR, and many varieties can be obtained. For example, for 2D nanostructures, when the metal is evaporated vertically, individual nanocaps on the nanospheres can generate LSPR signal; after the removal of the nanospheres, nanoholes on the substrate can also be used for LSPR; and when the metal is evaporated at a small evaporation angle without forming a 3D nanostructure, ellipsoidal nanoholes on the substrate are obtained after the removal of the nanospheres and can be employed for LSPR detections. In the next step, the evaporated metal nanostructures can be further anisotropically dry etched vertically or at an oblique angle θe, or isotropically wet etched to etch away the metal on the substrate and obtain single nanoparticles [12, 13, 17-20]. Some such fabricated nanostructures for LSPR include nanocrescents obtained by dry etching [12, 13], and clustered metal nanostructures by wet etching [20]. In dry etching, the substrate and the nanosphere might be also etched at a different etching rate compared with the metal [17]. The right parts of Figures 1(c)-1(e) demonstrate some 2D and 3D nanostructures after dry etching, when supposing the substrate and the nanosphere are etch resistant.

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Xiaodong Zhou, Kai Yu Liu and Nan Zhang Table 1. Fabrication processes to obtainable metal nanostructures by NSL Description Step 1. Arrange nanospheres on the substrate

Step 2. Evaporate metal onto the nanospheres Step 3. Etch metal

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Sept 4. Remove nanospheres

Dispersed NSL for dispersed nanostructures Randomly disperse the nanospheres on substrate.

Perpendicularly or obliquely evaporate the metal once, or obliquely evaporate the metal several times at different angles. Dry or wet etch the metal with different thickness. Dry etching is anisotropic, and wet etching is isotropic. For polystyrene nanospheres, dissolve by 5 to 10 min of sonication in toluene or ethanol, or burn at 350°C for 90 min.

Closely packed NSL for periodic nanostructures Closely pack the nanospheres hexagonally or quadrangularly in one to several layers, and the nanospheres can further be size reduced by heat shrinkage or plasma etching. Evaporate metal at any angle, the metal might be only on the nanospheres, or on the substrate, or on both, depending on the evaporation angle. Not necessary

For polystyrene nanospheres, dissolve by 5 to 10 min of sonication in toluene or ethanol, or peeled off with transparent tape. The peel-method can be used to remove silica nanospheres.

The final step is to remove the nanosphere after metal etching. This is optional, but the existence of the nanospheres will affect the spectral of the LSPR signal. The nanospheres are removed by dissolution in toluene or ethanol with 5 to 10 min of sonication, or burning at 350°C for 90 min. As dispersed NSL provides metal nanostructures with numerous varieties in shape, it is important to conduct the profile simulation for the nanostructures based on various factors including nanosphere size; metal evaporation number; metal evaporation angles and thicknesses; conformal or non-conformal metal deposition; dry or wet etching, etching angle, thickness, and relative etching rates for the nanosphere and the substrate; etc. In this section, the simulation formulas are derived, experimental results are presented to support our simulation results, and some complicated simulation examples are demonstrated. Our simulation method can be used to predict and design the profile of metal nanostructures for LSPR application, and this is more important for the nanostructures with profiles beyond simple imaginations when sophisticated engineering processes are involved.

2.1. Simulation Formulas for Dispersed NSL Space geometry is used to simulate the metal nanostructures obtained by NSL [17-20]. In order to simulate the nanostructure around a nanosphere after metal evaporation, five layers, namely “bottom”, “top1”, “top2”, “middle” and “top3” are calculated to form the whole complicated metal nanostructure, as indicated in Figure 2(a) for conformal metal deposition and Figure 2(b) for non-conformal metal deposition. They respectively represent the metal on the substrate, on the lower and top parts of the nanosphere, and on the lower and top parts of the metal deposition outline. In the software, each layer is a data matrix, and they are drawn together to form the 3D profile of the metal nanostructure.

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Metal Nanostructures with Controlled Profiles Fabricated by Nanosphere…

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Figure 1. NSL process with dispersed nanospheres and the fabricated nanostructures. (a) is the fabrication process with dispersed nanospheres, (b) is the condition for the formation of 2D or 3D nanostructures, (c) shows the profiles of gold deposition at r = 100 nm, t = 50 nm, θ = 20˚, before and after 47 nm of gold etching, (d) shows the profiles of gold deposition at r = 100 nm, t = 50 nm, θ = 60˚, before and after 25 nm of gold etching, when the deposition is assumed to be conformal, and (e) is with the same conditions as (d), except that the deposition is assumed to be non-conformal. The gold etching thickness is set to exactly etch away all the gold on the substrate. The light blue layer is the original substrate, and the gray ball represents the nanosphere

2.1.1. Nanostructure profile after metal evaporation a. Nanostructure profile for conformal metal evaporation For the convenience of calculating the profile of the nanostructure after metal evaporation, four inter-transformable coordinate systems are introduced: the original coordinate system xo-yo-zo, where the gold is evaporated at the angles of θ and φ (Figure 2(a)); the coordinate system xφ-yφ-zφ (where yφ = yo) with φ = 0, θ ≠ 0 (Figure 2(a)); the coordinate system xθ-yθ-zθ (where zθ = zφ) with φ = 0, θ = 0 (Figure 2(c)); and the coordinate system xφc-yφc-zφc for non-conformal gold deposition (Figure 2(b)), where the non-conformal angle θc is the angle between yθ and yφc. θc can be positive or negative depending on the materials and evaporation conditions. When θc is negative, the non-conformal part forms an undercut instead of the extension in Figure 2(b). When the metal is evaporated several times at different angles and thicknesses, the profile will look like Figure 2(d).

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Xiaodong Zhou, Kai Yu Liu and Nan Zhang

a

yo (yφ) top3

zφ zo

θ r φ top1 xφ

top2 θc xφc middle 0 xφ top1 t zφ (zφc)

middle xo t

bottom

bottom yθ

yφ

d

yo

c

t θ

top3 θc

top2 0

b

yφ

yφc

xφ

tk

t

0

0

xθ

xo

zφ (zθ) zo

ye

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yo

yo

e

f

θ

xo θe 0

xe

0 t

zo

θ r φ

xo

zo

Figure 2. Metal nanostructure around a nanosphere after oblique metal evaporation and etching. (a) is 3D conformal metal nanostructure after metal evaporation; (b) is 3D non-conformal metal nanostructure when the non-conformal angle θc is positive; (c) is for calculating the thickness of the metal after evaporation in the xθ-yθ-zθ coordinate system, where the metal looks as if evaporated from the top of the nanosphere; (d) is the nanostructure after multiple metal evaporations; (e) is the coordinate system for oblique dry etching; and (f) is the profile after isotropic wet etching.

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Metal Nanostructures with Controlled Profiles Fabricated by Nanosphere… According to Figure 2(c), the metal evaporated on the nanosphere has the shape of [18]

x  2

y2 t  1   r 

2

 z2  r 2

(1)

where t is the thickness of the metal, r is the radius of the nanosphere. When the metal is deposited at a direction with θ ≠ 0 and φ = 0 in the xφ-yφ-zφ (where yφ = yo) coordinate system, by making the coordinate transformation

 x  x cos  y sin    y  x sin   y cos

(2)

the metal deposited on the nanosphere after one deposition at the angle θ is

Ayo2 2Bx yo  Cx  z  r 2  0 2

2

(3)

where A  1  T cos2  , B  T sin  cos , C  1  T sin 2  , and

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2  2t t  T   k  k2   r r  

2

 tk  1   . r 

If φ ≠ 0, one more coordinate transformation of Eq. (3) will yield the metal profile on the nanospheres with θ ≠ 0 and φ ≠ 0. Because

 z    x o sin   z o cos   x  x o cos  z o sin    y  y o

(4)

Eq. (3) can further be derived as

A1 y o2 2B1 yo  C1  0

(5)

2 2 2 2 with A1  1  Ta 11 , B1  Ta 11 a 22 , C1  xo  z o  r  Ta22 , a11  cos  , and 2

a22  sin  xo cos   zo sin   .

The areas fulfill x2  z  r 2 has no metal evaporated on the glass substrate, while 2

other areas of the substrate have a metal deposition thickness of t cos .

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Xiaodong Zhou, Kai Yu Liu and Nan Zhang

For multiple metal evaporations, the gross thickness on the substrate is calculated by adding up the thickness of each deposition at each point. The thickness on the nanosphere is by adding the gold thickness tk in Figure 2(c) along the direction of each deposition. tk can be calculated as

 x  x cos  r 2  x 2  z 2 sin        2 2 2  tk  t (k ) r  x  z r under the condition x 

(6)

r 2  z2 , so in the xe-ye-ze coordinate system, the increments of

the thickness along the xe, ye, and ze directions are respectively

 zae  tk sin  sin    xae  tk sin  cos  yae  tk cos 

(7)

The new point in the xe-ye-ze coordinate system after one more metal evaporation is (xe’, ye’, ze’), which can be calculated by

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 xe '  xe  xae   y e '  y e  yae  z '  z  zae e  e

(8)

b. Nanostructure profile for non-conformal metal evaporation The above formulas are for conformal metal deposition. To calculate the metal profile for non-conformal evaporation, the coordinate system xφ-yφ-zφ is transferred into xφc-yφc-zφ by

 xc  x cos c  y sin c   yc   x sin c  y cos c z  z   c   x  xc cos c  yc sin c  y  x sin  y cos c c c c    y  yo 

(9)

inserting Eq. (9) into Eq. (3), we get

A0 y2c  2B00xc yc  C00 x2c  z 2  r 2  0

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(10)

Metal Nanostructures with Controlled Profiles Fabricated by Nanosphere… where A0  1  T cos2 (  c ) , B00 

9

T sin2  2c  , C00  1  T sin2 (  c ) , and 2

 xc  xo cos cos c  yo sin  c  zo sin  cos c   yc   xo cos sin c  yo cos c  zo sin  sin c   zc  z   xo sin   zo cos

(11)

xφc, yφc, and zφ are obtained by inserting Eq. (4) to Eq. (9). According to the gold profile on the nanosphere derived in Eq. (10), its tangent surface for the non-conformal structure is the surface that has only one intersection point with the gold profile in the xφc-yφc-zφ coordinate system, i.e., Eq. (10) should only have one solution for yφc. So we obtain the equation for calculating the non-conformal structure as 2 2 B00 xc  A0 (C00 x2c  z 2  r 2 )

(12)

which can be simplified and expressed in the xφc-yφc-zφc coordinate system as

(1  T ) x2c  A0 z2c  A0r 2  0  2t k t k 2   T   2  where A0  1  T cos (  c ) ,  r r   2

(13) 2

 tk  1   , θ is the gold evaporation r 

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angle, θc is the non-conformal angle in Figure 2(b).

c. Non-conformal metal evaporation at a vertical angle Base on our experimental observation under AFM, when gold was vertically evaporated onto the dispersed nanospheres, there was a slope on the gold nanoholes after the removal of nanosphere. This means the vertical metal evaporation is non-conformal with an angle  formed in the xo-yo-zo coordinate system. Depending on the degree of this angle, two cases are shown in Figure 3(a), where in the case of angle 1, only a metal slope is formed; while in the case of angle 2, one part of the metal profile follows the slope, while the other part fills around the bottom of the nanosphere. With the non-conformal angle  , in the xo-yo-zo coordinate system, the slope of the metal evaporation can be expressed as

ctg 

r  xo2  z o2  r  t  yo

As shown in Figure 3(b), the critical condition for the case of angle 2 is

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(14)

10

Xiaodong Zhou, Kai Yu Liu and Nan Zhang

a

yo

yo t

t

r -(r-t) yo

γ zo

r xo

0

A+

γ

-(r-t)

xo

0

A

t angle 1

b

t

γ rcosγ zo

angle 2

Figure 3. Non-conformal vertical gold evaporation for calculating the nanohole with slant sidewall. (a) shows two cases for different non-conformal angle  , (b) shows the critical angle of  for the angle 2 case to appear

r  r cos  r sin   t sin 

(15)

The point A+ in Figure 3(a) can be found by the solution of

ctg (r  x 2  z 2 )  r  t  y o o o  2 2 2 2  xo  z o  y o  r

(16)

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Equation (16) is actually



r 2  xo2  z o2  (r  t  r  ctg )  xo2  z o2  ctg



2

(17)

Let F  r  t  r  ctg , the solution of A+ is solved to be

A  F sin  cos  sin  r 2  F 2 sin 2 

(18)

The condition for Eq. (18) to have a solution is F sin   r , which is exactly the same as Eq. (15). In the case of angle 2, for a specific point (xo,zo), if A  xo2  zo2  r , the metal profile of this point is calculated by Eq. (14); if 0 

xo2  zo2  A , the point is calculated by

the profile of bottom part of the nanosphere.

2.1.2. Nanostructure profile after metal etching a. Nanostructure profile after vertically metal etching For vertical metal etching, the thickness of the metal is reduced by the etching thickness from the top to the bottom of the substrate. Depending on the etching process, the etching can

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be categorized as selective etching, i.e. the nanospheres and the substrate are not etched during etching as in the case of inductively coupled plasma (ICP) etching, and non-selective etching, i.e. the nanospheres and the substrate are etched as in the case of argon (Ar) milling. In our simulation, two relative etching ratios for the nanospheres and the substrate to the metal are set. During the etching, the secondary sputtering usually creates another layer of metal around the nanospheres. The secondary sputtering is not considered in our simulation, because the effect of the secondary sputtering is difficult to judge. The only way to eliminate the secondary sputtering is by etching the metal continuously for some time after the removal of the nanospheres. The continuation of the etching is simulated in our software, because it can be used to get rid of the secondary sputtered metal layer, and trim the 3D metal nanostructures after the removal of the nanospheres.

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b. Nanostructure profile after oblique metal etching Generally we can suppose the oblique etching has the azimuthal angle φe = 0, because the azimuthal angle is already variable in the simulation for each deposition. If the etching direction and yo–axis (in the xo-yo-zo coordinate system) forms an oblique etching angle of θe, we transform the coordinate system xo-yo-zo into xe-ye-ze as shown in Figure 2(e), where the etching appears to occur perpendicularly from top to bottom, while the substrate is obliquely placed. For oblique etching, the calculation starts from the point (x,y,z) in the xe-ye-ze coordinate system, so all of the above equations should be transformed into this coordinate system. The relationship between the xe-ye-ze and xo-yo-zo coordinate systems is

 x  xo cos e  yo sin  e  y  x sin   y cos o e o e  x  x cos   y sin   o e e  y   x sin   y cos e e  o  zo  z

(19)

By Eq. (19), the gold profile after deposition on the nanosphere has the same expression as Eq. (5), except a11 and a12 should be replaced by

a11  cos cos e  sin  sin  e cos  a22  x(sin  cos e cos  cos sin  e )  z sin  sin 

(20)

To calculate the non-conformal 3D nanostructure, we combine Eqs. (19) and (4) to obtain

 x  ( x cos e  y sin  e ) cos  z sin    y   x sin  e  y cos e   z  ( x cos e  y sin  e ) sin   z cos

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(21)

12

Xiaodong Zhou, Kai Yu Liu and Nan Zhang On further substituting Eq. (21) into Eq. (9), we have

  xc  (cos e cos cos c  sin  e sin  c ) x  (sin  e cos cos c  cos e sin  c ) y  z sin  cos c    z   x cos e sin   y sin  e sin   z cos

(22)

Eq. (13) is valid for non-conformal 3D structure calculation in this oblique etching coordinate system, as long as it uses the xφc and zφ deduced in Eq. (22) instead of the ones in Eq. (11). For the metal deposited on the substrate, it is known that when x2  z 2  r 2 is fulfilled, there is no metal deposited on the substrate. For oblique etching, xθ and zφ in x2  z 2  r 2 should be expressed by (x,y,z) in the xe-ye-ze coordinate system, which is  x  (cos e cos cos  sin  e sin  ) x  (sin  e cos cos  cos e sin  ) y  z sin  cos   z   x cos e sin   y sin  e sin   z cos

(23)

derived from Eqs. (21) and (2). Through Eqs. (5) and (20), Eqs. (13) and (22), the condition x2  z 2  r 2 and Eq. (23), a complete solution of the metal profile deposited on the nanosphere and the substrate for oblique etching calculation is acquired.

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c. Nanostructure profile after wet etching We suppose the wet etching is isotropic that all points exposed to the etchant are etched by a thickness of te along the normal direction of each point. For a quadric a11 x 2  a22 y 2  a33 z 2  2a12 xy  2a13 xz  2a23 yz  2a1 x  2a2 y  2a3 z  a4  0 (24) 2 2 2 2 2 2 a11  a22  a33  a12  a13  a23  0 , the normal to the surface at a point N 0 ( x0 , y0 , z 0 ) on this surface is [30]

where

x  x0 y  y0 z  z0   a11 x0  a12 y0  a13 z 0 a1 a12 x0  a22 y0  a23 z 0 a 2 a13 x0  a23 y0  a33 z 0 a 3

(25)

If the gold being etched is on the conformal part, based on Eq. (1), in the xθ-yθ-zθ coordinate system, the normal to the surface at a point P(xθ, yθ, zθ) is [20]

1  t r  ( y  y )  1 ( z  z ) 1 ( x  x )    x y z 2

(26)

After wet etching, point P turns into a new point P’(xθ’, yθ’, zθ’) on the wet etched surface, with the relationship of Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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x  x '2   y  y '2  z  z '2  te2

13 (27)

P’ is also on the normal to the surface that xθ’, yθ’ and zθ’ satisfy Eq. (26). Based on Eqs. (26) and (27), P’ can be calculated by

  x '  x  te  x     y '  y  te  y     z '  z  te  z 

x2 

y2  z2 4 1  t k r 

  y2  1  t r 2 x 2   z2  k  4   1  t k r    x2 

(28)

y2  z2 4 1  t k r 

If the gold being etched lies on the non-conformal part, by Eq. (13), in the xφc-yφc-zφc coordinate system, the equation of the normal to this surface at the point P(xφc,yφc,zφc) is

x  xc z  zc  (1  t ) xc A0 zc

(29)

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After wet etching, the new point P’(xφc’,yφc’,zφc’) is on the normal to the surface, it satisfies Eq. (29) as well as the condition

x

c

 xc '  zc  zc '  te2 2

2

(30)

So P’(xφc’,yφc’,zφc’) is obtained by

(1  t ) xc te   xc '  xc  (1  t ) 2 x2c  A02 z2c   A0 zc te z '  z  c c  (1  t ) 2 x2c  A02 z2c 

(31)

Since the calculated P’(xθ’, yθ’, zθ’) or P’(xφc’,yφc’,zφc’) can be converted back to the xoyo-zo coordinate system, Eqs. (28) and (31) can be used to calculate the profile of the gold nanostructure after wet etching.

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Figure 4. SEM and AFM of the fabricated nanostructures. (a) is the SEM of 50 nm thick of gold evaporated at 70˚ onto 170 nm diameter nanospheres dispersed on glass substrate, the inset on the top left is a magnified view. (b) is the obtained 3D nanostructures after RIE etching of the gold on the substrate and removal of the nanospheres

Figure 5. Simulated profiles of the gold nanostructures after gold deposition and gold etching, with the same conditions as Figure 4, where light blue represents the glass substrate, the gray ball represents the nanosphere. The top and bottom rows show the top and side views of the nanostructures, respectively. (a) is the profile of a 170 nm diameter nanosphere after 50 nm of gold is deposited at 70˚, (b) is the magnification of (a), and (c) is the gold nanostructure after 17nm of perpendicular gold etching

2.2. Experiment Verifications 2.2.1. 3D gold nanostructures fabricated by dispersed NSL Simple 3D gold nanostructures were fabricated according to the processes illustrated in Figure 1(a) [18]. The polystyrene nanospheres in diameter of 170 nm were drop coated and dispersed on the substrate, and 50 nm of gold was evaporated at a deposition angle θ = 70˚. Figure 4(a) is a SEM picture showing the nanospheres dispersed on the glass substrate after gold deposition. The gold film on the substrate was perpendicularly etched with Oxford Plasmalab 80 Plus reactive ion etching (RIE) etcher until no gold existed on the substrate. The gold thickness on the substrate is about 17 nm, calculated by t cos . After gold etching, the

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nanospheres were dissolved by 10 min of sonication in toluene, and the leftover nanostructures were observed with AFM as shown in Figure 4(b). According to the condition r sin  t cos  r , 3D gold nanostructures were formed, and Figure 4(b) proves that the nanostructure is non-conformal. Figure 5(a) gives the simulated top view and side view of the 170 nm diameter nanosphere after 50 nm of 70˚ gold deposition, Figure 5(b) is the magnification of Figure 5(a), and Figure 5(c) is the top view and side view of the nanostructure after 17 nm of perpendicular etching. In the simulation, the slope of the non-conformal structure is set to be θc= 10°. The top view in Figure 5(a) looks the same as Figure 4(a), and it also provides the side view which is indiscernible in the SEM top view. The 3D non-conformal nanostructure in Figure 5(c) is similar to Figure 4(b). Second sputtering sometimes happens during the gold etching, and its severity is subject to the etching method. Because this is not considered in our simulation, a slight discrepancy between the fabricated and simulated 3D nanostructures exists, probably due to the second sputtering and some loss of detail in morphology of the AFM picture.

2.2.2. Real time LSPR spectra during the wet etching Another experiment was to wet etch the sample in an undiluted gold etchant, and measure its LSPR spectra at different etching intervals to control the etching process [20]. Silica nanospheres were used. As silica nanospheres adhere to the glass substrate tightly due to the capillary force and remain the same after wet etching, the sample used to measure the LSPR spectrum was with the silica nanospheres on. After evaporating 50 nm of gold onto the nanospheres of 175 nm in diameter at 70°, the obtained sample is as shown in Figure 6(a). The gold evaporation under this condition forms a 3D nanostructure around the nanosphere. The gold is etched in an etchant formulated with 95% MilliQ water, 4% potassium-iodine and 1% iodine. At the etching durations of 1, 2, 3, 4, 5, 6, 8, 10, 15, 20, 30 min, the sample was taken out, rinsed with DI water and dried for LSPR measurements, with the resultant spectra presented in Figure 6(b). The peak wavelengths of the LSPR spectra after 1-6 min of wet etching are respectively 543.4, 531.6, 537.5, 540.1, 541.4 and 588.2 nm, as plotted in the insert of Figure 6(b), and the LSPR peak disappeared after 8 min. Because it is difficult to inspect the gold nanostructures under the silica nanospheres by SEM or AFM, we use some profile simulations to explain the trend of the LSPR shift.

a Figure 6. Continued

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c t = 0 min

d t = 1 min

e t = 2 mins

f t = 3 mins

g t = 4 mins

h t = 5 mins

i t = 6 mins

j t = 8 mins

Figure 6. Continuous wet etching of gold nanostructures in the etchant. (a) is SEM image of the sample before wet etching, which is obtained by evaporating 50 nm of gold onto the nanospheres of 175 nm in diameter at 70°, (b) shows the LSPR spectra of the sample after being wet etched for 1, 2, 3, 4, 5, 6, 8, 10, 15, 20, and 30 min, (c)-(j) illustrate the shapes of the gold nanostructure before etching and after being etched 1, 2, 3, 4, 5, 6, and 8 min, respectively.

Empirically, we know 3D gold nanostructure fabricated by gold evaporation is nonconformal with an angle θc. By fitting of the simulated profile, the possible gold nanostructures should have a non-conformal angle θc = -10°. Because under this condition, after etching 8.55 × 8 = 68.4 nm of gold, only a little gold remains on the substrate as demonstrated in Figure 6(j), thus no LSPR spectrum is distinguishable. Taking θc = -10°, we further simulated the nanostructure profiles after 0, 1, 2, 3, 4, 5, and 6 min of etching as illustrated in Figures 6(c)-(i). Comparing the size and shape variations of these nanostructures by time, the blue shift in the first 1-2 min was due to the quick size reduction [31, 17]; the peak wavelength kept almost unchanged at 3-5 min, because the reduced size of the nanostructure tended to blue-shift the spectrum, while the reduced thickness to width ratio red-shifted the spectrum [31, 32], two effects cancelled out and did not exhibit obvious spectra shift; at 6 min, the LSPR spectrum red-shifted a lot, because the thickness to width ratio of the nanostructures was greatly reduced when the gold on the nanosphere is etched

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away. This continuous peak tuning is an important feature for LSPR biosensing. In this experiment only 60 nm of LSPR wavelength shift was observed, because the original size of the silica nanospheres was only 175 nm. We expect to have larger LSPR wavelength shift range with larger silica nanospheres and thicker gold deposition. As drawn in the insert of Figure 6(b), the extinction of the LSPR spectrum reduces with nanoparticle size reduction, because absorbance scales with the volume of the nanoparticle and scattering scales with the volume squared [1].

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2.2.3. Nanoholes obtained with non-conformal gold evaporation Nanoholes were fabricated by dispersing 110 nm-diameter polystyrene nanospheres on the glass substrate [19], evaporating 40 nm of gold film onto the nanospheres, and removing the nanospheres by sonication in water for 40 seconds. The SEM images of the fabricated nanoholes are shown in Figure 7(a), where the density of our nanoholes is about 3/μm2 and the average interval of the nanoholes is 2 times of the diameter. The AMF image in Figure 7(b) indicates that the nanohole’s cross section is not a square but a slope. This is due to the non-conformal gold evaporation of the thermal evaporator, as the capillary force attracted gold atoms to deposit under the PS nanospheres during gold evaporation. Such a profile is different from previously reported nanoholes [15, 33], and it benefits the bonding of more biomolecules in the plasmonic enhanced area.

Figure 7. SEM and AFM images of 110 nm gold nanoholes. (a) is the SEM image, (b) shows the AFM image and the cross section of the nanoholes, (c) is the simulated profile after 40 nm gold evaporation at a non-conformal angle of 45˚, (d) is the shape of the nanohole on the substrate after removing the nanospheres

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The simulated gold profile after gold evaporation is presented in Figure 7(c), and the nanohole obtained after the removal of the nanosphere is shown in Figure 7(d). In these simulations, we suppose the non-conformal angle γ is 45˚. It can be observed that Figure 7(d) has the same profile as the one measured by AFM in Figure 7(b).

2.3. Complicated Simulation Examples

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2.3.1. The 3D multiple depositions and oblique etching A nanostructure with 3 depositions is calculated with our program [18]. The metal is deposited 3 times at φ = 0°, 120°, 240°, θ = 60°, 60°, 60°, with the same thickness of 40 nm, onto a nanosphere of 105 nm in radius. The top and side views of the metal profile after 3 depositions are shown in Figure 8(a), and the metal profiles after etching under various etching conditions are presented in Figures 8(b)-8(i).

Figure 8. The metal profiles obtained after three times of metal depositions and after subsequent etching. The deposition angles are θ = 60°, 60°, 60°, the azimuthal angles are φ = 0°, 120°, 240°, the radius of the nanosphere is 105 nm, the gold thickness for each deposition is 40 nm. The light blue plane represents the original substrate, and the gray represents the nanosphere. (a) is the side and top views after the depositions, (b) is after 80 nm of perpendicular etching, (c) is after 80 nm of 30° oblique etching, (d) is after 150 nm of perpendicular etching, (e) is after 150 nm of 30° oblique etching. (b)-(e) assume the nanosphere and the substrate are not etched. (f)-(i) have the same etching conditions as (b)(e), except they assume that the nanosphere and the substrate are equally etched as the metal

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As discussed in the simulation methodology, depending on the etching method and agent, the nanospheres and the substrate may or may not be etched during the metal etching. So we have simulated two sets of nanostructures etched under the same conditions, except that the assumption made in Figures 8(b)-8(e) is that the nanosphere and the substrate are unetchable, while the assumption made in Figures 8(f)-8(i) is that the nanosphere and the substrate are etched at the same rate as the metal. In Figures 8(b)-8(e), Figure 8(b) is simulated with 80 nm of perpendicular etching, Figure 8(c) is simulated with 80 nm of 30° oblique etching, Figure 8(d) is simulated with 150 nm of perpendicular etching, and Figure 8(e) is simulated with 150 nm of 30° oblique etching. When the nanosphere and the substrate are unetchable, the areas under the nanosphere are well protected, and thus a hemispherical cup exists in these figures. Their counterparts presented in Figures 8(f)-8(i) show that when the nanosphere is etchable, it cannot completely protect the underlying metal structure, and it will be etched away together with the metal layer; when the substrate is etchable, the substrate is etched and a metal deposition pattern is thus transferred onto the substrate. Without oblique etching, the asymmetric nanostructures only can be achieved by varying the deposition angles. By oblique etching, the asymmetric structure can be obtained through varying either the deposition angle θ or the azimuthal angle φ. Oblique etching provides us the freedom of one more dimension, and it can be used to enhance nanostructures at one side; or to design some 3D nanostructures with sharp corners. The nanostructures in Figures 8(c), 8(e), 8(g) and 8(i) are etched at 30°, with all other conditions kept identical to those in Figures 8(b), 8(d), 8(f), and 8(h). Compared with the perpendicularly etching, oblique etching strengthens one side of the nanostructures, and breaks the original symmetry generated from the original three depositions. Such a structure is able to redistribute the electrical field in LSPR, due to its localization of the electromagnet fields at the corners of the metal nanostructures [34].

2.3.2. The mixture of 3D and 2D depositions This example demonstrates a mixture of 2D and 3D depositions [18]. Figures 9(a) and 9(b) are the nanostructure profiles obtained by two times of metal deposition onto a 105 nm radius nanosphere. For each deposition, the metal thickness is 40 nm. The deposition angles are respectively θ1 = 30° and θ2 = 60°. θ1 = 30° deposition yields a 2D structure, and θ2 = 60° deposition yields a 3D non-conformal nanostructure. The profiles shown in Figures 9(c)-9(f) are obtained after metal etching of 80 nm at various etching angles, assuming that the nanosphere and the substrate are etch resistant. The difference between Figures 9(a) and 9(b) is that Figure 9(a) has an azimuthal angle of 90° between the two depositions, versus that of 180° in Figure 9(b). Depending on the directions of the depositions, the two metal depositions will be superimposed in some areas. After the 80 nm perpendicular etching of Figures 9(a) and 9(b), depending on the overlap area of the two depositions, the metal on the nanosphere deposited by the 2D deposition may either remain as a part connected to the 3D deposition as presented in Figure 9(c); or it will be removed together with the dissolution of the nanospheres as presented in Figure 9(d). The 80 nm metal etching of Figures 9(a) and 9(b) at 30° oblique angle gives out totally different nanostructures as plotted in Figures 9(e) and 9(f), with sharp corners remaining only at one side.

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Figure 9. The metal profiles obtained by double depositions before and after etching. The diameter of the nanosphere is 105 nm, thickness for both depositions is 40 nm, θ = 30°, 60°, the nanosphere and the substrate are assumed to be etch resistant. The light blue area represents the original substrate, the gray represents the nanosphere. (a) is after the depositions when the separated azimuthal angle between the two depositions is 90°, (b) is after the depositions when the separation azimuthal angle between the two depositions is 180°, (c) and (d) are after 80 nm perpendicular etching of (a) and (b), (e) and (f) are after 80 nm 30° oblique etching of (a) and (b)

This example demonstrates that nanosphere lithography is a powerful and versatile method to fabricate fine and delicate nanostructures, and our program can be used to design the nanostructure profiles with various deposition and etching parameters. However, the nanospheres should preferably be dissolved gently by chemicals without sonication, such as in heated ethanol or toluene, so as to prevent the delicate nanostructures from damage or breakage. For example, the leftover nanostructure in Figure 9(c) has some junction part of the 2D and 3D depositions, which is fragile and will probably break in ultrasonic treatment. By profile simulation, we can try to avoid such a weak structure by designing the deposition and etching parameters to yield rigid 3D nanostructures.

2.3.3. Comparison between the wet and dry etching Figure 10 compares the wet etching and dry etching of a metal nanostructure originally obtained with 4 times of metal evaporation, as presented in Figure 10(a) [20]. After wet etching 60 nm of metal with the nanosphere on the substrate, as shown in Figure 10(b), the metal on the nanosphere and some part on the substrate are etched away, and only 4

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connected cones left around the nanosphere. When 30 nm of metal is etched after removing the nanospheres, the etching is double side, and the leftover cones in Figure 10(c) are about a half size of the ones in Figure 10(b). Figure 10(d) is with 60 nm of anisotropic dry etching. The leftover metal nanostructure on the nanosphere after 60 nm of dry etching is much larger than that of wet etching, because in dry etching the size of the nanostructure only reduces 60 nm from the top to bottom, while in wet etching, the 60 nm of metal is reduced in all directions; but the metal on the substrate is etched at the same depth for wet and dry etching, because in wet etching, the metal on the substrate only experiences one-side etching. In Figures 10(b) and 10(c), clusters of metal nanoparticles are left on the substrate. But in dry etching in Figure 10(d), clusters on the substrate are not generated. The clusters are interesting nanostructures in plasmonics, since it is reported that a dimer of nanoparticles scatters much higher electrical field than a single nanoparticle [35], the narrow gaps between the clustered nanoparticles are expected to strongly enhance the plasmonic signal. Because there are too many factors, such as the size of the nanosphere, metal evaporation angle and thickness, etching method and etching thickness, influence the profile of the metal nanostructures obtained by dispersed NSL, our profile simulation program provides a useful tool to control the fabrication of the metal nanostructures by dispersed NSL. We regard this simulation as important, also because some 3D nanostructures after etching are hidden under nanospheres and are hard to be measured by SEM or AFM. Even the substrate can be tilt or diced to observe its side view in SEM, the 3D nanostructures cannot be fully inspected, and some delicate nanostructures might have been damaged during the dicing process. On the other hand, the electrical charges in SEM are high when glass substrate is used for LSPR applications, the substrate has to be evaporated with a few nanometers of gold to eliminate the electrical charges in order to view the metal nanostructures, and this layer of gold evaporation will deform the actual shape of the nanostructures on the substrate. Thus sometimes we would rather use the simulation to substitute the experimental observation of the complicated 3D nanostructures obtained by dispersed NSL, and the simulation tends to give more detailed profile than SEM or AFM observations.

a

b

c

d

Figure 10. Comparison for the profiles of the metal nanostructures obtained by wet etching and dry etching. (a) shows the nanostructure after evaporating 40 nm-thick metal film 4 times at the angles of θ = 60°, 60°, 60°, 60° and φ = 0°, 90°, 180°, 270°, when the metal evaporations are non-conformal with θc = 10°. (b) is after 60 nm of wet etching without removing the nanosphere, (c) is after 30 nm of wet etching after removing the nanosphere, and (d) is after 60 nm of anisotropic dry etching, assuming the nanosphere and the substrate are not etched during the dry etching

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3. NANOSTRUCTURES OBTAINED BY CLOSELY PACKED NSL Closely packed NSL has the advantages of cost-effectively obtaining periodic metal nanostructures on substrate, or harvesting different kinds of metal nanostructures in solutions by resolving the nanospheres after metal evaporation. In closely packed NSL, all nanospheres are either hexagonally or quadrangularly packed in mono or multiple layers. Prior to the metal deposition, the nanospheres can optionally be heated in an oven to shrink its size, so as to make some gaps between the nanospheres. After metal evaporation, the metal nanostructures on the nanospheres can be utilized for LSPR, or the nanostructures on the substrate can be used for LSPR after the removal of the nanospheres. For the NSL mask comprises of multiple layers of nanospheres, after metal deposition, the nanospheres can also be peeled off layer by layer with tapes to collect the metal nanostructures in solutions for LSPR detection, and the nanostructures on each layer have a different shape. By vertically evaporating metal on the nanospheres, the metal nanocaps on the nanospheres can be used for LSPR generation [36, 37]. When the nanospheres are removed, nanotriangles are left on the substrate, which has been fabricated by many research groups as a classic metal nanostructure for LSPR measurement [4, 5, 26, 31]. The shapes of the nanocaps and the nanotriangles can be trimmed by evaporating the metal at a small oblique angle. When the oblique angle is large, maybe no nanostriangle will be formed on the substrate due to the shadow of the closely packed nanospheres. To evaporate the metal at a large oblique angle is called glancing angle deposition, where no metal will exist in the flaws of the substrate and only patchy nanoparticles are generated on the top of the nanospheres due to the shadowing effect of several nanospheres. We demonstrated that the patchy gold nanoparticles on polystyrene nanospheres render sensitive LSPR responses. However, when the metal is evaporated at an oblique angle between the range of near vertical deposition and glancing angle deposition, depending on the nanospheres’ size, shrinkage, alignments including layers and angles, the metal might be on anywhere of the nanospheres and the substrate with various sizes. Since the shapes of the nanostructures are sensitive to the fabrication conditions and hard to be imagined, we simulate the 3D profiles of the nanostructures on each layer of the nanospheres and the substrate, with all of the above varieties considered. The simulations are verified with our experiments, and several examples are calculated to exemplify how 3D profile of the nanostructures on each layer of the nanospheres is extrapolated through the simulation.

3.1. Simulation Formulas for Closely Packed NSL This work bases on space geometry and coordinate transformation to simulate the metal nanostructures fabricated on each layer, when the nanospheres are closely packed in one or two layers [29]. 5 calculation layers are set for monolayer NSL simulation and 8 calculation layers are set for 2-layer NSL simulation, as shown in Figures 11(a) and 11(b), where “bottom” represents the metal deposited on the substrate, “top” and “down” represent the metal nanostructure on the top and bottom of a nanosphere, and the “topball” and

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“bottomball” represent the top and bottom parts of a nanosphere. To differentiate “top” and “bottom” is to ensure that these calculated layers are monotropic so as to avoid data missing caused by overwriting. Although only up to 2 layers of closely pack nanospheres are simulated, the simulation for 3, 4 or more closely pack nanospheres layers are similar except being more complicated, where the shadows of the nanospheres on several adjacent layers should be considered.

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3.1.1. Metal structure nearby a single nanosphere The metal is deposited onto the nanospheres at a deposition angle θ and an azimuthal angle φ. As described in Section 2.1.1, in order to calculate the metal thickness deposited on the nanospheres and the substrate, two coordinate transformations are carried out: the first is to transform the xo-yo-zo coordinate system with φ ≠ 0, θ ≠ 0 into xφ-yφ-zφ (where yφ = y) with φ = 0, θ ≠ 0; the second is to transform from xφ-yφ-zφ to xθ-yθ-zθ (where zθ = zφ) with φ = 0, θ = 0. In the coordinate system xθ-yθ-zθ, the metal appears as if vertically deposited, and the metal deposited on each nanosphere can be expressed as Eq. (1). Consequently, the metal evaporated in the xo-yo-zo coordinate system is calculated by Eq. (5).

Figure 11. Calculated layers for simulating the metal nanostructures fabricated by closely packed NSL. (a) is for monolayer and (b) is for double layer NSL. θ and φ are angles of the metal deposition, and t is the metal deposition thickness

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Figure 12. The range of the area on the substrate that is covered by the shadow of a nanosphere. (a) shows the calculation of D, and (b) shows the range of the nanosphere shadow along the xo and zo axes

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The above calculated metal shape together with the shadows of several adjacent nanospheres, determine the final profile of the metal nanostructure on a nanosphere. The shadow of each nanosphere is considered as following: if the areas fulfill

x2  z  r 2 , it 2

will be shadowed by the nanosphere and no metal will be deposited; otherwise the area will not be affected by the shadow of the nanosphere. In order to calculation the range of the area that will be covered by the shadow of a nanosphere, the shadow is plotted in two planes, i.e., xφ-yφ plane in the xφ-yφ-zφ coordinate system as drawn in Figure 12(a) and xo-zo plane in the xo-yo-zo coordinate system as drawn in Figure 12(b). According to Figure 12(a), the distance from the edge of the shadow to the origin of the xo-yo-zo coordinate system is D  r tan   r / cos , where θ is the deposition angle and r is the radius of the nanosphere. When the ellipsoidal shadow of the nanosphere is observed in the xo-zo plane as in Figure 12(b), it is obvious that the maximum ranges of the shadow are (  D cos   r sin   , r ) along the xo axis and (  D sin   r cos   , D sin   r cos ) along the zo axis, where φ is the azimuthal angle.

3.1.2. The influence of adjacent nanospheres The simulations carried out are illustrated in Figure 13: hexagonally and quadrangularly closely pack nanospheres in monolayer or double layer, all were experimentally reported in literatures [21-25, 27-28]. It should be noted that the diameter of the nanospheres might be smaller than their interval, because after the nanospheres are closely pack, they can be shrunk

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by etching and heating prior to metal evaporation [21, 22]. As long as enough adjacent nanospheres are considered, the angle φ to be considered for the hexagon arrangement is in the range of 0 to 30°, for the quadrille arrangement is in the range of 0 to 45°, other angles of φ will repeat theses profiles. The monolayer simulation is as drawn in Figures 13(a) and 13(b), where only the shadows of nanospheres from the same layer are considered. For 2layer simulation, the shadows from both layers of the nanospheres should be considered. For the hexagonal arrangement in Figure 13(c), the vertical distance between the centers of the



upper and lower nanospheres is d y  rup  rdown 2  2d arrangement in Figure 13(d), d y 

r

up

3



2

, for the quadrangular

 

2  rdown   2d , and dy should fulfill the condition 2

of d y  rlower  rupper , where rupper and rlower are the radii of the nanospheres in the upper and

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lower layers, and 2d is the interval of the nanospheres. Because the nanospheres are etched or shrunk after closely packed, the intervals for the two layers are the same.

Figure 13. Different arrangements of the closely packed nanospheres considered in our simulations, when the nanospheres are (a) hexagonally packed in monolayer, (b) quadrangularly packed in monolayer (c) hexagonally packed in double layer, (d) quadrangularly packed in double layer. Gray represents the lower layer, and orange represents the upper layer

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Figure 14. The adjacent nanospheres to be considered in calculating the metal nanostructure on a specific nanosphere “Ori”, when the nanospheres are in the same layer. Besides the nanosphere “A”, either adjacent 2 × 2 (in purple), 3 × 3 (in amber plus the ones in purple), or 4 × 4 (in pink plus the ones in purple and amber) arrays of nanospheres can be considered. (a) is for hexagonal arrangement, (b) is for quadrangular arrangement

a

b

For 3X3

B Ori A 1

4

4

φ xo

For 2X2 For 4X4

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zo

Ori

For 2X2

φ xo

1

For 3X3 For 4X4

zo

Figure 15. The influence of the nanospheres on the adjacent layer to a specific nanosphere “Ori” under calculation. (a) is for hexagonally packed nanospheres to count the influence of upper layer to lower layer. Besides nanospheres “A” and “B”, either adjacent 2 × 2 (in purple), 3 × 3 (in amber plus the ones in purple), or 4 × 4 (in pink plus the ones in purple and amber) arrays of nanospheres can be considered; (b) is for hexagonally packed nanospheres to count the influence of lower layer to upper layer, where adjacent 2 × 2, 3 × 3, or 4 × 4 arrays of nanospheres can be considered. (c) and (d) are for quadrangularly arranged nanospheres with the same meaning of (a) and (b)

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Depending on the metal evaporation angles as well as the size and interval of the nanospheres, enough number of adjacent nanospheres should be considered to attain an authentic profile of the metal nanostructures on a nanosphere. As presented in Figure 14, to calculate the metal nanostructure on a particular nanosphere specified as “Ori”, for both hexagonally and quadrangularly packed nanospheres in the same layer, at least 5 nanospheres around it should be counted: besides the one beneath the “Ori” denoted as “A”, the other 4 in purple form a 2 × 2 matrix for shadow calculations. Or in addition to the nanosphere “A”, the other 9 nanospheres around it are calculated, which form a matrix of 3 × 3. Similarly, a 4 × 4 or 5 × 5 matrix can be calculated and so on. For the 2-layer arrangements, besides considering the influence of the adjacent nanospheres in the same layer, the influence of the lower layer to the top layer should be considered, and the influence of the upper layer to the lower layer should also be considered. Figures 15(a) and 15(c) identify the upper nanospheres that influence the nanosphere “Ori” on the lower layer, with numbers being 6, 11, or 18, respectively; and Figures 15(b) and 15(d) identify the lower nanospheres that influence nanosphere “Ori” on the upper layer, with numbers being 4, 9, or 16, respectively.

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3.2. Experimental Verifications Some closely pack gold nanostructures were fabricated to compare with simulation [29]. The polystyrene nanospheres we used were with a diameter of 500 nm, purchased from Duke Scientific Corporation, and diluted 100 times with deionised water before usage. To closely pack the nanospheres on glass substrate, the solution was processed by first diluting Triton X100 with methanol in a ratio of 1:400 by volume, then diluting this solution with the nanosphere solution in a ratio of 1:1 by volume [26]. Figure 16(a) gives out the SEM image of monolayer closely pack nanospheres on a glass substrate, after evaporating 50 nm of gold film at θ = 41°, φ = 3°. The gold nanostructures observed under SEM looks smaller than simulation, because the thickness of the gold nanostructure is much thinner at the shadow side of the metal evaporation as indicated by the thickness of each nanostructure simulated in the inset. Similar phenomenon was reported by reference 27, where the simulated area covered by gold was smaller than the area inspected by SEM. In closely pack NSL, defect is a severe issue and it causes variation in the shapes of the gold nanostructures fabricated through the nanosphere mask. Figure 16(b) depicts a two-layer closely pack sample with many defects when gold is evaporated at 81°. Since the gold was evaporated at a large angle, the gold nanostructure is very thin at the shadow side, and that part only shows as some dots in SEM. The dots form a line in red in Figure 16(b), which is vertical to the direction of gold evaporation. The defects cause the change in φ, and each renders a different shape of gold nanostructures. For example, areas A, B, and C have the azimuthal angles of 0°, 20° and 15° for the gold evaporation, with local directions of the nanospheres drawn in yellow lines, thus their shapes are quite different. The simulated 3D profiles in the insets coincide well with experiments.

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Figure 16. Profiles of gold nanostructures after depositing 50 nm of gold onto closely packed nanospheres of 500 nm in diameter. (a) shows monolayer nanospheres with gold evaporation of θ = 41°, φ = 3°; (b) shows 2-layer nanospheres with gold evaporation of θ = 81°, and φ = 0, 20, 15° in areas A, B, and C; the white arrow indicates the gold evaporation direction, gray and amber in simulation represent the bottom and top layers. The simulations are top view and thickness of the gold nanostructures

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c

Figure 17. Simulated metal nanostructures fabricated on closely pack nanospheres. (a) is quadrangular with r = 105 nm, t = 50 nm, θ = 89.5°, φ = 30°, from left to right are the top view after metal evaporation, metal on the top of the nanosphere, metal on the bottom of the nanosphere when 26 adjacent nanospheres are considered, metal on the substrate, metal on the bottom of the nanosphere when adjacent 5 and 10 nanospheres are considered, respectively. (b)-(e) are for two layer nanospheres with interval of 300 nm, the lower layer has r1 = 130 nm, the upper layer has r2 = 110 nm, and 50 nm of metal is evaporated at φ = 10°. From left to right are the top view after metal evaporation, metal on the top of the upper layer nanosphere, on the bottom of the upper layer nanosphere, on the top of the lower layer nanosphere, on the bottom of the lower layer nanosphere, and on the substrate. (b) is hexagonal with θ = 89.5°, (c) is hexagonal with θ = 60°, (d) is hexagonal with θ = 0°, (e) is quadrangular with θ = 60°

3.3. Simulation Examples The metal nanostructures underneath the nanospheres are not accessible for SEM or AFM, while they are convenient to be obtained by simulation. It is important to know the shape and size of the nanostructures on the bottom the nanospheres, because these

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nanostructures will influence the plasmonic signals, or diversify the size and shape of the collected metal nanoparticles in solution when NSL is used as a means to fabricate them. Some examples are calculated in Figure 17 [29]. Figure 17(a) demonstrates the metal nanostructures when 50 nm of metal is evaporated on a quadrangularly closely pack nanosphere array of 105 nm in radius, at the angle of θ = 89.5°, φ = 30°. The metal nanostructures on the top of the nanospheres, on the bottom of the nanosphere and on the substrate are presented. Depending on the metal evaporation angle and the arrangement of the nanospheres, for some layers many adjacent nanospheres should be considered in order to get the correct profile. In this simulation, the shadows of up to 26 adjacent nanospheres are considered for calculating the nanostructures on each nanosphere. Compared with the results of considering only 5 or 10 adjacent nanospheres, the shapes of the metal nanostructures on the top of the nanosphere and on the substrate show no difference, while the metal nanostructure on the bottom of the nanosphere is different when only 5 or 10 adjacent nanospheres are considered as demonstrated on the right two columns of Figure 17(a). To avoid such error, in all calculations, at least 17 adjacent nanospheres for the nanospheres on the same layer, and 10 adjacent nanospheres on the neighbor layer are considered. In NSL, the metal nanostructures on the nanospheres are usually utilized when the metal evaporation angle is oblique [27, 28], and the metal nanostructures on the substrate are employed when the evaporation angle is small [21-26]. This simulation is suitable for any metal evaporation angles. Etching or heating to shrink the nanospheres prior to metal evaporation increases the diversity of the metal nanostructures obtainable by NSL. After etching the top of the closely pack nanospheres, both the nanosphere sizes of the two layers will be reduced. Figures 17(b)-(e) mimic this situation by setting the interval of the nanospheres to be 300 nm, while the lower and upper layers of the nanospheres have diameters of 260 nm and 220 nm respectively. In Figures 17 (b)-(e), the metal nanostructures on the top and bottom of the two layers of the nanospheres, as well as the metal shape on the substrate are provided. Although not shown in Figure 17, the thickness of each nanostructure is also available in the data output of the 3D profile. Several conclusions can be drawn according to the simulations: the metal nanostructure on the substrate is influenced by all layers of the nanospheres, it has to be obtained by calculating each shadow of each nanosphere; the metal nanostructure on each nanosphere is influenced by its adjacent nanospheres in different layers, and enough numbers of adjacent nanospheres must be considered to attain the authentic profile of the nanostructure; the metal nanostructures not only exist on the top of the nanospheres, but also on the bottom of them in some circumstances, which are unable to be observed by SEM and AFM. In closely packed NSL, the experiments exhibit high consistency with simulated profiles. However, the shape of the metal nanostructures is very sensitive to the azimuthal angle, the defects in the closely packed arrangements should be reduced to minimum in order to obtain identical metal nanostructures for LSPR.

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4. SOME NANOSTRUCTURES FABRICATED BY NSL FOR LSPR This section demonstrates some of our LSPR experiments conducted with some nanostructures uniquely used in our group. The LSPR spectrum, optical polarity, and detection sensitivity are different for these structures depending on the profile of these nanostructures, which is determined by the fabrication process in NSL.

4.1. 3D Nanostructures For the gold nanostructures shown in Section 2.2.1, when 3D nanostructures are formed on the dispersed polystyrene nanospheres, they can be directly used for LSPR detection.

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4.1.1. Polarization dependence of the 3D nanostructures The LSPR spectral is polarization dependent. Figure 18 presents the LSPR spectra measured with both polarized and unpolarized light.

Figure 18. UV-vis extinction spectra of the nanostructures with (a) polarized light and (b) unploarized light. U means the shadows of gold nanostructures are vertical, C means the shadows are horizontal; p is with vertical electrical field, and s is with horizontal electrical field

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In Figure 18(a), p-polarized light is the light with vertical electrical field and s-polarized light is with horizontal electrical field; U means the shadows of the nanospheres after gold evaporation are vertical, and C means the shadows of the nanospheres are horizontal. s-U and p-C in Figure 18(a) have the same LSPR light extinction peak around 590 nm, because the polarization of the light is perpendicular to the gold deposition direction; while s-C and p-U have the same LSPR peak around 730 nm, because the polarization of the light is parallel to the gold deposition direction. It should be noted that the slight difference between s-U and pC, as well as s-C and p-U is a result of ex-situ LSPR measurements, while the LSPR spectra will be somewhat different if the gold evaporation angle is different at various locations of the sample. LSPR is so sensitive that in situ test will greatly improve the LSPR sensor’s sensitivity and accuracy. Figure 18(b) demonstrates that the LSPR spectrum with unpolarized light shows the same spectrum for C- and U- shaped nanostructures, since it is the superposition of the two spectra of s-C and p-C, or the superposition of s-U and p-U. The unpolarized light can be used for LSPR sensing, but polarized light can tune the extinction peak of LSPR up to hundreds of nanometers, which is difficult to achieve through changing the shape of the gold nanostructures, thus this can be regarded as an advantage of the polarization dependent gold nanostructures for LSPR.

4.1.2. LSPR sensitivity versus the 3D nanostructures The LSPR spectra are measured to compare the LSPR function of various nanostructure shapes when the adjacent medium of the gold nanostructures is air and water, and the results are shown in Figure 19. Figure 19(a) shows the spectra when 50 nm of gold was evaporated onto the nanospheres at 70˚. Its extinction peak of LSPR shifts 104 nm when the medium is changed from air to water, and thus has a sensitivity of 317 nm/RIU, RIU is the refractive index unit. Figure 19(b) presents the LSPR spectra when the gold nanostructures were fabricated by evaporating 30 nm of gold onto the nanospheres at 70˚, with a LSPR chip the same as in Figure 18. Under this condition, the LSPR spectra yield a LSPR sensing sensitivity of 242 nm/RIU, while the spectra of the gold nanostructures have narrower line-width. Figure 19(c) presents the LSPR spectra of the same gold nanostructures of Figure 19(b), when the polystyrene nanospheres were burnt up at a high temperature in oven. In this case, the linewidth of the LSPR spectra is further reduced, but it renders a sensitivity of only 170 nm/RIU. It seems that when the gold thickness is reduced or the polystyrene nanospheres are removed, the LSPR spectra can be sharpened. But the decrease in the bandwidth of the LSPR spectra reduced the sensitivity of the LSPR sensing. Even so, in the optimization of our gold nanostructures fabrications, we aim at narrow LSPR spectra, because this gives clearer identification of small LSPR spectrum shift and overall we expect it to provide a higher signal to noise ratio of the LSPR detections.

4.2. Nanoholes Nanoholes described in Section 2.2.3 are used for LSPR detection of water, bovine serum albumin (BSA) adsorption, and biotin-streptavidin immunoassay [19].

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Figure 19. LSPR spectra of air and water for various shapes of gold nanostructures. (a) 50 nm of gold was evaporated onto the polystyrene nanospheres at 70˚; (b) 30 nm of gold was evaporated at 70˚; (c) when the nanospheres in (b) were removed. The LSPR sensitivities for (a), (b) and (c) are respectively 317, 242, and 170 nm/RIU, RIU is the refractive index unit

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4.2.1. LSPR sensitivity of the nanoholes The LSPR extinction spectrum of a nanohole sample in air and water were measured with a gold thin film, which has the same thickness of the nanoholes, as a reference [19]. In our experiments, as the nanoholes were almost identical on different locations of the sample, LSPR was measured at different locations, and the LSPR peak wavelengths were obtained by calculating the mean value of the peak wavelengths at different locations of the sample. The LSPR spectra of the nanoholes in air and water, as well as their mean values and error bars (based on standard deviation) are shown in Figures 20(a) and 20(b). According to the LSPR peak of 574.5 nm in air and 586.3 nm in water, the LSPR sensitivity of the nanoholes is calculated to be 36 nm/RIU. The lower sensitivity compared to 100 nm/RIU for 110 nm-dia nanoholes in Ref. [14] is due to the lower aspect ratio of our nanoholes, as the gold is thicker in our samples.

Figure 20. LSPR test of the nanoholes. (a) shows the LSPR extinction spectra in air and water, (b) shows their mean values and standard deviations of several measurements at different locations of the sample

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Before studying with BSA adsorption and biotin-streptavidin immunoassay, LSPR extinction spectra of nanoholes after incubation in water, ethanol and phosphate buffered saline (PBS) were measured respectively as control experiments, and no extinction spectra change was observed before and after the incubation of these kinds of solvent. Therefore we exclude the influence of the solvent on LSPR in the following BSA and streptavidin (SA) detections.

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4.2.2. BSA test of the nanoholes For the investigation of BSA binding onto the nanoholes, the LSPR spectrum of a nanohole sample was first measured [19]. Then the sample was incubated with BSA in PBS buffer at the concentration of 1 mg/ml for 20 hours at room temperature. The sample was nitrogen dried without rinse and the LSPR extinction spectrum was measured. Afterwards, the sample was rinsed with PBS and water repeatedly and nitrogen dried, with LSPR spectrum measured.

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The detection of BSA adsorption was demonstrated in Figure 21. After incubation into BSA in PBS, the peak shifted from 571 nm to 598 nm and indicated a 27 nm redshift. Since BSA has no selective absorption in UV–Vis region, this wavelength shift came from the nanohole LSPR in response to its local dielectric change induced by BSA, which physically adsorbed to the gold surface and the glass surface inside the nanoholes. As the plasmonic field is concentrated around the circumference of the nanoholes, it is mainly the BSA proteins adsorbed onto these areas caused the peak shift, and we estimate multilayer of BSA molecules accumulated on the surface. After subsequent rinsing with PBS, ethanol and DI water, it is found that the LSPR peak returned to 579.5 nm, and further sonication made the average wavelength of the sample down to 575 nm, very close to the wavelength before BSA adsorption. This confirms that the LSPR peak redshift was due to the physical adsorption of BSA molecules. The rinse and sonication of the sample did not bring the LSPR extinction peak back to its original position, due to the incomplete removal of BSA. After sonication, the standard deviation of the peak wavelengths for the measurements at different locations was high, indicated that BSA was removed more thoroughly at some areas than other locations. We also noticed the strong intensity increase after BSA adsorption. However, we avoid using intensity as the indicator of signal detection because light intensity is sensitive to many factors. Our BSA test has 4 times more LSPR wavelength shift than the nanoholes in Ref. [15], while their bulk reflective index sensitivity is 110 nm/RIU, i.e., 3 times higher than ours. Besides the influences come from different nanohole size, thickness and distribution, we believe the shape of our nanoholes is more beneficial for BSA bonding, as our nanoholes are in cone shape and thus larger BSA adsorbed areas are covered with electromagnetic field.

4.2.3. Biotin-streptavidin test To investigate the specificity and adhesion of analyte to the surface of nanoholes, biotinstreptavidin immunoassay was studied with a nanohole sample in several subsequent steps, and the LSPR spectrum was taken after each step [19]. Firstly, the sample was immersed into biotinylated thiol at the concentration of 374 μM in ethanol for 20 hours at room temperature to form a self assembled monolayer of biotin, then the sample was rinsed with ethanol and DI water and nitrogen dried. Secondly, the sample was immersed into 1 μM streptavidin in PBS for 20 hours at room temperature, dried without rinse. Thirdly, the sample was rinsed repeatedly with ethanol, PBS and DI water and nitrogen dried. Fourthly, the sample was exposed under a UV cleaner for 30 minutes to break the chemical bond between the thiol and gold, then the sample was rinsed with ethanol, PBS and DI water repeatedly and nitrogen dried. Fifthly, the sample was incubated with biotinylated thiol in ethanol for 20 hours at room temperature again, rinsed and dried. For the biotin-streptavidin immunoassay, its LSPR extinction spectra are plotted in Figure 22. The nanohole sample modified with biotin had a LSPR peak at 567 nm. After incubation in streptavidin and dried, the LSPR peak red-shifted to 578 nm and exhibited a prominent 11 nm shift, due to the local dielectric change induced by streptavidin bonding. Repeated rinse of ethanol, PBS and DI water only blue-shifted the wavelength 3 nm, because some unspecific bonded substances on the sample were washed away. Biotin and streptavidin were supposed to be chemically bonded only on gold, and the LSPR peak shift was mainly contributed from the streptavidin at the circumference of the nanoholes. As the density of our nanoholes was about 3/μm2, the area illuminated by the detection light was 5 mm in diameter,

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based on the size/shape of the nanohole and streptavidin (SA) [4], there were about 218 SA around each nanohole, therefore, the effective number of streptavidin under detection for a sample was estimated to be 1.28 × 1010, i.e., 21.3 fMol/sample. This is about 3 times of the limit of detection (LOD) reachable by silver nanotriangles measured in nitrogen gas environment, where 4.6 × 109 SA/sample, equivalent to 7.6 fMol/sample, was reported by Northwestern University [4]. After UV exposure and rinse, the LSPR peak turned to 561.7 nm, as the UV exposure broke the bond between biotinylated thiol and gold. No obvious extinction intensity change at this step was observed, which inferred the biotin and streptavidin were not washed away although they were not bonded on the nanoholes and were already out of the localized plasmonic field of 5 ~ 15 nm high from the metal surface. When we incubated the UV exposed sample again with biotinylated thiol in ethanol, both intensity and wavelength of the sample’s LSPR extinction peak returned to an even lower wavelength of 560 nm, because the streptavidin nearby the nanoholes were dragged away by excessive biotin in ethanol, and a new layer of biotin was anchored onto the sample. This means the sample is reusable in our experiments, thus the cost of the device can be further reduced.

Figure 22. Biotin-streptavidin immunoassay measured by the nanoholes. (a) shows the extinction spectra, (b) shows the error bars for the LSPR peaks at different locations of the sample

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4.3. Patchy Gold Nanoparticles

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Although nanotriangles fabricated by closely packed NSL has been intensively investigated, we demonstrate that patchy gold nanoparticles on the top of polystyrene nanospheres can be utilized for LSPR detections with quite high sensitivity. The original 500 nm-diameter nanosphere solution, purchased from Duke Scientific Ltd. with 10 wt%, was diluted with Triton X-100 and methanol (1:400 by volume), and dropped onto a clean glass substrate to self-assemble into a monolayer of hexagonally close packed array as photographed in Figure 23(a). The samples were deposited with 50 nm of gold in a thermal evaporator, at a deposition angle θ and an azimuthal angle φ. In this work, θ was investigated from 40° to 85°, and Figures 23(b) and 23(c) show the nanostructure arrays obtained under small (60°) and large (85°) deposition angles, respectively. The inserted simulation results are consistent with the SEM images. However, we have to concern another factor, the azimuthal angel φ, which determines the orientation of the nanosphere. As shown in Figure 23 (d), when the deposition angle θ is 85°, the orientation of the red line is different from that of the blue line, so that the nanostructures obtained are also different.

Figure 23. (a) SEM image of the closely packed nanospheres with a diameter of 500 nm, (b) and (c) SEM images of the nanospheres with gold nanostructure arrays fabricated, by evaporating 50 nm of gold at small and large deposition angles, respectively. Insets are simulated shape. (d) SEM image of the nanostructures arrays with different orientation

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Figure 24. LSPR measurement of Au nanostructure array on the nanosphere with media of different refractive indices. The insert shows the wavelength shift of the LSPR peak with the refractive index increase and its linear fit.

In closely packed NSL, defects tend to happen at a critical length longer than 10 – 100 µm. The defects change the orientation of the nanospheres and thus vary the nanostructure shape. According to Kretzschmar’s group’s study [27], at small deposition angle θ, the shape of the patch on the nanospheres are not so sensitive to the azimuthal angle φ, while only at large deposition angle, the shape of the patchy varies dramatically at different φ. Thus in our experiment, we used patchy gold nanoparticles obtained at 85° for LSPR measurement. To examine the LSPR sensitivity of the patchy nanostructures, the refractive index of the chip’s adjacent medium was adjusted by using glycerol solutions at different concentration. In the experiments, the solution and the LSPR chip were confined in a transparent fluidic chamber, the chip’s transmission spectra were taken as shown in Figure 24. The LSPR extinction peak red-shifted linearly with the refractive index variation, varied from 645 to 670 nm, and rendered a LSPR sensitivity of 258 nm/RIU.

5. CONCLUSION This chapter described in detail on the application of nanosphere lithography for fabricating metal nanostructures for LSPR detections. Our work is aiming to pave the way for NSL to be a practical and reliable tool for LSPR applications. Since the nanostructures fabricated by NSL have an abundance of different shapes, and each shape renders a different LSPR spectrum and LSPR sensitivity, a complete set of 3D profile simulation that covers all possible fabrication approaches in NSL is conducted to control the shape of desired metal nanostructures in LSPR. Our profile simulations are verified with the nanostructures fabricated and inspected by SEM and AFM. Besides being a tool to predict and control the simulation process, the software can simulate much more complicated metal nanostructures beyond our imaginations such as multiple gold

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evaporations at various angles and followed by oblique angle dry etching. Moreover, the simulation provides 3D information and some profile details that are hard to be extrapolated from SEM and AFM. Both dispersed NSL and closely packed NSL have been applied for LSPR applications by many research groups. In this chapter, LSPR detections with three kinds of nanostructures, which are first used in our group, are described in detail. For dispersed NSL, we proposed 3D gold nanostructures on polystyrene nanospheres, gold nanoholes with slant sidewalls; and for closely packed NSL, we employed patchy gold nanoparticles on polystyrene nanospheres. All of these nanostructures are cost-effective and have demonstrated satisfactory sensitivities on the detections of BSA, biotin-streptavidin immunoassay, or glycerol solutions at different concentrations.

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[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

Bohren, CF; Huffman DR. Absorption and Scattering by Small Particles; Wiley Interscience: New York, 1983. Hutter, E; Fendler, JH. Adv. Mater., 2004, 16, 1685-1706. Sepulveda, B; Angelome, PC; Lechuga, LM; Liz-Marzan, LM. Nano Today, 2009, 4, 244-251. Haes, AJ; Van Duyne, RP. J. Am. Chem. Soc., 2002, 124, 10596-10604. Riboh, JC; Haes, AJ; McFarland, AD; Yonzon, CR; Van Duyne, RP. J. Phys. Chem., B 2003, 107, 1772-1780. Anker, JN; Hall, WP; Lyandres, O; Shah, NC; Zhao, J; Van Duyne, RP. Nat. Mater., 2008, 7, 442-453. Stewart, ME; Anderton, CR; Thompson, LB; Maria, J; Gray, SK; Rogers, JA; Nuzzo, RG. Chem. Rev., 2008, 108, 494-521. Fleischmann, M; Hendra, PJ; Mcquillan, AJ. Chem. Phys. Lett., 1974, 26, 163-166. Kneipp, K; Kneipp, H; Itzkan, I; Dasari, R; Feld, M. Chem. Rev., 1999, 99, 2957-2975. El-Sayed, IH; Huang, X; El-Sayed, MA. Cancer Lett., 2006, 239, 129-135. Larson, TA; Bankson, J; Aaron, J; Sokolov, K. Nanotechnology, 2007, 18, 1-8. Shumaker-Parry, JS; Rochholz, H; Kreiter, M. Adv. Mater., 2005, 17, 2131-2134. Rochholz, H; Bocchio, N; Kreiter, M. New J. Phys., 2007, 9, 53. Prikulis, J; Hanarp, P; Olofsson, L; Sutherland, D; Kä1l, M. Nano Lett., 2004, 4, 10031007. Gao, D; Chen, W; Mulchandani, A; Schultz, JS. Appl. Phys. Lett., 2007, 90, 073901. Lu, Y; Liu, GL; Kim, J; Mejia, YX; Lee, LP. Nano Lett., 2005, 5, 119-124. Zhou, X; Virasawmy, S; Knoll, W; Liu, KY; Tse, MS; Yen, LW. Plasmonics, 2007, 2, 217-230. Zhou, X; Knoll, W; Liu, KY; Tse, MS; Oh, S; Zhang, N. J. Nanophotonics, 2008, 2, 023502. Xiang, G; Zhang, N; Zhou, X. Nanoscale Res. Lett., 2010, 5, 818-822. Zhou, X; Zhang, N; Tan, C. Nanoscale Res. Lett., 2010, 5, 344-352. Yang, SM; Jang, SG; Choi, DG; Kim, S; Yu, HK. Small, 2006, 2, 458-475.

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[22] Tan, BJY; Sow, CH; Koh, TS; Chin, KC; Wee, ATS; Ong, CK. J. Phys. Chem., B 2005, 109, 11100-11109. [23] Zhang, G; Wang, D; Möhwald, H. Nano Lett., 2005, 5, 143-146. [24] Zhang, G; Wang, D; Möhwald, H. Chem. Mater., 2006, 18, 3985-3992. [25] Zhang, G; Wang, D; Möhwald, H. Nano Lett., 2007, 7, 127-132. [26] Jensen, TR; Malinsky, MD; Haynes, CL; Van Duyne, RP. J. Phys. Chem., B 2000, 104, 10549-10556. [27] Pawar, AB; Kretzschmar, I. Langmuir, 2008, 24, 355-358. [28] Pawar, AB; Kretzschmar, I. Langmuir, 2009, 25, 9057-9063. [29] Zhou, X; Liu, KY; Knoll, W; Quan, C; Zhang, N. Plasmonics, 2010, 5, 141-148. [30] Polyanin, AD; Manzhirov, AV. Handbook of mathematics for engineers and scientists; Chapman & Hall/CRC Press: Boca Raton–London, 1st ed., 2006, 152-153. [31] Yonzon, CR; Jeoung, E; Zou, S; Schatz, GC; Mrksich, M; Van Duyne, RP. J. Am. Chem. Soc., 2004, 126, 12669-12676. [32] Bukasov, R; Shumaker-Parry, JS. Nano Lett., 2007, 7, 1113-1118. [33] Rindzevicius, T; Alaverdyan, Y; Dahlin, A; Höök, F; Sutherland, DS; Käll, M. Nano Lett., 2005, 5, 2335-2339. [34] Hao E; Schatz, GC. J. Chem. Phys., 2004, 120, 357-366. [35] Zou, S; Schatz, GC. Chem. Phys. Lett., 2005, 403, 62-67. [36] Endo, T; Kerman, K; Nagatani, N; Hiepa, HM; Kim, DK; Yonezawa, Y; Nakano, K; Tamiya, E. Anal. Chem., 2006, 78, 6465-6475. [37] Endo, T; Kerman, K; Nagatani, N; Takamura, Y; Tamiya, E. Anal. Chem., 2005, 77, 6976-6984.

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Chapter 2

STRATEGIES FOR THE ENHANCEMENT OF SURFACE PLASMON RESONANCE IMMUNOASSAYS FOR HIGH SENSITIVITY BIOSENSORS John S. Mitchell and Yinqiu Wu The New Zealand Institute for Plant & Food Research Ltd, East St, Hamilton, New Zealand

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ABSTRACT Surface plasmon resonance (SPR) has achieved widespread use as a transduction technique for biosensors whereby binding interactions on a noble metal surface can be followed in real-time by examination of minute changes in the refractive index of the surface. For a fixed surface, the amount of bound mass on the sensor surface is reflected by changes in the index of refraction at the surface, which in turn changes the angle at which peak plasmon resonance occurs. This transduction technique lends itself to incorporation with immunoassays, where either antibodies or their antigen binding partners are immobilized on the sensor surface and changes in antibody / antigen binding are used to reflect free antigen concentrations. The sensitivity of such assays is critically dependent upon the format of the sensor optics, the structure of the underlying metal surface, the chemical functionalization of the sensing interface and the format of the binding events that compose the assay. As many of the important biomolecules targeted by immunoassays are present only in very low concentrations in real samples, there are stringent sensitivity requirements. Many researchers have examined ways of enhancing the signal response from SPR immunobiosensors both through adding mass to the surface and / or through cooperative plasmon enhancements between the noble metal surface and a noble metal colloid or nanoparticle. In this chapter, we examine a wide range of cutting-edge techniques used to improve signal sensitivity and assay performance for SPR immunobiosensors. This includes using proteins as high mass labels in sandwich and competitive immunoassay formats; noble metal nanoparticle enhancement through high mass and cooperative plasmon effects; use of other high mass labels; the effects of sensor surface construction on sensitivity and enhancement; and the use of plasmonic enhancement techniques for plasmonic immunoassays where the sensing surface is not fixed but takes the form of a free-floating

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John S. Mitchell and Yinqiu Wu nanoparticle. Furthermore, we examine how fundamental changes to the structure, shape, and distance from nanoparticle to sensor surface affect SPR signal strength and look briefly at how this plasmonic theory can be applied beyond conventional SPR immunosensors into fluorescence, acoustic and electrochemical biosensors. We also assess how well these immunosensors have been applied to measurements in complex biological matrices and where these sensor technologies may lead in the future.

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INTRODUCTION Surface plasmon resonance (SPR) is a quantum optical-electrical phenomenon that has been extensively harnessed as a means of transduction in biosensor technology [1]. When a photon of light is incident on a noble metal surface, it can couple with the electrons of the metal, transferring its energy to them and causing them to move as a single electrical entity known as a plasmon, a quantum of collective charge oscillation. This plasmon generates an electrical field that extends out from the metal surface and has an intensity that exponentially decays with distance from the surface. For sensing purposes, it is typically of significant strength up to about 300 nm from a planar gold surface. Plasmon coupling will occur when the momentum of the incoming photon equals that of the plasmon. Chemical changes to the sensing surface result in changes in the wavelength of the photon needed to achieve resonance. Where the noble metal surface is spatially fixed, SPR can be defined by the resonance angle (r), which is the angle of incidence of the photon to the surface required for plasmon coupling. Any chemical changes occurring on or very close to the noble metal surface will perturb the surface plasmons and result in a shift in r. In essence, this describes a very sensitive refractometer, where small changes in bound mass on the sensor surface can produce significant changes in r. This is typically utilized in a Kretschmann configuration [2] where the noble metal thin film is coated onto glass and the evanescent wave resulting from illumination from the back-side with p-polarized light penetrates through the metal film (Figure 1). Clearly, SPR lends itself well to the transduction of binding events upon the noble metal surface (gold or silver) where the mass bound to the surface changes in response to the concentration of a target analyte by the analyte either facilitating or inhibiting binding of a high mass chemical species to the surface. The most significant binding interactions of this type are antibody / antigen [3], DNA / complementary DNA [4], receptor / target and host / guest binding interactions. Immunological interactions between antibodies and their target antigens were already widely used in radioimmunoassay (RIA) and enzyme-linked immunosorbent assays (ELISA) using radiochemical and enzyme colorimetric transduction respectively. It was entirely logical therefore to extend these immunoassays to an SPR transduction format. Commercial SPR instruments such as BIAcoreTM are now in widespread use for examining immunological binding interactions [5] and in immunoassays [6]. SPR immunoassay applications range from environmental testing to laboratory medicine [7], sports science [4], physiology [8] and food safety [9].

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Figure 1. A schematic of an SPR biosensor using the Kretschmann configuration [2]. Reprinted by permission from Macmillan Publishers Ltd: Nature Reviews Drug Discovery 2002, 1(7) 515-528, copyright 2002

Immunoassays for large molecules such as proteins typically utilize a “sandwich” assay format whereby an anchoring antibody is immobilized onto a solid surface and the target protein analyte is passed over the surface and bound by the antibody. Then a second antibody is added that recognizes a different part, or epitope, of the antigen and binds to the surfacebound antigen (Figure 2). In the case of RIA and ELISA, this second antibody will typically be conjugated to a radioactive or enzymatic tag that gives a signal proportional to the concentration of the bound analyte. In contrast, immunoassays for small molecules must use different formats as a small molecule lacks sufficient size to be recognized by two different antibodies simultaneously. In this case, there are two main options: i. immobilize the small molecule onto the sensing surface by chemical conjugation and then mix the sample containing the free analyte with its antibody in slight excess and then pass over the immobilized surface where unbound antibody will attach to the surface giving a signal inversely proportional to the concentration of analyte (Figure 2), or ii. immobilize the antibody onto the sensing surface and expose it to a mixture of the free sample antigen and antigen labeled with a high mass label; only the high mass-labeled antigen will give significant signal and this will be in inverse proportion to the free analyte concentration. Any of these immunoassay formats can be adapted for SPR signal transduction. Given that the amount of signal obtained is directly proportional to the bound mass on the surface, then to obtain a sensitive immunoassay using SPR one must bind either a high mass analyte and/or a high mass label to the surface. High mass analytes can include antibodies themselves by, for example, immobilizing the antibody’s target (e.g. glutamic acid decarboxylase) onto the surface and examining the binding of the target antibody [10]. In the case of small molecules (< 2 kDa), high mass labels need to be used. Some studies utilize direct, un-labeled detection of small molecules using state of the art SPR optics [11] but these are still insensitive techniques and not suitable for measurement of low concentrations of analyte. For

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large molecule analytes, this issue could be simply solved by a logical continuation of sandwich assay formats, [3] but for small molecules the use of high mass labels introduces significant challenges arising from the steric interactions between the large and bulky label and the antibody, thus inhibiting antigen / antibody binding. There are also issues, as we will see later, with keeping larger nanoparticles stable in the microfluidics often used in SPR immunosensors. In addition to these challenges, many target analytes are present in very low concentrations in the real-world samples of interest, so any biosensor must be able to discriminate these small concentration changes. For SPR, this means either being able to discriminate very small changes in refractive index through improved optics or increasing the specific signal change from any given change in analyte concentration by use of labeling technologies or improved sensor surface design. A further complication is that many of these samples are in complex biological matrices [7] where high mass components such as blood proteins and salivary mucins can interfere with binding of antibody to antigen and can coat non-specifically onto sensor surfaces, thus distorting SPR assay results. These significant challenges are now beginning to be addressed, with some techniques for small molecule SPR enhancement becoming standardized [6].

Figure 2. A schematic of sandwich (A) and competitive (B) immunoassay formats as applied to SPR Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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Whilst most of the research emphasis has been placed on the critical design of the sensing interface, research is still on-going to improve the fundamental optical performance of conventional SPR sensor systems. The role of median and skew rays under total internal reflection has been examined and it has been shown that use of a properly aligned nonLambertian fiber-optic side-emitting diffuser (FOSED) significantly improves signal quality [12]. Use of a higher incident angle region for detection has been shown to give greater resonance angle shifts by using a prism of lower refractive index [13]. Improvements in sensitivity of resonance angle detection of up to two orders of magnitude are also possible by changing the method of polarization measurement [14]. Another study has indicated that using spatial modulation phase detection (SMPD) can yield greater sensitivity than intensity measurements for array-based formats [15]. In this chapter, we will focus predominantly on the use of high mass labeling technology, especially noble metal nanoparticles, to enhance signal sensitivity. This approach to enhancement has received a great deal of attention in recent years and represents a major part of the effort to improve SPR biosensor performance. Increasingly, such techniques are being extended to small molecules [16]. Noble metal nanoparticles have the added advantage of being able to couple with the noble metal sensing surface, thus engaging in cooperative plasmon enhancement. As well as examining label functionalization, we will also look at how the design of the sensing surface can affect signal-to-noise and look at some of the latest surfaces being developed for plasmonic immunosensors. In addition to fixed surface SPR immunosensors where the assay components are flowed over the surface, usually in a flow cell microfluidic environment, one can also have plasmonic immunosensors where the noble metal is in the form of a free floating nano- or microparticle. These formats can be very useful for rapid, single addition assays in cuvettes or on test strips.

PROTEIN LABELLED ENHANCEMENT Sandwich Immunoassays When the target antigen is a large molecule (> 2 kDa) such as a protein, then a simple adoption of the sandwich immunoassay format will often provide the necessary enhancement of SPR signal. In this case the analyte binding an immobilized antibody is sufficiently massive to generate its own significant SPR signal, but to achieve greater sensitivity and lower limits of detection (LOD), a second antibody binding to the analyte can be used. For example, the protein fibronectin has been measured using an acousto-optic tuneable SPR immunosensor both through direct measurement of the binding of fibronectin to an immobilized antibody and through a sandwich assay format, achieving a five-fold improvement in the lower concentration limit of the assay when applying the sandwich format [17]. In another example, human complement factor 4 was detected using a wavelength modulation SPR biosensor, and in this case a 10-fold improvement in the lower concentration limit of the assay was achieved through the simple introduction of the sandwich immunoassay format [18]. Whilst the sandwich assay format does involve an extra step and so consumes more reagent and time, the benefits in terms of sensitivity enhancement make it well worthwhile for many applications. There is also the option of conjugating a high mass

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label onto the second antibody to give still further enhancement of SPR signal and this is covered in later sections. The sandwich assay enhancement has also been applied in some complex media, such as serum. Ferritin detection by SPR in real-time has been demonstrated in buffer achieving LOD of 15 ng/mL in a regenerable assay [19], (Figure 3), and attempted in diluted serum but with high non-specific binding. Another feature of the sandwich assay format is that using two different antibodies improves the specificity of the binding response, as a final high signal response will only be achieved with an antigen that possesses both the required epitopes for binding. This can be particularly useful in complex biological media where structurally similar and potentially interfering proteins may be present [19]. Another example is the detection of recombinant dust mite allergen using a polyclonal antibody signal enhancement strategy; this gave an LOD of 32 ng/mL in fetal bovine serum [20]. The sandwich immunoassay technique has also been extended to other transduction methods where high mass labels can enhance signal response, such as in acoustic resonance devices [21].

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Small Molecule Immunoassays Whilst SPR enhancement for large molecule immunoassays could be achieved by simply extending the sandwich assay formats, there were much greater challenges in developing small molecule SPR immunoassays. The enzyme and radiolabel tags used in conventional immunoassays to attach to small molecules lack the necessary mass to function as effective enhancement labels in SPR formats. This meant that either they would need to be replaced by tailored high mass labels or the antibody itself would have to provide the necessary bound mass. The latter option requires robust conjugation techniques to immobilize the small molecule to the SPR sensor surface. As the molecule is small, it must be spaced from the surface to ensure that antibodies are sterically able to access the immobilized antigen. The main techniques used for such an immobilization are: (i) Forming a conjugate of the antigen directly to the sensor surface through covalent attachment of a linker that terminates in a functional group that can react with either the gold or a self-assembled monolayer or polymer coating over the gold sensor surface, and (ii) Forming a conjugate between the antigen and an intermediate binding agent, e.g. a protein or biotin, which can then attach itself to the surface through either covalent bonding or strong intermolecular interactions to binding partners such as streptavidin. Immobilization of the small molecule onto the sensing surface has been widely used in the ultrasensitive detection of steroid hormones [8]. Derivatives of progesterone [22], estradiol [23], cortisol [8] and testosterone [24] have been prepared with linkers terminating in primary amine groups. These can be used to form stable amide linkages to carboxymethyl dextran hydrogels coated onto gold SPR surfaces. They use oligoethylene glycol linkers to space the antigen from the sensor surface. As the linkers are hydrophilic, they project the antigen into the polar aqueous mobile phase and allow optimal antibody binding. Surface bound primary antibody is then in turn bound by a secondary antibody at high concentrations allowing for signal enhancements of 6–8 times the primary antibody response. This enhancement is much higher than what one would expect for simple 1:1 binding of the secondary antibody and suggests that a complex, interlinked structure of bound antibodies has

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formed on the surface. This high level of signal enhancement (compare with 2.5-fold enhancement for ferritin sandwich immunoassay [19]) allows the concentrations of primary antibody used to be greatly reduced, meaning that smaller concentrations of free steroid will produce significant inhibition of antibody binding to the surface and thus lower concentrations can be discriminated. This approach has permitted LOD as low as 25 pg/mL for steroid hormones and has opened the way to non-invasive detection in human saliva [8]. As the functionalized surfaces use covalent immobilization of the small molecule targets, they can be regenerated and re-used for many measurements. Regeneration, i.e. removal of bound antibodies to return the surface to its original state ready for the next measurement, can be achieved by exposing it to a mixture of high pH (NaOH) and a chaotropic reagent (acetonitrile). Despite exposure to these harsh conditions, the surfaces remain intact for in excess of 1100 binding and regeneration cycles, thus allowing reliable long-term use [22]. Another study has covalently immobilized clenbuterol onto a CM5 dextran chip using a thiophosgene coupling method and utilized secondary antibody enhancement to improve assay signal resulting in an LOD for clenbuterol of 10 ng/g in human hair [25]. Later studies have utilized the antigen-immobilized format for different types of small molecules. The toxin 2,4-D has been detected using physisorption of a 2,4-D-bovine serum albumin (BSA) protein conjugate onto the gold sensing surface and signal enhancement with an interesting combination of avidin and biotinylated BSA forming a complex network of interlinked BSA molecules attached to the primary antibody by biotinylation [26] (Figure 4). Here, 10-fold signal enhancements were achieved and LOD for 2,4-D of 8 pg/mL reported. Protein conjugates of the small molecule analyte can be used in competitive assay formats to generate the needed signal and this format has been applied to detection of dinitrotoluene (DNT) using a self-assembled monolayer (SAM) surface [27]. Signal enhancements of three-fold were achieved with use of a secondary antibody [27].

Figure 3. Sensorgrams for ferritin sandwich immunoassay in buffer showing binding of ferritin (a) and secondary antibody labeling (b) with regeneration (c) at varying concentrations of ferritin [19]. Reprinted from Talanta, 60(1), Cui, X. Q.; Yang, F.; Sha, Y. F.; Yang, X. R., Real-time immunoassay of ferritin using surface plasmon resonance biosensor, 53-61, Copyright 2003, with permission from Elsevier.

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Figure 4. Detection of 2,4-D using immobilized 2,4-D-protein conjugate with primary antibody binding enhanced by a biotin-BSA / avidin cross-linked network [26]. Reprinted from Biosensors and Bioelectronics, 23(5), Kim, S. J.; Gobi, K. V.; Iwasaka, H.; Tanaka, H.; Miura, N., Novel miniature SPR immunosensor equipped with all-in-one multi-microchannel sensor chip for detecting lowmolecular-weight analytes, 701-707, Copyright 2007, with permission from Elsevier

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NOBLE METAL NANOPARTICLE ENHANCEMENT Theory of Noble Metal SPR Enhancement Gold nanoparticles possess a very high mass concentrated into a comparatively small volume on the nanoscale. Their size is ideal for conjugating proteins, DNA and small molecules to their surface via simple chemistry. This can include simple charge-based adsorption of proteins or by utilizing the dative bonds formed between gold and thiol groups when thiolated derivatives are formed of the target analyte or antibody. As such, they lend themselves well to being a type of high mass label to achieve enhancement of plasmonic signals in SPR immunobiosensors. However, in addition to these valuable properties, noble metals can undergo SPR themselves. Their plasmon electric fields can be many orders of magnitude higher than incident fields and are typically constrained to a small distance (< 300 nm) from the nanoparticle surface [28]. This produces a very strong field gradient in which very interesting optical phenomena are observed, such as enhancement of the emission properties of otherwise weakly fluorescing molecules and the Surface Enhanced Raman (SERS) effect. Where one has two gold surfaces that approach each other, e.g. a gold nanoparticle approaching a gold sensing surface, then the electric fields of the surfaces can interact and one can observe a coupling of the plasmon modes of the surfaces. This coupling produces a large shift in

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refractive index and so a plasmon signal is obtained that is much larger than that expected for a simple mass increase. It is reported that these shifts, or enhancements, are about 25-times larger than would otherwise be observed without gold labeling [29]. In theory, if one were to bind a large enough number of gold nanoparticles close to the surface, such that the nanoparticles can couple to each other, then a continuous evanescent wave will result that would pass through both the gold colloids and the gold sensor surface. These phenomena apply for silver nanoparticles also, but there is much less work done on this as silver colloids are notoriously difficult to produce reliably and silver surfaces tend to oxidize very quickly. Silver deposition has been used in electrochemical transduction of immunoassay signals from antibody-immobilized electrodes [30] but silver nanoparticle label enhancement is much rarer. Research has shown that plasmon coupling produces a broadened plasmon resonance, an increase in minimum reflectance and a large shift in plasmon angle, all of which yield improved SPR sensitivity [31]. The coupling interaction is very dependent upon the distance between the nanoparticles and the sensor surface. If they are too far apart then it is not possible for the electric fields to interact and so no enhancement is observed. This distance dependence has been calibrated by spacing colloids from gold film using polyion multilayer spacers [32]. This has led to researchers optimizing the distance between nanoparticle and sensor film by using for instance, a 30 nm SiO2 layer immobilized onto a gold film of 50 nm thickness [33]. When SiO2 layers are used to alter the distance incrementally between a planar gold surface and gold nanoparticles, an optimal distance for maximum SPR angle shift is obtained at about 32 nm, though the exact distance is thought to be dependent upon the dielectric used and interestingly was found to be independent of gold surface coverage at coverages of < 10% [34]. The interparticle distance is also critical to the contribution of surface plasmons to SERS. This distance can be “tuned” by coating silver films onto templates formed using nanosphere lithography combined with reactive ion etching and in this case both local and delocalized surface plasmon (SP) modes contribute to the total enhancement [35]. The size and shape of the nanoparticles also has a critical effect on their performance for plasmon enhancement. As the colloidal gold diameter increases from 30-59 nm when used with a 47 nm thick gold film, plasmon angle, minimum reflectance and curve breadth all increase with increasing nanoparticle size until a plateau is reached where the SPR enhancement effect is maximized [36]. Also of importance is the relationship between nanoparticle size and the mean free path (MFP) of the plasmon electrons. The plasmon bandwidth increases as colloid size decreases below the MFP and increases with increasing size above the MFP [37]. The extent of the sensing distance range out from the surface of a gold nanoparticle can be adjusted by tuning the composition, shape and in-plane width and out-of-plane height of the nanoparticles [38]. The interactions between the SPs of the planar metal surface and the localized plasmons (LPs) of metal nanoparticles have been carefully examined for both gold and silver [39].

Metal Nanoparticle Enhancement: Functionalization and Assay Performance The first class of analyte molecules detected by gold-enhanced SPR immunoassay was the proteins. They have the advantage that sandwich immunoassay formats can be achieved

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and so gold labels can be conveniently substituted for enzyme or radiochemical labels and the large size of the analyte does not introduce steric crowding problems. Seminal work in this field was done by Lyon and colleagues, who were the first to demonstrate colloidal goldenhanced SPR immunoassay [29]. They utilized 11 nm gold nanoparticles to enhance 25-fold the signal for immunoassay of human IgG and obtained picomolar LOD [29], (Figure 5). Gu and colleagues utilized larger 40 nm colloids to enhance IgG binding interactions with immobilized Fab’ fragments demonstrating that enhancement was proportional to the concentration of the gold-labeled secondary antibody [40]. From then on, many papers were devoted to exploring this approach and this format is now quite well established and has been applied across a wide range of proteins. Beta-amyloid (1-40) has been detected using this format with the antibody paratope specifically directed away from the sensing surface for optimal binding [41]. Antibody fragments have been immobilized onto gold nanoparticles and the resulting conjugate used in nanoparticle-enhanced immunoassays [42]. The second antibody in the sandwich can sometimes be biotinylated and the strepatavidin conjugated nanogold bound onto it in a separate step [43]. Not content with simply enhancing the sandwich immunoassay response with gold nanoparticle tags, some groups have further improved the signal enhancement by incorporating enzymes onto the nanoparticle surface. This has been done by use of thiolterminating oligoethylene glycol chains, which provide a base for immobilizing the antibody also. In one example, the enzyme catalyzes the deposition of insoluble products from 3,3’diaminobenzidine onto the gold surface [44]. This approach has allowed detection of glutamate decarboxylase down to 30 pg/mL [44]. A similar approach uses gold nanoparticles conjugated to anti-prostate specific antigen (PSA) antibody functionalized with horse radish peroxidase to reduce a 3,3’-diaminobenzidine (DAB) substrate giving enhancement of an immunoassay for PSA-alpha(1)-antichymotrypsin complex and achieving an LOD of 27 pg/mL [45]. This compared to earlier work where use of a mixed SAM surface with sandwich assay format and no gold enhancement gave LOD only as low as 10.2 ng/mL [46]. Gold nanoparticle tagging has also been applied to detection of microorganisms where two antibody-gold conjugates are used to amplify signal response upon sandwich binding for Escherichia coli, resulting in a three orders of magnitude improvement in detection sensitivity [47].

Figure 5. Plot of % reflectance v. time for binding of unlabelled antibody and antibody-gold conjugates to human IgG in a sandwich assay format [29]. Reprinted with permission from Lyon, L. A.; Musick, M. D.; Natan, M. J. Anal. Chem. 1998, 70, 5177-5183. Copyright 1998 American Chemical Society

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Whilst gold nanoparticle enhancement has received widespread application in immunoassays of large molecules, its effective application to small molecules (molecular weight < 2 kDa) posed a number of additional challenges. Metal nanoparticles are very bulky relative to small molecules and they can sterically impede the approach of antibodies to the small molecules and thus significantly reduce levels of antibody binding, resulting in much less sensitive assays. Furthermore, the removal of bound gold nanoparticle labels from the sensing surface is necessary to achieve a re-usable or ‘regenerable’ sensor. Where attempted, this typically requires quite severe regeneration solutions, which can damage protein-coated surfaces. As small molecule immunoassays using SPR had typically relied on immobilized antibodies and detection via protein-labeled antibodies, it was clear that a new approach was needed in surface construction and nanoparticle labeling. The steroid hormone progesterone was conjugated to a hydrophilic oligoethylene glycol linker via its 4-position to provide attachment at a point distant from existing steroid functional groups and sufficient spacing from the sensor surface to allow optimal antibody binding [22]. This conjugate was then in situ immobilized on a dextran hydrogel coated on a planar gold surface, forming a strong covalent linkage that could withstand in excess of 1100 binding and regeneration cycles, thus making a very robust and reusable sensor surface [22]. By using a hydrophilic linker, optimal antigen projection into the aqueous mobile phase could be achieved. A small molecule immunoassay was then designed around this SPR surface by mixing sample with a primary antibody and then passing the mixture over the surface. Unbound primary antibody then bound to the immobilized progesterone and this binding interaction could be enhanced by use of a secondary antibody or a secondary antibody conjugated to a gold nanoparticle. Nanoparticle diameter was optimized at 25 nm and this labeling yielded signal enhancement of 13-fold [22]. LOD for progesterone of 8.6 pg/mL was achieved, compared with 23 pg/mL for secondary antibody enhancement alone and about 1 ng/mL for assays constructed without enhancement [22], (Figure 6). This enabled a significant improvement in detection capabilities for small molecule SPR, which previously often suffered from unstable surfaces and limited sensitivity. The technique was further extended to the androgenic hormone testosterone. Analogous conjugation chemistry was employed and 25 nm gold nanoparticles coupled to secondary antibodies enhanced the sensor signal 12.5-fold enabling the antibody concentration to be reduced sufficient to obtain LOD for testosterone of 3.7 pg/mL in buffer and 15.4 pg/mL in stripped human saliva [24], (Figure 7). Without the use of gold nanoparticle enhancement, this assay system would not have been able to detect the very low concentrations of testosterone found in human saliva (29-290 pg/mL [48]); a significant analytical challenge in developing plasmonic biosensor devices for non-invasive steroid detection. An analogous approach was later adopted by Yuan and colleagues who used a mixed SAM / steroid-protein conjugate surface combined with 10 nm gold nanoparticles to enhance the inhibition immunoassay of progesterone [49]. LOD was reduced from 373 pg/mL to 5 pg/mL upon 9-fold signal enhancement by the gold, similar to the LOD achieved using covalently immobilized dextran surfaces. This approach has now been extended to different classes of important small biomolecules but often using different types of sensor surface design [9]. Ochratoxin A has been detected using 40 nm gold nanoparticles for enhancement. In this case the limit of detection was improved from 1.5 ng/mL to 42 pg/mL, resulting from a 17-fold signal enhancement [9].

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Figure 6. SPR immunoassay curves for an un-enhanced immunoassay of progesterone (●) and a 25 nm gold enhanced immunoassay (○), showing the large improvement in detection [22]. Reprinted from Analytical Biochemistry, 343(1), Mitchell, J. S.; Wu, Y.; Cook, C. J.; Main, L., Sensitivity enhancement of surface plasmon resonance biosensing of small molecules, 125-135, Copyright 2005, with permission from Elsevier

Figure 7. Schematic showing the assay steps employed in gold nanoparticle-enhanced immunoassay of the steroid testosterone, including binding, enhancement and regeneration steps [24]. Reprinted from Biosensors and Bioelectronics, 24(7), Mitchell, J. S.; Lowe, T. E., Ultrasensitive detection of testosterone using conjugate linker technology in a nanoparticle-enhanced surface plasmon resonance biosensor, 2177-2183, Copyright 2009, with permission from Elsevier

A critical new area in SPR enhancement for immunoassays is in localized surface plasmon coupled emission (LSPCE) [50]. In this technique, a metal nanoparticle is used to generate localized SPs, which can couple with dye molecules to generate an enhanced fluorescence response when the fluorescent dye couples with the evanescent field of the

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metal. SPCE is highly directional and occurs at the plasmon angle of the emission [51]. As coupling is distance dependent, the technique is only sensitive to dye molecules close to the metal surface. Fluorescent dye-labeled antibody can have a gold nanoparticle attached via protein A and this fluorescence probe is used in an immunoassay of mouse IgG in a sandwich format with antibody immobilized on a bare optical fiber which provides the incident light [50]. The fluorescence is detected by a photomultiplier tube and the assay gave a very low LOD for mouse IgG of 1 pg/mL [50].

Effects of Nanoparticle Size, Shape and Distance on Critical Assay Parameters

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Fundamental studies on the effects of parameters such as nanoparticle size, shape and distance have yielded some interesting results of central importance to the design of goldenhanced SPR immunoassays. It has been shown that the SPR wavelength increases as the size of the nanoparticle increases and there is a corresponding red shift in the spectrum [52], (Figure 8). A shift to longer wavelengths is also observed when the distance between the nanoparticles and the gold sensor surface decreases [52]. The red shifts are also greater for gold nanoparticles than for silver ones [52]. SERS studies have also examined the effects of size and distance and have found that for longer distances between nanoparticle and substrate, increasing colloid size increases enhancement up to at least 80 nm, but for shorter distances enhancement is optimized at nanoparticle sizes of 70 nm [53]. To model the effects of increasing gold colloid diameter when coated with a dielectric layer, Khlebtsov and colleagues have found that gold diameters of 40-80 nm are optimal for conjugate efficiency when considering the translation of binding events on the gold surface into changes in extinction and light scattering from the colloid [54].

Figure 8. Calculated spectra of gold nanoparticles of varying size with conjugated Raman reporter molecule at 1.2 nm distance from the surface. Inset shows comparison between calculated and measured max values [52]. Reprinted with permission from Driskell, J. D.; Lipert, R. J.; Porter, M. D. J. Phys. Chem. B. 2006, 110, 17444-17451. Copyright 2006 American Chemical Society

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Whilst gold colloids are the most common metal labels used for SPR immunoassay, increasing interest is being shown in examining the effects of using gold core shell particles. These are particles consisting of a thin shell of gold around a core of another material, typically silica. Khlebstov and colleagues have shown that using a silica core of 180 nm with a 15 nm gold shell can give a detection limit of 0.25 ng for rabbit IgG, compared to 15 ng when using a conventional 15 nm gold colloid [55] and is still superior to smaller silica core diameters.

Non-Specific Binding

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Whilst much of the research on enhanced SPR immunoassays has focused on novel assay formats, elegant surface construction and improved assay sensitivities, less attention has been devoted to the practical application of these sensors to real-world biological samples. Matrices such as blood, saliva and food products are very complex and contain high concentrations of interfering substances that can impede antibody binding and markedly worsen assay sensitivity. Furthermore, high mass components of samples can non-specifically bind to sensor surfaces. SPR techniques devoted to immunosensing in these matrices typically use either chemical extraction, chemical pre-treatment or heavy dilution of samples to overcome these matrix effects, [56] but there are alternatives. Recent work on SPR immunoassay in human saliva has demonstrated that the steroid hormone cortisol can be detected to very low physiologically relevant levels (LOD of 49 pg/mL), without the need for complex sample pre-treatment or chemical extraction, by the simple addition of sodium dodecyl sulfate to the antibody diluent [8]. This keeps non-specific binding to a low and consistent level so that the high molecular weight mucins don’t interfere with the SPR signal and don’t give incorrect concentration data. As saliva is a very viscous medium with high mucin content, this may have utility in other biological media.

Related Techniques In addition to its use in standard SPR immunoassays, SPs have been applied to enhance sensitivity in a number of related techniques. One such approach is to generate plasmons in an aluminum substrate and use them to enhance chemiluminescence through microwave irradiation [57]. Researchers have reported the ability to control a triggered chemiluminescence spatially and temporally using microwave pulses resulting in an up to 25-fold enhancement [57]. Fluorescence can also be enhanced by use of SPs. Typically, noble metal nanoparticles are deposited on a sensing surface and fluorescent dyes positioned 5-45 nm away, where the distance is optimized by balancing the close spacing needed for plasmon enhancement with the longer spacing needed to avoid metal quenching of the fluorescence [58]. These surface enhanced fluorescence (SEF) immunoassays can be applied in a chip format for potential high-throughput screening applications [58]. An earlier example looks at assessment of human chorionic gonadotropin using indirect SPR fluoroimmunoassay, which was shown to be an improvement on either SPR or total internal reflection fluorescence on their own [59]. When utilizing gold nanoparticles for SEF, the distance between the

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nanoparticle and the fluorescent dye, the size of the nanoparticle and even the solvent employed can have an effect on the levels of enhancement achieved [60], (Figure 9). 10-fold enhancement has been observed when using an optimized combination of 5 nm gold at 2 nm spacer distance and with ethanol solvent and applied to measurement of cardiac marker compounds [60]. When applying these enhancements in a fiber optic biosensor, the metal nanoparticles serve not only to generate a strong local field to enhance fluorescence but also improve scattering of fluorescent emissions and also increase the far-field signals that can be detected by the fiber optic [61]. Silver nanoparticles have also been used, for example, in the detection of insulin in a microtiter plate format using antibody fragments, where a four-fold increase in fluorescence was observed [62]. Microtiter plates have also been coated with silver colloids which then enhance the fluorescence from labeled antigens binding to the antibody functionalized surface in a single step assay [63]. When a hydrogel immobilization matrix is used, even more enhancement of fluorescence is observed [64]. Long-range surface plasmons (LRSPs) are a type of plasmon mode that propagates with much lower damping than conventional SPs and so has a much stronger electromagnetic field that can extend for micrometers out from the metal film. These plasmon modes have been harnessed for use in surface plasmon-enhanced fluorescence and combined with a micrometer thick dextran hydrogel to gain much higher immobilization capacity within the extended range of the electromagnetic field [65], (Figure 10). This combination has resulted in about a four orders of magnitude improvement in LOD for detection of free PSA into the femtomolar concentration range [65].

Figure 9. Plots showing the relationship between enhancement (%) and thickness of the SAM layer (a) or size of the gold nanoparticle (b) for surface enhanced fluorescence [60]. Reprinted from Biosensors and Bioelectronics, 21(7), Hong, B.; Kang, K. A., Biocompatible, nanogold-particle fluorescence enhancer for fluorophore mediated, optical immunosensor, 1333-1338, Copyright 2006, with permission from Elsevier

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Figure 10. Long-range surface plasmon (LRSP) format for detection of prostate-specific antigen by a sandwich immunoassay with surface plasmon-enhanced fluorescence [65]. Reprinted with permission from Wang, Y.; Brunsen, A.; Jonas, U.; Dostalek, J.; Knoll, W. Anal. Chem. 2009, 81, 9625-9632. Copyright 2009 American Chemical Society

Perhaps the most significant related technique is SERS. Whilst the exact mechanism of the SERS effect is still a matter of debate, the electromagnetic theory holds that excitation of SPs by incident light gives rise to the Raman enhancement when the plasmons oscillate perpendicular to the metal surface, and so scattering occurs and enhancement is greatest when the incident radiation is in resonance with the plasmon frequencies in the metal. SERS is increasingly being applied as a bioassay platform [66] and consideration is being given to how surfaces might be designed to maximize enhancement. An example of the use of immunogold conjugates in SERS is the detection of Hepatitis B virus surface antigen, where a silica or quartz substrate has antibody immobilized, captures the antigen and binds the second antibody attached to a gold nanoparticle that is also attached to a SERS active molecule in a sandwich format before silver staining is used to further enhance the signal. LOD of 0.5 g/mL could be obtained with such an assay design [67]. Gold nanoparticle enhancement can also be applied in more distantly related transduction techniques. Scanning Tunneling Microscopy (STM) has been used instead of SPR to detect beta-amyloid (1-42), a marker for Alzheimer’s disease, where the gold nanoparticle labels antibody in a sandwich format [68]. Other techniques that employ noble metal colloids to enhance signal, include the construction of standard sandwich immunoassays using enzyme tags that biocatalyze the deposition and subsequent growth of silver nanoparticles in a

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microwell format [69]. Transduction is often by anodic stripping voltammetry and has detected IgG down to 30 pg/mL [69]. In another variation, silver deposition occurs on a gold nanoparticle label to improve the enhancement from the nanoparticle further in an electrochemical transduction format, giving an LOD as low as 0.2 ng/mL for cardiac troponin I [70-72]. Enzyme catalyzed deposition of mass onto the SPR sensor surface was originally applied in a cuvette format, where glucose oxidase enzyme labeled with horse radish peroxidase was immobilized in the bottom of the cuvette on the SPR surface and glucose was added, mixed with a colorimetric substrate solution, which precipitated product onto the surface in proportion to the amount of hydrogen peroxide generated by the enzymatic oxidation of the glucose [73]. Gold nanoparticles have also been useful as simple high mass labels when using quartz crystal microbalance (QCM) transduction [74], where the mass of the gold alters the frequency of the microbalance significantly, allowing a linear range of 1 ng/mL - 10 g/mL for streptavidin detection, compared with LOD of 50 ng/mL without gold labeling [74]. Another transduction technique examined is microgravimetric analysis using gold nanoparticle enhancement of immunoreactions and silver metal deposition, which give a two orders of magnitude improvement in detection of human IgG [75]. Interferometry is another transduction system sometimes used and immunoassays based on this have employed a polyamidoamine (PAMAM) dendrimer to immobilize the antibody, achieving LOD for PSA of 1 ng/mL on silica surfaces [76].

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OTHER HIGH MASS LABEL ENHANCEMENTS Whilst the use of gold nanoparticles as enhancement labels is heavily favored because of its dual contributions of high mass and cooperative plasmon enhancement, other high mass labels have also been used, generally in the earlier days of research into SPR immunoassay enhancement. Liposomes are tiny vesicles enclosed with a membrane that is usually a phospholipid. They have been used as high mass labels of SPR binding interactions by biotinylating their surface and attaching them to biotinylated secondary antibodies employed in a sandwich immunoassay using avidin as a bifunctional bridge [77]. This enhancement technique based on bound mass alone improved LOD from about 1 g/mL to 100 pg/mL for detection of interferon- [77]. Latex nanoparticles of 127 nm diameter have also been utilized for enhancement of sandwich immunoassay of thyroid stimulating hormone and yielded LOD of 3 pg/mL compared with 7 ng/mL without enhancement [78]. Using such latex particles is said to increase the mass loading on the surface by a factor of 5000 times [78]. Clearly such nanoparticles are better suited to sandwich immunoassay formats for larger molecules, where the steric constraints are not as severe as for small biomolecules.

EFFECTS OF SURFACE CONSTRUCTION Fixed Planar Gold Surfaces Self-assembled monolayer (SAM) structures are perhaps the most common approach adopted for SPR biosensors that have been developed in-house by individual research teams.

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They are very simple to prepare and can contain the necessary end-groups for conjugation to a wide range of important target antigens. For example, the small molecule histamine has been immobilized via an 11-carbon chain SAM with detection by simple primary antibody binding [79]. These approaches do have the disadvantage that the SAMs are typically very vulnerable to the regeneration conditions used and so they are either applied for nonregenerable surfaces [79] or have limited lifetimes, typically fewer than 100 binding and regeneration cycles. It is now typical to use a mixed SAM where a mixture of straight chain compounds terminating in thiol end groups are used both to immobilize antibody or antigen and to passivate the gold surface. Some researchers advocate having 10% of the SAM functionalized for immobilization of proteins and 90% to serve as a blocking agent [43]. When such a surface is applied in combination with nanogold enhancement in the detection of PSA, LOD of 10 ng/mL is reported [43]. This ratio of functionalized to blocking SAM has also been applied to small molecule immunoassay by immobilizing a protein conjugate of the analyte onto the SAM and enhancing the immunoassay with 10 nm or 40 nm gold nanoparticles, where sensitive assays were obtained for progesterone [49] (Figure 11) and chloramphenicol [80] respectively. For SAM surfaces, it has also been shown that the affinity of the immobilized derivative for the antibody can have a marginal effect on the sensitivity of the assay, with immobilization of lower affinity derivatives of the analyte resulting in slightly lower LOD for trinitrotoluene (TNT) [81]. Capture efficiency can also be boosted by using higher affinity interactions as the surface binding step. Biotinylated antibodies have been bound to Cryptosporidium oocysts and the resulting antibody complex captured onto the SPR surface using streptavidin, a much higher affinity interaction than simple antibody/antigen surface binding, thus improving sensor signal [82]. The type of SAM linker chain can vary, with some groups favoring the use of polyethylene glycol chains with alkanethiol termini to immobilize benzaldehyde and develop competitive immunoassays, using gold nanoparticle enhancement or high mass labeling with avidin [83]. Another novel approach to improve sensor sensitivity slightly is to introduce ethanol to the running buffer to change the baseline position of the assay to a longer plasmon wavelength [84]. This has resulted in 10-fold improvement in LOD for detection of heat shock protein 70 [84]. Antibody immobilization to planar gold surfaces is often achieved by using protein A or protein G to orient the antibodies or antibody fragments to optimize binding with large molecule targets, and this has been applied to PSA sandwich immunoassay with 20 nm gold nanoparticle enhancement, achieving detection in the pM region [85]. In some cases, using multiple layers of neutravidin and biotinylated antibodies has been advocated as an improved sensing surface, but recent studies have indicated that these surfaces do not always provide improvements in sensitivity [86]. The exact formulation of the SAM layer and control of antigen surface coverage have significant influence on assay sensitivity when using these surfaces, with up to 10-fold improvement in LOD for small molecule immunoassay reported for use of SAMs when this formulation is carefully considered [87]. When using coupling agents such as streptavidin, special attention needs to be paid to the pH conditions employed, as these can significantly affect levels of immobilization and assay sensitivity [10]. Covalent immobilization, as previously discussed, has significant stability benefits, with some surfaces lasting more than 1100 binding and regeneration cycles [22]. The position of attachment to the small molecule antigen and the nature of the intermediate linker chain used are also critical to assay performance, as they affect the ability of the antibody to recognize

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and bind to the antigen functionalized surface. High sensor signal is desirable to achieve low LOD and so maximizing the specific antibody binding to the surface is important. The steroid hormone 17-estradiol has been immobilized onto an SPR substrate using hemisuccinate linkage to the 3-position through an existing hydroxyl group, Mannich conjugation through the 2-position and thioether conjugation through the 4-position [23]. Antibody binding to the 2-position conjugate was very low because of modification of a critical functional group on the antigen but strong binding was achieved to both 2- and 4-position conjugates (Figure 12) and immunoassay using secondary antibody enhancement could achieve LOD of 25 pg/mL [23].

Figure 11. Mixed self-assembled monolayer on a gold SPR sensor surface for immobilization of progesterone – linker – protein conjugates and subsequent immunogold-enhanced immunoassay [49]. Reprinted from Biosensors and Bioelectronics, 23(1), Yuan, J.; Oliver, R.; Li, J.; Lee, J.; Aguilar, M.; Wu, Y., Sensitivity enhancement of SPR assay of progesterone based on mixed self-assembled monolayers using nanogold particles, 144-148, Copyright 2007, with permission from Elsevier

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Figure 12. Plot of response v. primary antibody concentration for estradiol conjugated to an SPR sensor surface at positions 3- (♦), 4- (▲) and 2- (■) showing the importance of conjugation position in surface design [23]. Reprinted from Steroids, 71(7), Mitchell, J. S.; Wu, Y.; Cook, C. J.; Main, L., Estrogen conjugation and antibody binding interactions in surface plasmon resonance biosensing, 618-631, Copyright 2006, with permission from Elsevier

The composition of the metal sensing surface itself has interestingly been shown to have a significant effect on immunoassay performance. When the planar gold surface is coated with silver, LOD drops to as much as eight times lower than that achieved with gold alone in immunoassays for IgGs [88]. Coating with mercury gives approximately four-fold reductions in LOD compared with use of uncoated gold [88]. The use of such bimetallic surfaces is postulated to improve the assay performance by moving to a longer resonant wavelength and, in the case of silver, combining the better resolution of the reflectivity minimum and improved evanescent field enhancement of silver with the chemical stability of gold [88, 89]. Overcoatings of silver have been applied to gold SPR substrates using an intermediate dithiol attachment matrix and the silver mirror reaction, and this format has resulted in an 8-fold improvement in LOD for detection of human IgG [90].

Figure 13. Optical extinction spectrum of a hybrid SPR substrate with nanospheres, lozenges and bipyramids, showing the effect of shape on wavelength [96]. Reprinted with permission from Lee, S.; Mayer, K. M.; Hafner, J. H. Anal Chem. 2009, 81, 4450-4455. Copyright 2009 American Chemical Society.

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Nanoislands

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When using a continuous planar metal surface, SAMs can be patterned in nanoislands and used for immobilization of antibodies [91]. Their binding to target proteins can then be detected using imaging SPR (SPRi), where changes in refractive index can be monitored in two dimensions [91]. The metal substrate itself can also be deposited as a nanoisland [92]. This technique utilizes the localized surface plasmons (LSPs) within the metal islands and has the key advantage over planar surfaces that no complex total internal reflection optics are required, so the surface construction is very amenable to portable and low-cost biosensors and with lower metal surface area used in plasmonic sensing, they can potentially detect smaller quantities of analyte. Localized surface plasmon resonance (LSPR) has a wide range of potential applications in biosensors, including use in cell sorting and tagging and point-ofcare diagnostic devices [93]. Silver nanoisland films have been applied to LSPCE assay formats [94] using Rhodamine Red-X dyes, where immunoassay signal could be enhanced by up to 10-fold when the nanoislands are supported by glass and up to 50-fold when supported by metal [94]. These results can be explained both in terms of the plasmon coupling occurring (where the underlying metal provides a surface plasmon polariton coupling medium linking LSP dipoles in the islands), and by an increase in the spontaneous emission rate and thus the quantum efficiency of the dye in the vicinity of the plasmon field from the metal [94]. Silver nanoislands have provided substantial enhancements of fluorescence when applied to metalenhanced fluorescence immunosensing [95]. For detection of myoglobin, the high plasmon field intensities of the islands have given 7.8-fold enhancement with silver islands on glass and 50.5-fold enhancement of fluorescence when coated on metal [95]. The shape of the nanoislands can have a significant effect on performance, with acute angles typically favored for their high plasmon field intensities. An example is the coating of gold bipyramids onto silanized surfaces where they were found to give higher immunoassay sensitivities than either gold nanospheres or gold nanorods [96], (Figure 13).

Nanohole Arrays It has been known for some time that dielectric optical cavities have artificial resonance that can be used to enhance the detection sensitivity of an evanescent wave biosensor when a fluorescent label binds to the functionalized surface [97]. Modeling indicated that one could expect improvements in sensitivity of more than one order of magnitude over conventional planar waveguide substrates using equivalent sensor geometries [97] and that this was related to resonant coupling of power into the optical cavity. Gold nanowells have been fabricated to act as SERS substrates and are shown to have high electromagnetic fields localized both within and outside the wells [98]. Nanohole arrays have recently attracted a great deal of attention for their potential as biosensor substrates [99-101]. They offer several advantages over conventional monodisperse arrays of discrete metal nanoparticles, which include a reduced functionalization area and hence higher sensitivity, optical field concentration resulting in enhancement effects, and a simplified optical set-up using transmission measurement based on well-defined spectral features rather than broad plasmon peak absorptions. A gold nanohole array format has been applied to immunodetection of a small

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molecule analyte, the steroid cortisol, using specialized hydrophilic linker technology with a thiol terminus for immobilization to the gold [99], (Figure 14). The system was regenerable and demonstrated low non-specific binding. Gold nanoparticle labeling could also be used to achieve a three-fold enhancement of wavelength shift [99]. The technique offers the possibility of portable SPR biosensors using a much simpler optical format.

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Noble Metal Colloids Directed assembly of gold nanoparticles onto underlying gold substrates has been an area of increasing interest for its potential in further improving the plasmon shifts obtained upon binding of high mass molecules to the functionalized surface. Research has shown that when gold nanoparticles are assembled via thiol attachment to a planar gold surface and then used for conjugation of protein A via gold-binding protein, a generic binding surface is set up that can be applied to immobilization of antibodies [102]. Such a surface ensures optimal orientation of the antibodies whilst obtaining the benefit of an increased surface area and thus more antibody immobilization. Ten-fold improvements to sensitivity were observed when applied to detection of the pathogen Salmonella typhimurium, [102] though this could be due to increased surface area rather than plasmonic enhancement from the colloid, as it was an unlabelled assay format. Others have also utilized a similar format and claimed nano-gold enhancement of signal response by using nanoparticles thiol-linked to the underlying gold for the immobilization of antibodies, but the enhancements have been less than 2-fold over conventional un-labeled sandwich assay and could be accounted for by greater surface area for immobilization [103]. When gold nanoparticles are immobilized onto organosilane coated planar gold substrates, either through direct electrostatic adsorption or via thiol SAM attachment, the resulting surfaces give improved plasmon shifts when used in immunoassay compared with bare gold alone [104], (Figure 15). This has been attributed to the silica layer providing an increased tunneling barrier for electrons from the gold nanoparticles, thus increasing charging in the particles and inducing a plasmon shift due to coupling between the localized plasmons of the nanoparticles and the SPs of the gold substrate [104]. This translated to greatly improved detection of PSA with the thiol approach, giving up to 5-times higher binding response than the direct electrostatic format [104]. LSPR substrates are now commonly prepared by adsorption of silver nanoparticles onto glass substrates with appropriate thiol passivation to enable either attachment of biomolecules or use as a low specificity detection matrix for organic volatiles [105]. When immobilizing gold nanoparticles onto microtiter plates for LSPR sensing, the interparticle distances were found to be crucial and a larger distance was favorable for improved sensitivity in immunosensing; this was said to be related to the larger sensing volume [106]. LSPR substrates utilizing gold nanoparticles immobilized onto silanized surfaces have been applied to low-cost detection of stanozolol [107]. Variation of the shape of the gold nanoparticles immobilized can also affect the sensitivity and a recent report has indicated that gold nanorods may offer better sensitivity than comparable nanosphere fabrication for LSPR sensors, as indicated by monitoring streptavidin / biotin binding [108]. Further studies on such gold nanorod formats have found comparable sensitivity to conventional SPR formats but using much simpler optics [109], (Figure 16). There has been some interest in utilizing

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gold as a nanocomposite with other materials to construct metal-coated SPR sensing surfaces. ZnO-Au nanocomposites have been coated onto an underlying gold surface via thiol coupling and antibody attached to the nanocomposite via glutaraldehyde coupling [110]. This approach has the advantage of increasing the surface area: volume ratio of the metal surface, thus improving absorption and emission of light and electron transfer properties [110]. This format resulted in an approximate doubling in the signal response relative to conventional planar gold surfaces [110]. Metal-enhanced fluorescence can be adapted to use on beads, which enables the development of immunoassays where the metal-enhanced fluorescence of a metal-coated immobilized bead is altered upon binding of the target species to the antibody immobilized on the bead. This approach has been investigated with a view to developing flow cytometry bead-based formats using silica beads coated in nanostructured silver [111]. In this case, up to 10-fold higher fluorescence intensities could be obtained [111].

Figure 14. Cortisol SPR nanohole array immunobiosensor format with nanogold enhancement (a) and the cortisol-linker derivative used (b) [99]. Reprinted with permission from Sharpe, J. C. ; Mitchell J. S.; Lin, L.; Sedoglovich, N.; Blaikie, R. Anal. Chem. 2008, 80, 2244-2249. Copyright 2008 American Chemical Society.

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Figure 15. The effects of gold sensor surface modification on resonance wavelength shifts in immunoassay of prostate specific antigen (PSA). Silica-1: gold film/silica layer, Silica-2: gold film/silica layer/gold nanoparticle layer [104]. Reprinted from Analytica Chimica Acta, 651(1), Jung, J.; Na, K.; Lee, J.; Kim, K. W.; Hyun, J., Enhanced surface plasmon resonance by Au nanoparticles immobilized on a dielectric SiO2 layer on a gold surface, 91-97, Copyright 2009, with permission from Elsevier.

Figure 16. Plot of wavelength shift v. time for chemical activation of sensor surface (a-b), immobilization of rabbit IgG (c-d) and addition of varying concentrations of goat anti-rabbit IgG (eend) on gold nanorods [109]. Reprinted with permission from Mayer, K. M.; Lee, S.; Liao, H.; Rostro, B. C.; Fuentes, A.; Scully, P. T.; Nehl, C. L.; Hafner, J. H. ACS Nano. 2008, 2, 687-692. Copyright 2008 American Chemical Society.

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COLLOID AND BEAD-BASED PLASMONIC IMMUNOASSAY

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Plasmon Shift-Based Immunoassays Given that noble metal nanoparticles exhibit LSPR, it is possible to use the nanoparticles as the plasmonic sensing surface in place of a planar gold substrate. This extends the utility of the metal nanoparticles beyond just acting as metal SPR enhancement agents and opens the way to a suspension-based approach where a fixed sensor surface is no longer needed and LSPR shifts are monitored using simpler optics such as UV-Visible spectrophotometry. This significantly reduces the cost of the assay and often reduces the sampling time. When the interparticle distance between noble metal colloid nanoparticles changes, their absorption wavelength also changes and so changing distances can be detected using simple UV-visible spectrophotometry or resonance light scattering (RLS). This technique is essentially a directed self-assembly of metal nanoparticles using the antigen as the bridge between them. This approach has been applied extensively to oligonucleotide detection and has been expanded into homogeneous immunoassay [112]. An interesting example of the colloid-based approach is the detection of Giardia cysts in water using antibody immobilized gold nanoparticles [113]. Here, the cysts serve as a link between the immunogold nanoparticles, creating larger complexes that are separated out. The gold nanoparticle size is then increased through catalytic growth of gold on the surface, before being read in a spectrophotometer [113]. Another example is the detection of PSA using a combination of magnetic antibody conjugated beads and immunogold [114]. Here, the PSA forms bridges between a magnetic bead and multiple gold nanoparticles and the resulting complex can be magnetically separated from the sample and the bound gold nanoparticles expanded by growth of gold using the original nanoparticles as seed particles (Figure 17). The resulting color change as the nanoparticles increase in size can be either visually determined or spectroscopically analyzed [114]. Titania sol-gel matrices have also been investigated to provide protective layers for the gold nanoparticles in this sort of assay format, so that they have reduced non-specific binding to the magnetic beads, [115] although this precludes additional growth of the gold nanoparticles. Colloid-based RLS techniques have been applied to sensitive detection of glutathione with an LOD of 4.7 ng/mL [116]. When using magnetic beads as carriers for the antibody, they can be held into a fixed position using a magnetic pillar above the prism and binding to the bead surfaces can be examined [117]. Whilst this does use a fixed sensor surface, it is a useful extension of the bead approach. Another ingenious method for using gold nanoparticles in suspension-based assays involves mixing an antibody-gold conjugate with its immune protein antigen [118]. This forms a high mass interconnected network of antibody-gold and antigen that can be centrifugally separated from un-bound antibody-gold conjugate. This un-bound immunogold can then act to catalyze the deposition of further gold onto the existing gold nanoparticles by mixing with HAuCl4 and ammonium hydroxide [118], where the starting gold label acts as a seed particle. This has proven very effective in detecting the immune protein complement component 3 down to 1.52 ng/L using resonance scattering and has also been applied to measurements in human serum [118].

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Figure 17. Prostate-specific antigen (PSA) is mixed with immunogold and polyclonal antibody immobilized magnetic beads. PSA bridges between bead and gold and is magnetically separated, immunogold is detached and then its diameter expanded by growth of gold colloid [114]. Reprinted from Biosensors and Bioelectronics, 24(5), Cao, C.; Li, X.; Lee, J.; Sim, S. J., Homogeneous growth of gold nanocrystals for quantification of PSA protein biomarker, 1292-1297, Copyright 2009, with permission from Elsevier.

Colorimetric Immunochromatographic Strip Test Assays Gold nanoparticles have been used as the colorimetric label in immunochromatographic test strips, where a color change on an absorbent strip dipped into the sample indicates the presence and sometimes the concentration of the analyte. Normally such techniques are at best semi-quantitative because of problems with the reliable passage of fluid up the strips and the need to detect color changes either with the naked eye or using a rudimentary reader. Conventional formats employing gold nanoparticles tend to have an immobilized primary antibody line and then pass gold-labeled secondary antibody over the top, mixed with the sample that carries it up the strip. This then gives a red color in the primary antibody line with an intensity that is directly proportional to the concentration of the large molecule antigen. Recent work has demonstrated that when some of the immobilized primary antibody is first bound to gold nanoparticles before immobilization, then when the gold-labeled secondary antibody binds it generates a stronger color signal, thus giving an enhancement effect [119].

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When applied to measurement of total PSA concentration, an enhancement of up to 2-fold is obtained at higher concentrations [119].

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CONCLUSION The immunoassay formats that have proved so successful in RIA and enzyme immunoassay (EIA) translated smoothly to SPR transduction. From these simple beginnings it soon became apparent that ultrasensitive immunoassays using SPR would require practical techniques for signal enhancement to allow detection of ever smaller quantities of key biomolecules. From simple protein-based enhancement using protein conjugates and secondary antibodies, the technology swiftly moved to conjugation of high mass labels such as latex beads and liposomes. A major step came in the use of gold nanoparticle labels, where high mass was combined with cooperative plasmon coupling to generate high signal enhancement. Gold nanoparticles worked well for signal enhancements in sandwich immunoassays of large molecules, but significant steric effects had to be overcome to extend them to small molecule detection. Using appropriate linker and conjugation technology has enabled gold labeling chemistry to be extended to analytes such as the steroid hormones, and now detection of very low physiologically relevant concentrations has been realized even in a complex biological matrix. Careful consideration has been given to sensor surface design and this has yielded yet further improvements in assay performance both in sensitivity and improved surface stability, whilst suspension-based formats have helped to simplify plasmonic immunoassays. In future we hope to see current robust techniques for SPR signal enhancement become more standardized and applied routinely in the analysis of real-world samples, both in the laboratory and on-site. There is significant potential in exploiting enhancements from networks of interlinked gold nanoparticles rather than simply relying on discrete individual nanolabels. The construction of new surfaces tailored to achieve more intense plasmon fields may well lead the way to further improvements in assay sensitivity. The use of LSPR and nanohole array systems offers hope for in-expensive and portable SPR biosensors with sufficient sensitivity to meet today’s rigorous analytical challenges.

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Chapter 3

PLASMON DISPERSION AND DAMPING IN TWO-DIMENSIONAL ELECTRON GASES ON METAL SUBSTRATES Antonio Politano* Università degli Studi della Calabria, Dipartimento di Fisica, Rende, Italy

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ABSTRACT Herein we report on high-resolution electron energy loss spectroscopy (HREELS) measurements on surface plasmon dispersion in systems exhibiting quantum well states, i.e. Na/Cu(111), Ag/Cu(111), and Ag/Ni(111). Our results demonstrate that the dominant coefficient of surface plasmon dispersion for thin and layer-by-layer Ag films presenting quantum well states is quadratic even at small q ||, in contrast with previous measurements on Ag semi-infinite media and Ag thin films deposited on Si(111). We suggest that this behavior is due to screening effects enhanced by the presence of quantum well states shifting the position of the centroid of the induced charge less inside the geometrical surface compared to Ag surfaces and Ag/Si(111). For ultrathin Ag films, i.e. two layers, the dispersion was found to be not positive, as theoretically predicted. Annealing of the Ag film caused an enhancement of the free-electron character of the quantum well states, thus inducing a negative linear term of the dispersion curve of the surface plasmon. Moreover, we report the first experimental evidence of chemical interface damping in thin films for K/Ag/Ni(111). As regards Na/Cu(111), we found a different dispersion curve compared to thick Na films, thus confirming the enhanced screening by Na quantum well states. Results reported here should shed light on the influence of quantum well states on dynamical screening phenomena in thin films.

Keywords: Electron energy loss spectroscopy (EELS), Surface plasmons,

* Corresponding

author: E-mail: [email protected]

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INTRODUCTION

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The reduced dimensionality may induce the appearance of novel electronic states. In semi-infinite media (surfaces), surface states are formed within energy gaps of the projected bulk band structure due to the broken symmetry at the solid/vacuum interface. Electrons occupying such states are confined at the top layer of the bulk sample and form a twodimensional nearly free-electron gas. On the other hand, in thin films discrete states are formed, characterized by a notable dependence of their energy on thickness. Such states, called quantum well states (QWS), describe standing electron waves confined within the film. Recent experimental [1-8] studies have demonstrated the existence of variations with film thickness for properties such as the electronic density of states, electron-phonon coupling, chemical reactivity, superconductivity, magnetism, surface energy, and thermal stability. Further information on the physical and chemical properties of thin films can be obtained from their electronic collective excitations and, in particular, from the dispersion relation of the surface plasmon (SP) [9-14]. Nevertheless, the influence of electron quantum confinement on the dispersion relation of electronic collective excitations in nanoscale thin films has not been investigated yet. Moreover, the different distribution of occupied and unoccupied electronic states in thin films with respect to surfaces would imply enhanced damping mechanisms for collective excitations by creating electron-hole pairs [15-18]. High-resolution electron energy loss spectroscopy is a suitable technique for such aims as it allows measuring the dispersion curve of the frequency and the line-width of the SP [1924]. Herein HREELS measurements of systems exhibiting electron quantum confinement, i.e. Ag/Ni(111), Ag/Cu(111), and Na/Cu(111) are reported. Our results should shed light on the influence of QWS on dynamical screening phenomena in thin films.

FORMATION OF QWS The electronic structure of films has been widely investigated by photoemission spectroscopy in recent years. These studies focused particularly on two main spectral features: the occupied Shockley-type surface state resulting from the termination of the crystal [25-42], and the QWS due to the quantum confinement of the sp valence electrons in the adlayer [43-56]. The binding energy of the Shockley state depends critically on film thickness. Figure 1 summarizes experimental photoemission data acquired at 300 K for Ag/Cu(111) by Mathias et al. [57] as a function of silver film thickness. Moreover, the film morphology was found to be a very important parameter influencing the electronic structure. Recently, it has been demonstrated that the dispersion of QWS changes upon annealing the adlayer [58]. Angle-resolved photoemission experiments showed that Ag QWS on Au(111) have flat inplane dispersion in a disordered film and a nearly free-electron-like dispersion in an annealed and well-ordered film. Accordingly, the sp density of states of the film may be tuned by annealing. As an example, Figure 2 shows photoemission data recorded at room temperature for a silver film (of nominally 15 ML) on Cu(111), both before and after the annealing procedure. The peak position of QWS shifted slightly to a higher binding energy. Moreover,

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annealing reduced the broadness of the peak as a consequence of a higher flatness of the adlayer (Figure 3).

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Figure 1. Energy of the ν = 1 to ν = 4 QWS and resonances as a function of nominal film thickness as determined from photoemission spectra from non-annealed silver films (adapted from Ref. [57])

Figure 2 (a). Photoemission map of a 15 ML thick silver film (nominal thickness) before (a) and after (b) annealing at 450 K; (c) normal emission spectra taken from (a) and (b) clearly visible are the modifications in the spectral shape of the quantum wells state (QWS) and in the energy region, where the Shockley surface state (SS) is observed. S1 and S2 indicate the two distinct Shockley surface states appearing in the spectrum after annealing; the two Lorentzians in the bottom graph result from a fit into the QWS-peak and S1-peak of the bottom spectrum. Also shown as a solid line is the total fit result. A Gaussian was used to reproduce the low energy tail of the spectrum and a Lorentzian to fit the peak S2 (adapted from Ref. 14 [57]).

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Figure 3. Schematic summary of the morphological changes induced by the heat treatment of the silver film (Adapted from Ref. [57])

EXPERIMENTAL

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Measurements were carried out in a UHV chamber operating at a base pressure of 5x10-9 Pa, equipped with standard facilities for surface characterizations, described elsewhere [5962]. HREEL experiments were performed by using an electron energy loss spectrometer (Delta 0.5, SPECS). The samples were single-crystal surfaces of Ni(111) and Cu(111) with a purity of 99.9999% which were cleaned by repeated cycles of ion sputtering and annealing at 800-900 K. Surface cleanliness and order were checked using Auger electron spectroscopy measurements and low-energy electron diffraction (LEED), respectively. Silver was deposited onto the Ni(111) and Cu(111) surface by evaporating from an Ag wire wrapped on a tungsten filament. Well-ordered Ag films could be obtained at very low deposition rates (≈0.05 ML/min). Alkalis (Na, K) were evaporated in the UHV chamber by means of a well outgassed dispenser (Saes Getters). A particular care has been dedicated to avoid CO contamination [63-69]. The occurrence of the p(1x1)-Ag, (3/2x3/2)-Na, and p(2x2)-K LEED patterns was used as the calibration point of θAg=1.0 ML θNa=0.44 ML, and θK=0.25 ML, respectively. A constant sticking coefficient was assumed to obtain other desired coverage. Coverage has been also controlled through X-ray photoemission spectroscopy, whose analyzer is described in Ref. [63]. The energy resolution of the spectrometer was degraded to 7 meV so as to increase the signal-to-noise ratio for off-specular spectra. The angular acceptance α of our electron analyzer was ±0.5°. All depositions and measurements were made at room temperature.

RESULTS AND DISCUSSION 1. Ag Films on Ni(111) Ag surfaces are characterized by a strong lowering of the SP energy, which follows a positive dispersion as a function of the parallel momentum transfer. Such behavior was ascribed to the presence of filled d bands [12, 20, 64-69]. However, experimental studies on low-dimensional Ag systems, such as ultrathin films on metal [9, 15, 16, 70-74] and semiconductor [18, 75, 76] substrates, nanowires [77] or nanoparticles [78, 79] are less common.

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Ag on Ni(111) offers the possibility of investigating the relationship between quantum electron confinement and the damping of collective excitations in ultrathin films as confined electron states in the Ag adlayer exist. As a matter of fact, photoemission measurements [80] revealed the occurrence of Ag QWS for films with thickness from 0 to 15 layers. Ag on Ni(111) is an example of film growth in large (16%) mismatched materials. Such a large misfit determines the silver film to have the crystalline structure of bulk silver even for ultrathin Ag layers [81]. Moreover, the small solubility of Ag into Ni prevents from the dissolution of the silver layer into the nickel substrate [82]. As a matter of fact, the formation of a surface alloy was not reported for this system [53, 82, 83]. For coverages higher than three layers, a reconstruction of the silver overlayer was observed [81]. The deposited Ag layer exhibits a 7x7 moiré structure . Figure 4 shows HREEL spectra for thin Ag layers on Ni(111) as a function of Ag coverage. The spectrum of the Ni(111) surface is characterized by a broad peak at 1.0 eV. For less than 2 ML of Ag, the spectrum is extremely broad without a well-defined peak. A broad Ag SP at 4.2 eV arose for Ag coverages above 2 ML. However, its plasma energy was considerably higher than that of the SP of semi-infinite Ag, as also found for Ag/Si(111) [18, 76]. As the Ag coverage is increased, the energy of the plasmonic mode reduced and its line shape became sharper. Such finding has to be ascribed to the s-d polarization. For thin Ag films, the overall screening of the plasmon via the polarizable d electron medium diminishes and a higher SP frequency occurs [66]. Moreover, the spill-out region not affected by s-d polarization becomes more important, causing a further blue-shift of the SP frequency. Important information on the screening properties of metal systems is given by the dispersion relation of the SP.

Figure 4. HREEL spectra for Ag/Ni(111) as a function of Ag coverage

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To measure plasmon dispersion, values for the parameters Ep , impinging energy, and θi , the incident angle, were chosen so as to obtain the highest signal-to-noise ratio. The primary beam energy used for the dispersion, Ep =40 eV, provided, in fact, the best compromise among surface sensitivity, the highest cross-section for the plasmonic excitation, and q|| resolution. As,

the parallel momentum transfer q|| is determined by the values of Ep , Eloss, θi and θs :

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where Eloss is the energy loss and θs is the electron scattering angle [17, 84]. Hence, it is possible to estimate the integration window in reciprocal space:

where α is the angular acceptance of the apparatus [17, 84]. Under our experimental conditions, Δq||=0.012 Å-1, much less than the scanned range in the reciprocal space. Selected HREEL spectra for 10 ML of Ag as a function of the scattering angle are shown in figure 5. The Ag SP energy dispersed from 3.751 up to 3.880 eV. The measured dispersion curve Eloss(q||) in Figure 6 was fitted by a second-order polynomial given by: Eloss(q||)=A+Bq||+Cq||2 where A=(3.751±0.002) eV, B=(0.000±0.004) eV∙Å, and C=(1.57±0.04) eV∙Å2. Hence, the dispersion curve for 10 ML Ag/Ni(111) reported in Figure 5 is purely quadratic, as the linear coefficient B is null. Similar results were obtained for slightly different Ag thicknesses for which QWS were also observed to exist by photoemission spectroscopy [80]. At higher thicknesses at which the discrete electronic structure of QWS [80] is vanishing, the dispersion curve deviates from being purely quadratic. As expected [9, 15, 16, 18, 74, 85, 86], the energy of the SP changed with Ag coverage. It is worth mentioning that great efforts have been devoted to solve the controversy concerning the quadratic versus linear form of the dispersion curve. The conclusion of this long debate [68, 87-89] is that the dominant coefficient of the SP dispersion for small momenta is always linear. On the other hand, quadratic terms become important at higher values of the parallel transfer momentum q||. However, quadratic terms are nearly absent for Ag(100) [21, 68].

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Figure 5. HREEL spectra as a function of the scattering angle for 10 ML Ag/Ni(111) at T=300 K

Figure 6. SP dispersion as a function of the parallel transfer momentum for 4.5, 8.5, 10.0, and 35.0 ML Ag/Ni(111) at T=300 K. The solid line indicates the best-fit curve for experimental data

According to Feilbelman's model [90-98] of the SP dispersion, a direct relation between the linear coefficient of the dispersion curve and the position of the centroid of the induced charge, d┴, may be found [9, 99]. In Ag such centroid lies inside the geometrical surface (z0.2 Å−1 electrons may be promoted from occupied to unoccupied electronic states such that the corresponding plasmon peak would broaden considerably until decaying into the single-particle excitation continuum.

Figure 7. (a) HREEL spectra for 10 ML Ag/Ni(111) as a function of the primary electrons beam energy. (b) (■) Loss energy and (○) FWHM as a function of the inverse of the kinetic energy of primary electrons

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Figure 8. (a) FWHM of the Ag SP peak for 10 ML Ag/Ni(111) as a function of the energy of the Ag SP, as obtained in different experimental conditions (■) θ i s =55° fixed and Ep varying (○)θi =55°, Ep =40 eV fixed and θs varying. (b) FWHM of the SP peak for 10 ML Ag/Ni(111) as a function of the parallel momentum transfer q||, calculated using with the following experimental parameters: θ i =55°, Ep =40 eV. The abrupt increase of the FWHM beyond q ||=0.2 Å−1 indicates that an extra decay channel due to indirect single-particle transition between surface electronic states opens up

The width of the SP peak at q||=0 is a sensitive function of the lattice potential, i.e. it is influenced by the periodic potential of the bulk. Such decay mechanisms can be direct or mediated by the exchange of reciprocal lattice vectors or phonons [68]. However, at larger values of q|| the plasmon lifetime becomes less sensitive to the bulk lattice potential and the increasing of the damping is then caused by electron-hole pair excitations in the surface region. In real systems the lifetime of the plasmon is further limited by the scattering against crystallographic defects. For a thinner film, extra decay channels exist as compared with a thicker film. As a matter of fact, sp-sp interband transitions were found [16, 18, 100] to be more efficient in thinner films rather than in thicker ones. A comparison of the behavior of the FWHM of the Ag SP in 10 ML Ag/Ni(111), 10 ML Ag/Si(111), and Ag(111) is reported in Figure 9. Panel (a) shows the FWHM dependence on the Ag SP energy, while panel (b) on the parallel transfer momentum. For 10 ML Ag/Si(111), Yu et al. [18] found for different thicknesses an initial decrease of the FWHM as a function of the Ag SP energy, followed by an abruptly increasing. Such abrupt increase was ascribed to the opening of a new damping channel, i.e. intraband transitions between Ag 5sp-derived QWS. The initial decrease of FWHM was associated with the increased surface barrier due to the ionic pseudopotential of the crystal. On the contrary, the FWHM of Ag SP in 10 ML Ag/Ni(111) has an initial flat dependence on both Ag SP energy and q || and a stronger dependence beyond a critical value of both Ag SP energy and q ||. The notable difference

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existing in the behavior of the FWHM of Ag SP in 10 ML Ag deposited on Si(111) and on Ni(111) may be tentatively ascribed to substrate effects. The FWHM of the Ag SP in Ag(111) [21] has, in analogy with 10 ML Ag/Ni(111), an initial flat dependence on both Ag SP energy and q||. The critical values of Ag SP and q|| were found to be 3.78 eV and 0.15 Å-1, respectively. Beyond such values, the FWHM significantly increased. It is worth stressing that the FWHM of the Ag SP in Ag(111) was always notably lower than that of Ag SP in thin films. This finding is ascribed to enhanced damping processes via 5sp-5sp indirect transitions, due to the different electronic properties of thin films with respect to surfaces.

Figure 9. (a) FWHM of the Ag SP peak as a function of the energy of the Ag SP for (∆) 10 ML Ag/Ni(111); (■) 10 ML Ag/Si(111) (data taken from ref. [18]); and (+) Ag(111) (data taken from ref. [21] || for (∆) 10 ML Ag/Ni(111); (■) 10 ML Ag/Si(111) (data taken from ref. [18]); and (+) Ag(111) (data taken from ref. [21] .

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Figure 10. HREEL spectra for K/10 ML Ag/Ni(111) as a function of K exposure. (b) (■) Ag SP energy and (○) FWHM as a function of K coverage.

If the surface is exposed to chemically reactive atoms or molecules, the distribution of occupied and unoccupied electronic states changes. Accordingly, differences in damping processes and in the energy of the plasmonic excitation are quite expected. Rather than considering the new overlayer plasmon, we investigated the overlayer-induced modification of the substrate plasmon. Figure 10a shows Ag surface excitation spectra for increasing amounts of adsorbed K. At low K coverages, the Ag SP peak is only weakly affected. As the K coverage approaches one monolayer, the FWHM of the SP increased from 0.17 (clean Ag layers) up to 0.70 eV. Moreover, upon K adsorption the plasma energy of the Ag SP shifted from 3.80 down to 3.56 eV. The red-shift of the Ag SP energy may be ascribed to a charge transfer from the Ag substrate to the adsorbates. A reduced Ag SP frequency in the presence of electronegative coadsorbates was reported also for Ag single-crystal surfaces [14, 22]. A similar red-shift of the SP was revealed also for K/Ag(1 1 0) [104] and, hence, we can suggest that it is K-derived.

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It is more correct to describe the plasmonic excitation as the K/Ag interface plasmon rather than a red-shifted Ag SP. New adsorbate-induced electronic states arose at the interface. Hence, the significant plasmon broadening is due to new channels for decay into electron-hole pairs at the K/Ag interface. The interband transitions involving the overlayerinduced band below the Fermi level thus influences the SP energy and gives rise to a red-shift of the SP energy. Such SP broadening is ascribed to chemical interface damping. It was observed in absorption spectra of small metal particles embedded in a reactive matrix [105, 106], but it is the first experimental evidence of its occurrence in thin films.

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Ag/Cu(111) In Ag/Cu(111) silver islands appear for Ag coverages higher than two layers, which instead grow layer-by-layer. Around each spot of the p(1x1)-Ag, LEED measurements showed also the occurrence of a (9.5x9.5) reconstruction for the first Ag layer and of a (9x9) reconstruction in correspondence of θAg=2.0 ML, in excellent agreement with previous structural studies on this system [107, 108]. Hence, this system offers the opportunity to study the dependence of dispersion and damping dispersion of the SP as a function of the growth mode. Density-functional theory calculations based on s-d polarization model found an initial negative dispersion for two layers of Ag deposited on Al surfaces [66]. By contrast, experiments [18] carried out on Si(111) reported a positive behavior. It would be extremely useful to study dynamic screening processes in the case of a thickness of two layers. In fact, two layers are commonly accepted as a borderline between interface physics and thin-films physics [73, 79, 129-131]. As a matter of fact, such coverage constitutes the minimal thickness necessary to observe in the loss spectra well-distinct features assignable to collective excitations, while the very broad loss features observed from 0 to 2 ML are related to single-particle transitions [109]. However, experimental measurements on Ag nanoscale thin films deposited onto metallic substrates which could verify the effects of s-d screening have not yet been performed. On the other hand, the collective excitations in two alkali layers on metal substrates were extensively studied [73, 79, 129-132] as significant differences compared to thick films were found [110]. Contradictory results about the existence of the SP at small momenta were reported [81, 129, 130, 134]. The investigation of the behavior of collective excitations also for two Ag layers on metallic substrates should provide a significant advancement in understanding dynamic screening processes at metal surfaces. Moreover, a shifted bulk plasmon (BP) is expected to exist at small momenta, where the SP weight is vanishing [66]. Selected HREEL spectra for 2 ML of Ag as a function of the parallel transfer momentum are shown in Figure 11 (left panel). In contrast with all previous experimental works on Ag surfaces [20, 64, 66, 88, 99, 111115] and layer-by-layer Ag films on Si(111) [18], the frequency of the SP did not increase as a function of the parallel transfer momentum (Figure 12).

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Figure 11. HREEL spectra as a function of the scattering angle for 2 (left) and 5 ML Ag/Cu(111) at T=300 K

Figure 12. SP dispersion as a function of the parallel transfer momentum for 2 ML and 5 ML of Ag on Cu(111) at T=300 K

Present results well agree with theoretical calculations by Liebsch [66]. As the thickness of the Ag film is reduced, the overall screening of the charge associated to the SP via the polarizable d electronic medium diminishes, giving higher plasmon energy. Furthermore, the spill-out region not affected by s-d polarization becomes more important, causing a further blue-shift of the plasmon frequency. Moreover, the differences that we found with respect to measurements on 2.5 ML Ag/Si(111) [18] were ascribed to the enhanced screening properties of metal substrates compared to semiconductor surfaces and to more efficient screening processes due to electron quantum confinement [116, 117]. Interestingly, for two Ag layers, in the limit of small momenta, we observed the excitation of the BP, shifted by s-d screening at a frequency ωp*=ωp /√εd, where εd is the local dielectric function and ωp is the s-p BP frequency [66]. The real part of the dielectric function decreases as the film thickness is reduced, as a direct consequence of the occurrence of a less sharp onset of transitions involving d states in thin films compared to bulk Ag [66]. As a matter of fact, a significant energy step between small and high momenta exists due to the different nature of the excitation, i.e. BP and SP, respectively. The lack of the SP excitation at q||=0 was reported

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also for ultrathin alkali layers [109, 118, 119]. On the contrary, in ultrathin Ag layers on Si(111) [18] the SP was excited even at small momenta. Once again, such evidence should be taken as a fingerprint of very different screening processes between Ag/Si and Ag/Cu. However, increasing Ag coverage, i.e. 5 ML (right panel of Figure 11 and Figure 12), the SP was excited also at small values of q||, as generally found for thick alkali layers [120] and Ag/Si(111) [18]. In Figure 13 we compare the SP dispersion for different Ag systems. The behavior found for two Ag layers on metal substrates differs from the behavior found in all other Ag systems exhibiting, instead, a positive and quadratic dispersion. As a matter of fact, the dispersion curve calculated by Liebsch [66] for 2 ML Ag/Al well agree with present results for Ag on Cu(111).

Figure 13. Behavior of the dispersion curve of the Ag SP for Ag(111) (dashed-dotted line) [21], 2 ML Ag/Cu(111) (filled squares [72]), 2 ML Ag/Al (theory, continuous line) [66], and 2.5 ML Ag/Si(111) (dotted line) [18]. The values of the SP energy for the various dispersion curves were normalized to their respective SP energies measured at smallest value of q|| in order to put in evidence the different behavior

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The granularity of the as deposited film for coverages higher than two layers is argued from the behaviour of the dispersion of the collective excitation (Figure 14). The absence of dispersion below a critical wave vector, i.e. 0.15 A-1, indicates that the s electrons oscillate independently in the single (111)-oriented grains. Similar results were reported for Ag/Si(111) [76]. The critical wave vector was suggested [76] to be related to the average island size through the relation Qc=2π/d

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From the above equation, we estimate that the grain size in Ag/Cu(111) is about 30-40 Å The propagation of the SP can occur only for modes whose wavelength is smaller than the diameter of the single grain. Interestingly, the Ag grains behave like isolated clusters with respect to the plasmonic excitation. The behaviour of the SP dispersion well agrees with the Stranski-Krastanov growth mode of this system [143-147]. The increasing strain caused by adsorbed layers destabilizes the film and induces clustering [121]. This behavior arises from the large lattice mismatch between Ag and Cu (13%). Low-energy electron diffraction experiments showed the occurrence of a 9x9 over-structure of the deposited silver film, as previously reported [122]. It is worth noticing that we have not evidences for the existence of Mie plasmons within Ag islands. Two well-distinct Mie plasmons at 3.1 and 3.9 eV were revealed only for Ag deposited on metal-oxide surfaces [78, 79] and not for three-dimensional islands on metal [15, 74] and semiconductor [76] surfaces. As concerns metal/metal interfaces, the occurrence of Mie plasmons was invoked only for Na quantum dots on Cu(111) [4, 123] but only for a very restricted alkali thickness range. Mie plasmon merged into the ordinary SP already for two nominal Na layers.

Figure 14. Dispersion of the SP energy for different Ag coverages. The lack of a dispersion before a critical wave-vector indicates that the SP is confined within Ag grains

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In order to remove the SP confinement, an annealing of the film at 400 K was performed. Loss spectra in figure 15 provided evidences for drastic morphological changes in the film and a higher degree of ordering, as suggested by the analysis of the LEED pattern. Annealing the surface at 400 K caused significant changes in the dispersion curve and, in particular, the loss of the SP confinement. The measured dispersion curve Eloss(q||) of the annealed film, reported in Figure 16, was fitted by a second-order polynomial given by: Eloss(q||)=A+Bq||+Cq||2 (A=3.791±0.006 eV; B=-0.60±0.09 eV∙Å; and C=3.4±0.3 eV∙Å2)

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The linear coefficient was found to be slightly negative. This finding was ascribed to the enhanced sp density of states existing in thin Ag films, as a direct consequence of the presence of QWS. Increasing the free-electron character of the QWS by annealing [58] should imply the occurrence of a negative linear term.

Figure 15. HREEL spectra for as-deposited (left panel) and annealed (right panel) 22 ML Ag/Cu(111)

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Nonetheless, the value of the linear coefficient is still enough higher than the linear coefficient of the SP dispersion curve of alkalis [124], aluminum [23, 125, 126], or alkalineearth metals [127]. Such finding leads us to suggest that the centroid of the induced charge lies in the close vicinity of the jellium edge [93-95, 98, 124], in contrast with all other Ag systems [9, 17, 68, 115, 128], but not outside as for simple metals. Interestingly, the quadratic coefficient coincides with that of SP dispersion in Ag(111) [21], i.e. the surface with the same crystallographic orientation. Significant differences exist between spectra acquired for annealed (figure 15) and sputtered films (figure 17). The dispersion curve measured in a sputtered Ag film reported in figure 16 shows that the quadratic term is predominant: (A=3.760±0.004 eV; B=-0.08±0.06 eV∙Å; and C=2.5±0.2 eV∙Å2).

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Accordingly, an increased linear coefficient and a decreased quadratic term were obtained by fit procedure. Contrary to the sputtered Ag(100) surface [19] for the quadratic term the bulk value of 6 eV∙Å2 [129] was not recovered. In our opinion, the link proposed in ref. [19] between the value of the quadratic term of the SP dispersion and that of the bulk plasmon, related to bulk properties, should be revised. The occurrence of an increased linear coefficient suggests that sputtering induces a significant shift of the position of the centroid of the induced charge associated to the SP compared with that of SP in annealed films. It is worth mentioning that ion bombardment of a growing film was found to produce both bombardment-induced segregation normal to the film surface and an advancing nanoscale subsurface diffusion zone [130]. Such phenomena should be considered in theoretical studies on the electronic response of sputtered thin films (still lacking). Moreover, our results provide the grounds for angle-resolved photoemission experiments shedding light on the sputtering-induced modifications of the QWS.

Figure 17. HREEL spectra for a sputtered 22 ML Ag/Cu(111) surface

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Damping processes of the SP in Ag/Cu(111) Figure 18 (left panel) shows the thickness-dependence of the full-width at half maximum (FWHM) as a function of the parallel momentum transfer for Ag/Cu(111). An initial negative behavior of the FWHM was found for ultrathin films, while for higher coverages (22 ML) only a poor dispersion was found (Figure 16). A comparison among the dispersion of the SP line-width for 5 ML Ag/Cu(111), 5 ML Ag/Si(111) [18], and Ag(111) [21] shows (Figure 18, right panel) that for Ag films a critical wave-vector beyond which the dispersion became positive exists, while for the single-crystal surfaces the initial dispersion of the FWHM is nearly flat. However, the value of the turning point differs in 5 ML deposited on Cu(111) and Si(111) (0.19 and 0.08 Å-1, respectively). This behavior of the FHMM is well described by a theoretical model recently proposed [131] on plasmon lifetime in free-standing Ag layers. The behavior of line-width of the Ag SP as a function of the parallel momentum transfer was found to be characterized by a negative behavior of the line-width for small momenta up to a critical wave-vector. This finding was ascribed [131] to the splitting between symmetric and anti-symmetric excitation modes and the enhanced electron-hole pair excitation at small q||. For higher values of q||, a linear increase of the line-width was reported.

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For the as-deposited Ag film, the initial dispersion of the FWHM is negative (Figure 19). The behavior became positive after a critical wave-vector (0.10 Å-1) This finding is in agreement with results for Ag/Si(111) [18] and recent theoretical calculations for freestanding Ag slabs [131]. However, the behavior of the FHWM for single-crystal Ag surfaces is positive [68] and was recovered by annealing the Ag film. It is worth noticing that in all cases (as-deposited, annealed, and sputtered Ag film), the value of the FWHM is higher than for Ag semi-infinite media (for Ag(111) the FWHM at small momenta is 69 meV [21]). As discussed above, the different distribution of occupied and unoccupied electronic states in thin films with respect to surfaces would imply enhanced damping mechanisms for collective excitations by creating electron-hole pairs. The different behavior with respect to the case of Ag/Ni(111) should be ascribed to differences in both the growth mode and in the nature of QWS. In fact, due to the absence of a gap in Ni(111), the character of the quantum well states in such two systems differs substantially [102]. Quantum well states on Ag/Cu(111) are characterized by standing wave patterns, not supported on Ag/Ni(111). Furthermore, on Ag/Cu(111) the interfacial transmittivity is suppressed, with an enhanced specular reflectivity. The opposite occurs for Ag on Ni(111). Such significant differences in the electronic properties between these two bimetallic surfaces should in principle imply quite different Landau damping processes of the plasmonic excitation.

Figure 19. Behavior of the FWHM for as-deposited, annealed, and sputtered Ag films (22 ML) on Cu(111)

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It is worth noticing that sputtering induces a broadening of the SP line-width [74, 132]. The SP peak became progressively asymmetric, as indicated by the increasing of the skewness , i.e. the third standardized momentum [133, 134], upon sputtering.

Na/Cu(111) The (111) surface of noble metals (Cu, Ag, Au) exhibits a large confined gap within the

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projected bulk band structure centered at the  point of the surface Brillouin zone [30, 39, 135, 136]. The adsorption of Na on Cu(111) has been extensively investigated recently. The existence of Na 3pz-derived confined electron states has been well established [107, 162-170]. Multilayers of alkali metals are easily grown on Cu(111) even at room temperature (RT) while in other systems the growth of a second alkali layer is possible only at liquid nitrogen temperature [137]. A study reporting the dispersion curve of the SP in alkali thin layers at RT is still lacking. HREEL spectra of Na adsorbed at RT on Cu(111) as a function of coverage are shown in Figure 20. For the clean Cu(111) surface, the onset of collective excitations of valence electrons could be detected at 2.1 eV [4, 84, 138]. A broad loss feature, peaked at 3.70 eV and assigned to the Na SP, gradually arose in the spectrum as a function of Na coverage. Loss spectra in Figure 21 show the SP of two layers of Na/Cu(111) as a function of the scattering angle for Ep=20 eV. The spectrum recorded in the specular geometry is centered at 3.70 eV and it exhibits clear energy dispersion for off-specular angles.

Figure 20. HREEL spectra of Na/Cu(111) as a function of the alkali coverage. The incident beam energy Ep was 20 eV

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Figure 21. Electron energy loss spectra of 0.90 ML Na/Cu(111) at different scattering angles θ s. The incident beam energy Ep was held constant at 20 eV and all spectra were recorded at an incident angle of θi =55° with respect to the sample normal

As mentioned above, changing the primary beam energy of the electrons, it is possible to modify their mean free path in the solid and so their penetration length [103]. This allows a direct control of the surface sensitivity. Moreover, the modification of the electron penetration upon changing the impinging energy would imply, according to random-phase approximation calculations [139], a change in the position of the reflection plane at which probing electrons are scattered within the extended electron-density distribution. On the basis of the above result, a strong dependence of dynamic screening processes on impinging energy is expected especially for systems in which electrons are confined into a two-dimensional space. Hence, spectra were acquired also for a higher primary energy, i.e. 100 eV, in order to reveal such behavior.

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Multipole SP in Na/Cu(111) became quite evident in spectra taken with Ep=100 eV behind 3 degrees off-specular (figure 22). Its energy was found to be 4.70 eV, in excellent agreement with the value reported for bulk Na by Tsuei et al. [140], i.e. 4.67 eV. However, the multipole mode was not revealed for lower impinging electron beam energies, thus suggesting the existence of threshold primary beam energy. The measured dispersion curve Eloss(q||) for Ep=20 eV reported in Figure 23 was fitted by a fourth-order polynomial [68] given by:

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Eloss(q||)=a+bq||+cq||2+dq||3+eq||4 where a=3.99 eV, b=0.70 eV∙Å, c=-29.53 eV∙Å2, d=113.14 eV∙Å3, and e=-112.31eV∙Å4. The slope of the dispersion curve is negative up to 0.25 Å−1, then the loss energy of the SP increases with increasing q|| and the dispersion becomes definitively positive. It should be noticed that the SP was not damped into single-particle transitions until q||=0.53 Å−1, while the critical wave-vector of alkali SPs was found to be around 0.30 Å−1 both in thin [109, 118] and thick [124] alkali films. Furthermore, no loss features were revealed at the lowest values of parallel momentum transfer (below 0.10 Å−1). A similar result was found by Zielasek et al. [119] in Cs/Si(111)(7x7) in which the SP was suppressed for small wave vector. These findings are in agreement with Liebsch’s calculations [66] predicting that for two layers of alkali metals the spectral weight of the SP is vanishing at q||=0. In other words, at q||=0 only the excitation of the so called multipole SP is possible. In this regard, we notice that the Na/Cu(111) system behaves differently from K/Ni(111) [109] for which the excitation of the SP was found to be possible. The energy dispersion of the SP of two Na layers was 430 meV, while it was 130 meV for thick Na films. Such finding could be explained by screening effects that push the position of the induced charge density centroid more outside the substrate jellium edge than in thick Na films. We propose that electron quantum confinement in Na QWS may be the responsible of such result. In fact, the electron confinement of Na electrons on Cu(111) was found [100, 101] to drastically change the dynamical screening properties of this system. The occurrence of electron confinement in Na QWS, moreover, modifies the electron charge-density distribution. Hence, the reflection of external charges at different distances from the surface [139] should affect the electronic response of this system. The dispersion curve of the SP, measured using a primary electron beam of 100 eV (figure 24) is quite different from that in Figure 23. The energy dispersion of the SP is lower (70 meV vs. 430 meV). The measured dispersion curve Eloss(q||) for Ep=100 eV was fitted by a fourth-order polynomial [68] given by: Eloss(q||)=a+bq||+cq||2+dq||3+eq||4 where a=3.84 eV, b=-0.79 eV∙Å, c=6.22 eV∙Å2, d=-17.98 eV∙Å3, and e=-20.99 eV∙Å4. It showed an initial negative behavior that became positive above 0.15 Å−1, in analogy with the dispersion curve of SPs of other simple metals.

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Figure 23. Na SP energy as a function of q|| (Ep=20 eV)

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Figure 24. Na SP energy as a function of q|| (Ep=100 eV)

Figure 25. The SP dispersion versus the parallel momentum q || for: (■) two Na layers, Ep=100 eV ; (○) two Na layers, Ep=20 eV ; (-) thick Na film (data taken from ref. 57 [124]) Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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It should be noticed that for Ep=100 eV the SP could be excited even at small momenta. This is a combined effect of the higher dipole/impact scattering ratio of the excitation (see Figure 29 and its discussion) and of the increased integration window [68] in reciprocal space. The latter implies the collection of a finite range of momenta of the outgoing electron. A dispersion curve not dependent on the primary electron beam energy is expected only for the case of pure sheet plasmons [11, 175-186], i.e. two-dimensional plasmons strictly confined to the surface and with a square root-like dependence on q||. However, HREELS measurements revealed that the dispersion curve depends on the impinging energy. When the thickness of the film gets smaller, the tail of the wave function of the QWS can reach and interact with the substrate. The main effect is a hybridization interaction between overlayer and substrate states . We suggest that the occurrence of such interactions between overlayer states and substrate ones may lead to the observed differences in the two dispersion curves. Figure 25 shows a comparison between the dispersion curve obtained for two layers of Na on Cu(111) and the dispersion curve of the SP of a thick Na film [124]. The plasmon energy for a nanoscale thin Na film is lower because screening effects make lower the induced average charge-density and thus the SP energy, as found in two layers of K on Ni(111) [109]. The differences in the critical wave-vector of the SP for each dispersion curve are quite evident, thus suggesting strong differences about damping processes. The damping of the plasmon excitation is clearly revealed by the trend of the full width at half maximum (FWHM) versus q||, as shown in Figure 26 for Ep=20 eV. The width of the Na plasmon initially decreased, followed by a steep increase as a function of q||. Experimentally, a similar behavior was also observed for a variety of metal surfaces such as Ag surfaces , Mg(0001) [127], Al(111) [23], and Cu(111) [84]. On the contrary, the FWHM of the SP in thick films of Na and K [124] increases for increasing q||, while it was almost constant in two layers of K on Ni(111) [109]. Existing theories [69] predict that, with increasing of the momentum q|| parallel to the surface, the width of the SP rapidly increases due to the decay into electron-hole pairs (Landau damping). The behavior shown in Figure 26 could arise from a more efficient disexcitation channel of the SP by single-particle transitions at small momenta. Such assumption could explain a so high critical wave-vector (0.53 Å−1) of the plasmonic excitation too.

Figure 26. FWHM of the Na SP peak as a function of q || (Ep=20 eV)

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The FWHM of the SP for Ep=100 eV was characterized by a negative dispersion vs. q|| (figure 27). To the best of our knowledge, it is the first time that a FWHM was found to have a similar behavior which disagrees with existing theories. Such finding suggests a strong dependence of the dynamical response of electrons and of screening effects as a function of the impinging energy. Interestingly, the FWHM for thick Na films [124] has the opposite trend as a function of the parallel momentum transfer. This experimental result is a clear evidence that understanding of the broadening mechanisms of a SP requires a careful analysis of the band structure of the system.

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Figure 27. FWHM of the Na SP peak as a function of q || for (■) 0.90 ML Na/Cu(111), Ep=100 eV and for (○) thick Na film (data taken from ref. [124])

Figure 28. Intensity of the SP (■) and of the elastic peak (*) as a function of the off-specular angle (Ep=20 eV)

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Figure 29. Intensity of the SP (■) and of the elastic peak (*) as a function of the off-specular angle (Ep=100 eV)

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The intensity of the SP versus the off-specular angle for Ep=20 eV (Figure 28) clearly demonstrates that such collective mode was excited by impact mechanism because it is peaked at six degrees off-specular [68], while a dipolar loss would have the same behavior of the elastic peak as a function of the off-specular angle. Instead, the intensity of the SP for Ep=100 eV exhibited a maximum around 1.5 degrees off-specular (Figure 29). Thus, the SP excited by a higher primary beam energy has a substantially dipole character while the same mode with a lower impinging energy exhibits a notable impact character. The dependence of the dipole/impact scattering ratio in SP excitation as a function of primary beam energy is still an unexplored research field.

CONCLUSIONS In conclusion, measurements carried out on Ag layer-by-layer films grown on Ni(111) provided evidences of the relationship between the existence on Ag QWS and the linear term of the dispersion curve, which is null in contrast with all other Ag systems. Similar loss measurements for Ag/Cu(111), characterized by a Stranski-Krastanow growth mode, revealed SP confinement within islands in as-deposited Ag layers. Annealing caused an enhanced free-electron density of states of the Ag QWS, which renders negative the linear coefficient of the dispersion relation. The increased value of the FWHM in Ag films compared with Ag single-crystal surfaces suggest the occurrence of enhanced damping mechanism due to the opening of new decay channels of the SP in systems exhibiting QWS. For two layers of Na on Cu(111), the SP was not excited at small momenta, also found for two layers of Ag on the same substrate. A strong dependence of the dispersion curve on the energy of the primary beam was observed.. The FWHM of two Na layers behaves very differently with respect to that reported for a Na thick film. These differences were ascribed to different Landau damping mechanisms of the plasmonic mode in the two cases. Screening effects enhanced by electron quantum confinement and interactions between overlayer states and substrate states are suggested to be at the basis of these results. Such

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findings provide the grounds for theoretical studies aimed at characterizing of the nature and dispersion of the excitation modes in nanoscale thin films exhibiting confined electron states.

ACKNOWLEDGMENTS We want also to thank Dr. Stefan Mathias and Prof. Michael Bauer for having allowed to use their photoemission data, and, moreover, Dr. Vincenzo Formoso for many helpful discussions.

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[94] Feibelman, P. J. (1994). Comment on Surface plasmon dispersion of Ag. Physical Review Letters, 72(5), 788. [95] Feibelman, P. J. (1989). Interpretation of the linear coefficient of surface-plasmon dispersion. Physical Review B, 40(5), 2752-2756. [96] Feibelman, P. J. (1973). Sensitivity of surface-plasmon dispersion and damping to potential barrier shape. Physical Review Letters, 30(20), 975-978. [97] Feibelman, P. J. (1971). Dependence of the normal modes of plasma oscillation at a bimetallic interface on the electron density profile. Physical Review B, 3(9), 2974-2982. [98] Feibelman, P. J. & Tsuei, K. D. (1990). Negative surface-plasmon dispersion coefficient: A physically illustrative, exact formula. Physical Review B, 41(12), 85198521. [99] Rocca, M., Lazzarino, M. & Valbusa, U. (1992). Surface-Plasmon on Ag(110)Observation of Linear and Positive Dispersion and Strong Azimuthal Anisotropy. Physical Review Letters, 69(14), 2122-2125. [100] Silkin, V. M., Quijada, M., Muino, R. D., Chulkov, E. V. & Echenique, P. M. (2007). Dynamic screening and electron-electron scattering in low-dimensional metallic systems. Surface Science, 601(18), 4546-4552. [101] Silkin, V. M., Quijada, M., Vergniory, M. G., Alducin, M., Borisov, A. G., Muino, R. D., Juaristi, J. I., Sanchez-Portal, D., Chulkov, E. V. & Echenique, P. M. (2007). Dynamic screening and electron dynamics in low-dimensional metal systems. Nuclear Instruments & Methods in Physics Research Section B-Beam Interactions with Materials and Atoms, 258(1), 72-78. [102] Chiang, T. C. (2000) Photoemission studies of quantum well states in thin films. Surface Science Reports, 39(7-8), 181-235. [103] De Crescenzi, M. & Piancastelli, M. N. (1996). Electron Scattering and Related Spectroscopies, World Scientific, Singapore. [104] Moresco, F., Rocca, M., Hildebrandt, T., Zielasek, V. & Henzler, M. (1999). K adsorption on Ag(110), effect on surface structure and surface electronic excitations. Surface Science, 424(1), 62-73. [105] Persson, B. N. J. (1993). Polarizability of small spherical metal particles: influence of the matrix environment. Surface Science, 281(1-2), 153-162. [106] Hövel, H., Fritz, S., Hilger, A., Kreibig, U. & Vollmer, M. (1993). Width of cluster plasmon resonances: Bulk dielectric functions and chemical interface damping. Physical Review B, 48(24), 18178-18188. [107] Bendounan, A., Forster, F., Ziroff, J., Schmitt, F. & Reinert, F. (2005). Influence of the reconstruction in Ag/Cu (111) on the surface electronic structure: Quantitative analysis of the induced band gap. Physical Review B, 72(7), 075407. [108] Schiller, F., Cordón, J., Vyalikh, D., Rubio, A. & Ortega, J. E. (2005). Fermi Gap Stabilization of an Incommensurate Two-Dimensional Superstructure. Physical Review Letters, 94(1), 016103. [109] Chiarello, G., Cupolillo, A., Caputi, L. S., Papagno, L. & Colavita, E. (1997). Collective and single-particle excitations in thin layers of K on Ni(111). Surface Science, 377(1-3), 365-370. [110] Tsuei, K. D., Plummer, E. W., Liebsch, A., Pehlke, E., Kempa, K. & Bakshi, P. (1991). The Normal-Modes at the Surface of Simple Metals. Surface Science, 247(2-3), 302326.

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[130] He, J. H., Carosella, C. A., Hubler, G. K., Qadri, S. B. & Sprague, J. A. (2006). Bombardment-Induced Tunable Superlattices in the Growth of Au-Ni Films. Physical Review Letters, 96(5), 056105. [131] Yuan, Z. & Gao, S. (2008). Landau damping and lifetime oscillation of surface plasmons in metallic thin films studied in a jellium slab model. Surface Science, 602(2), 460-464. [132] Politano, A. & Chiarello, G. (2010). Sputtering-induced modification of the electronic properties of Ag/Cu(111). Journal of Physics D: Applied Physics. [133] Perri, S., Lepreti, F., Carbone, V. & Vulpiani, A. (2007). Position and velocity space diffusion of test particles in stochastic electromagnetic fields. Europhysics Letters, 78(4), 40003. [134] Stephanov, M. A. (2009). Non-Gaussian Fluctuations near the QCD Critical Point. Physical Review Letters, 102(3), 032301. [135] Chulkov, E. V., Silkin, V. M. & Echenique, P. M. (2000). Inverse lifetime of surface states on metals. Surface Science, 454, 458-461. [136] Steeb, F., Mathias, S., Fischer, A., Wiesenmayer, M., Aeschlimann, M. & Bauer, M. (2009). The nature of a nonlinear excitation pathway from the Shockley surface state as probed by chirped pulse two photon photoemission. New Journal of Physics, 11, 013016. [137] Bonzel, H. P., Bradshaw, A. M. & Ertl, G. (1989). Alkali Adsorption on Metals and Semiconductors, Elsevier, Amsterdam. [138] Palik, E. D. (1985). Handbook of Optical Constants of Solids. Academic Press, New York. [139] Nazarov, V. U. (1999). Multipole surface plasmon excitation enhancement in metals. Physical Review B, 59(15), 9866-9869. [140] Tsuei, K. D., Plummer, E. W., Liebsch, A., Kempa, K. & Bakshi, P. (1990). Multipole Plasmon Modes at a Metal-Surface. Physical Review Letters, 64(1), 44-47.

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Chapter 4

EVANESCENT COUPLING BETWEEN RESONANT PLASMONIC NANOPARTICLES AND THE DESIGN OF NANOPARTICLE SYSTEMS T. J. Davis CSIRO Materials Science & Engineering, CSIRO Future Manufacturing Flagship, Clayton South MDC, Victoria, Australia

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ABSTRACT Localized surface plasmon resonances in metallic nanoparticles arise from the interaction between the free electrons in the metal and their associated electromagnetic fields. The resonant frequencies depend on the geometry of the nanoparticle, its electric permittivity and the electric permittivity of the surrounding medium. When two or more nanoparticles come in close proximity, the evanescent electric fields associated with the surface plasmons interact with the surface charges, changing the resonant frequencies. In this chapter a theory of the interaction of plasmonic nanoparticles is presented. The theory is based on the “electrostatic" eigenvalue method that describes the localized surface plasmon resonances in nanoparticles of any shape provided they are much smaller than the wavelength of light. The theory leads to simple algebraic expressions for the coupling between nanoparticles that can be used to deduce the resonant properties of the nanoparticle ensembles. In particular, the theory is used to deduce the relative energies of the hybrid states associated with nanoparticle ensembles, it is applied to the understanding of dark modes and plasmon-induced transparency in metamaterials, and it is shown how the theory can be used to design and analyse plasmonic circuits.

INTRODUCTION Localized surface plasmon resonances (LSPR) in nanoparticles are finding applications in a wide variety of technologies from sensing [1-8], solar cell technologies [9-13] to cancer therapy [14, 15]. With the development of methods for fabricating nanostructures based on

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lithography [16-18] or chemical synthesis [14, 19-21] there has been increasing interest in tailoring the optical properties of nanoparticle systems. This can be achieved by designing systems of interacting nanoparticles. When two nanoparticles are placed close to one another, the evanescent fields associated with the LSPR induce charges on the neighbouring nanoparticles which alter the resonances. Coupling the nanoparticles together can lead to a variety of effects, such as the splitting of resonance frequencies [22-30], the formation of dark modes [31-34], plasmon-induced transparency [32, 35], Fano resonances [30, 36-38], as well as improved ability for sensing [39, 40]. Important in optimising the interactions is a method to allow the systematic design of the coupling to achieve desired optical properties. The response to an applied light field of an ensemble of nanoparticles is straightforward to calculate using any one of a variety of numerical procedures such as Finite Difference Time Domain [41, 42], the Boundary Element Method [43, 44], the Discrete Dipole Approximation [45] and Fourier Model Methods including Rigorous Coupled Wave Analysis [46-49]. However, with these methods it is often difficult to extract parameter relationships and one often has to use intuition to determine the optimum configuration. While it would be more advantageous to write down the analytical solution to the interaction problem, the Maxwell’s equations that describe the electromagnetic fields are generally very difficult to solve. As an alternative, a number of approximate analytical methods have been developed such as the molecular exciton coupling model [29] or the plasmon hybridization model [50-52]. In this article, we describe a particular method based on the “electrostatic” approximation [53-56] that models the LSPR in terms of the resonant modes, or eigenmodes, of the metallic nanoparticle. The advantage of this method is that the mathematics is independent of the shape of the particle and multiple resonance modes can be included in the model. Furthermore, the method leads naturally to a description of the interaction between nanoparticles in an ensemble and provides relatively simple relationships between the excitation amplitudes of the modes and the resonant frequencies. In the following sections, we describe the electrostatic eigenmode method and apply it to a number of simple situations involving the coupling between nanoparticles. This method then forms a means by which nanoparticle ensembles can be designed to have the desired optical properties.

ELECTROSTATIC” EIGENMODES FOR PLASMONIC NANOPARTICLES In this section we review the underlying theory that describes the LSPR in terms of surface-charges and their associated evanescent electric fields. The theory is based on an electrostatic approximation that ignores the effects of magnetic fields. In the limit where the nanoparticle size approaches zero, the electric and magnetic fields decouple and Maxwell’s equations take on the same form as in electrostatics. For practical purposes this size scale occurs where the nanoparticle is much smaller than the wavelength of light and it can be expressed by the requirement that

 b kd2  1 where  b

is the relative electric permittivity

of the background medium in which the nanoparticle is embedded, k is the wavenumber of the light in vacuum and d is a characteristic dimension of the nanoparticle. Implicit in the

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electrostatic method is that all processes take place instantaneously and it does not contain the effects of radiation of energy. The errors associated with this approximation arise from retardation [57], which is the effect of the phase difference that exists between fields emanating from one point on the nanoparticle and arriving a short time later at another point. The main effect of retardation is a red-shift of the wavelengths of the localized surface plasmon resonances. Within the electrostatic approximation, the electric potential obeys Poisson’s equation and many results can be derived from potential theory. The surface-charge and surface-dipole distributions are described by integral equations whose properties have been well studied [58, 59]. The integral equations are representations of the eigenvalue problem for the selfconsistent resonant charge distributions on the surface of the nanoparticle. These equations were used by Ouyang and Isaacson [53] to describe the localized surface plasmon resonances induced in metallic nanoparticles by electrons in electron microscopy and the theory was further developed by Mayergoyz et al [54, 55] to describe optical interactions. These methods form a class of eigenmode methods. In the context of metal particles, a differential form of the eigenmode method was developed by Bergman [60, 61] to study the effective dielectric constant of composite materials. In this method, Bergman examines the pole-spectrum which represents resonances associated with the metal particles in the composite material. The basic idea in the electrostatic eigenmode method is that the electric field surrounding a particle p can be determined by solving for the self-sustained distribution of surface-charge

 pk r 

or surface-dipoles

 pk r . These are the eigenfunction solutions for the kth mode of

the pth nanoparticle and are associated with eigenvalue

 pk . The eigenfunctions are

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representations of LSPR and are obtained from the integral eigenvalue equations

r  rp .nˆ  pk k    r    r dS p , p p 3 2  r r

(1)

rp  r .nˆ p  pk k  p r p  dS p , 3  2 r r

(2)

k p

p

and

 pk r  

p

ˆ p is the surface normal at r p . The interpretation of where nˆ is the surface normal at r and n (1) is quite simple. There is a surface-charge

 pk r 

all the electric fields arising from surface-charges

induced at r due to the contributions of

 pk r p  at all points r p

on the surface of

the nanoparticle. A similar explanation can be given for (2). Implicit in this description is the oscillatory time dependence exp  it  of the surface-charges and surface-dipoles. It is a general result from potential theory [58] that the eigenfunctions form a biorthogonal set that obeys the relationship

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T. J. Davis

  r  r dS   j p

The resonant frequencies

 pk

k q

pq

 jk .

of a nanoparticle are obtained from the eigenvalues

and are related to the real part of the electric permittivity    of the nanostructure by

 

Re  

k p

 1   pk  b 1  k p 

 ,  

(3)

 pk

(4)

where  b is the permittivity of the background medium. Note that we assume the background

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permittivity is real. This is a reasonable assumption in most cases since the nanoparticles are often embedded in transparent media, such as glass, air or transparent liquids. The resonant frequency is found by selecting the background medium in which the nanoparticle is embedded and then finding the frequency that gives a metal permittivity satisfying Eq. (4). It is a general property of the homogeneous integral eigenvalue equation that the eigenvalues are real and are greater than one. This means from Eq. (4) that resonances occur only for nanoparticles made from materials where the electric permittivity is negative. This situation arises in metals and in semiconductors under certain conditions. We usually find that the required permittivities occur for metals in the region of the visible and infrared spectrum so that the surface-charge oscillations (i.e. the localized surface plasmon resonances) occur at these frequencies.

Figure 1. The first nine surface-dipole eigenfunctions for a rectangular prism. The surface mesh and a rendered image of the prism are also shown, slightly tilted to indicate the three-dimensional nature of the prism. The prism is 1 unit thick, 2.5 units wide and 10 units long. The gray level indicates the relative strength of the surface-dipoles (as indicated by the scale)

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The solution of the eigenvalue problem is easily found using standard numerical methods. For example, the integral can be converted to a sum over surface elements which casts the equation into a standard matrix eigenvalue problem [54]. Examples of the surfacedipole eigenfunctions for a rectangular prism calculated using this method are shown in Figure 1. Each mode represents a particular localized surface plasmon resonance which appears as a surface-charge or surface-dipole standing wave on the structure. When a light beam is incident on the nanoparticle p, it can excite one or more of the resonant modes, depending on the frequency  and the polarization. The surface-charge distribution is then represented by a sum of the individual resonances

 p r,     a kp   pk r  .

(5)

k

The excitation amplitude a p   depends on the incident field according to [55] k

a    k p

2 pk  b      b 

 b  pk  1     pk  1 

 pk r p nˆ p .E 0  dS p ,

(6)

where E 0   is the electric field vector of the incident light. In the electrostatic approximation this field is constant over the surface of the nanoparticle and can be taken outside the integral in (6). Furthermore, if

 r nˆ .dS k p

p

p

 pk r 

is appropriately normalised, the integral

 p kp is related to the average dipole moment for the mode k. Then the

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excitation amplitude can be written in the form

a kp    f pk  p kp .E0   ,

(7)

where we have defined a “resonance” factor

f

k p

  

2 pk  b      b 

 b  pk  1     pk  1

.

(8)

The excitation amplitude becomes large when the real part of the denominator of (8) is zero. This is the condition for a resonance and leads to Eq. (4). The excitation amplitude is zero if the resonant mode does not have a dipole moment, p p  0 , or if the incident field has k

a polarization perpendicular to the dipole moment of the LSPR mode, p p .E 0    0 .

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k

116

T. J. Davis

Figure 2. The real and imaginary components of the excitation amplitudes for light polarized in the x and y directions incident on the rectangular prism shown in Figure 1. The polarization axes are defined in Figure 1. The prism is taken to be gold embedded in a medium with  b  1.77 . The positions of the resonant modes are also shown

As an example, we use the rectangular prism from Figure 1 and calculate the real and imaginary parts of the excitation amplitudes for incident light polarized in x and in y (perpendicular and parallel to the long axis respectively). The results are shown as functions of wavelength in Figure 2. The rectangular prism has the electric permittivity of gold [62] and

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it is embedded in a medium with permittivity  b  1.77 , similar to water. The excitation amplitudes show the typical resonance behavior: the amplitudes are in-phase with the incident field below the resonance, they pass through 90 degrees at resonance and approach 180 degrees out of phase above the resonance (at higher frequencies or shorter wavelengths). This property is very useful and we shall describe later a plasmonic system that exploits it to achieve high sensitivity to the presence of molecules. Note that some of the modes (for example, mode 2) cannot be excited by the incident light field because their dipole moments are zero. The dominant mode is mode 1 which has a strong dipole moment. The total dipole moment of the nanoparticle can be obtained from (5) on multiplying by rq , the position of a point on the surface, and then integrating over the surface. This leads to

p p     a kp  p kp ,

(9)

k

where the dipole moment of mode k is p kq   r qk r dS q . This is the same as

p kp    pk r p nˆ .dS p that we used previously provided that the eigenfunctions

 pk r 

and

 pk r  are appropriately normalized [63]. Once the dipole moment is known, it is possible to calculate the cross sections for scattering C sca and absorption C abs based on the relations [64]

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2

C sca

k4 p  , 6 E 0

 p C abs  k Im  E0

  . 

(10)

(11)

The cross sections give a measure of the scattering and absorption properties of the nanoparticles as functions of the applied light frequency and are useful for calculating the optical spectrum of the nanoparticle. This electrostatic eigenmode method has a number of useful features. The solution of the eigenvalue problem (1) or (2) is relatively straightforward to find using numerical procedures. Maygergoyz et al [54] discuss a simple method for doing this and we base our method on that discussion. Examination of (1) and (2) will show that the eigenvalue equations are scaleindependent. That is, all lengths in (1) or (2) can be multiplied by a scale factor and the equations remain the same. This means that once the eigenvalues and eigenfunctions are found for a nanoparticle of a given shape, they remain the same independent of the nanoparticle size. This is a consequence of the electrostatic approximation and we find that this breaks down for nanoparticles of sizes comparable to or larger than the wavelength of

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light, such that

 b kd2  1 no longer holds. Another useful property is that the eigenvalue

problem is independent of the material properties of both the nanoparticle and the medium in which it is embedded. These properties only affect the calculation of the resonant frequencies of the LSP modes, via (4). This means that once the eigenvalues and eigenfunctions are found for one nanoparticle of a particular shape, they never need to be recalculated. Most importantly, the electrostatic eigenvalue method provides analytical expressions that describe the resonant properties of the nanoparticles and the evanescent electric fields that are associated with the surface-charge distributions. This allows us to derive relationships between the parameters that describe the resonances and the LSPR modes, providing insight into the underlying physics. This is of immense value and is the primary reason that we have been using this method, despite its approximate nature.

COUPLING OF PLASMONIC NANOPARTICLES An interesting property of the electrostatic eigenmode method is that it can be applied to nanoparticle ensembles. The equations (1) and (2) make no requirement that the surfaces involved in the integrals are continuous. That is, they can be disjoint or separate. In this case the solutions to the eigenvalue problem yield the resonant modes and eigenvalues of coupled systems where the electric fields from each nanoparticle influence the resonances on the adjacent nanoparticle. The main problem with this approach is that, for numerical computation, the number of surface elements increases with every particle added. This increases the amount of computer memory required for the solution and dramatically increases the computational burden. As an alternative, we separate from (1) and (2) the contributions of the neighbouring particles and derive an associated eigenvalue problem for

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the coupling. Moreover, it is then possible to derive an expression for the excitation amplitudes for nanoparticles in an ensemble in terms of their excitations when isolated. This provides a very useful analytical approach to the coupling problem and will enable us to analyze a number of important problems in plasmonics. If a number of nanoparticles are brought within close proximity to one another, the electric fields from the surface charges interact with the neighboring nanoparticles, modifying the resonant modes. The modifications are described by changes to both the eigenfunctions and the eigenvalues that can be determined as follows. We represent the new charge distribution ~ r  of the ensemble of nanoparticles as a linear combination of the eigenfunctions of all the particles in the ensemble [65]

~r     qk  qk r  . q

(12)

k

This new charge distribution must obey the eigenvalue equation

 ~ r  r.nˆ  r dS  . 3 2  r  r

~r  

(13)

Substituting (12) for ~ r  yields

  lh lh r   Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

l

h

Eq. (14) is then multiplied by

 2

   qk  qk rq  q

 pj r 

r  r .nˆ

k

q

r  rq

3

dS q .

(14)

and integrated over all surfaces. The biorthogonal

condition (3) leads to

 pj 

 2

    r  k q

q

j p

p

k

nˆ p .r p  rq . r p  rq

3

 qk rq dS q dS p .

(15)

Defining

G pqjk     pj r p 

nˆ p .r p  rq . r p  rq

3

 qk rq dS q dS p ,

(16)

then (15) becomes

 pj 

 2

 G q

jk pq

 qk ,

k

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(17)

Evanescent Coupling between Resonant Plasmonic Nanoparticles… which is an eigenvalue problem for the expansion coefficients is easy to solve numerically to yield the new eigenvalues



j p

119

 pj . This eigenvalue problem

 and the expansion coefficients

. The new eigenfunctions (12) of the coupled nanoparticles are then obtained in terms of

the eigenfunctions of the uncoupled nanoparticles. The derivation can be repeated for the surface-dipole eigenfunctions yielding similar results. Note that the coupling coefficient includes the self-coupling of a nanoparticle. That is, we have that G pp  2 jj

this, multiply Eq. (1) by

 qj r 

 pj . To prove

and integrate over the nanoparticle surface. Applying the

biorthogonality condition (3) gives

 qk 2

k   q r 

nˆ .r  rq  r  rq

3

 qk rq dS q dS    qj r  qk r dS   jk ,

(18)

Gqqjk   jk 2  qk .

(19)

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which implies that

The advantage of this method is that the resonances and the surface-charge distributions associated with the coupled system can be found without requiring intense computation. Moreover, only the modes of interest need to be used. For example, we are usually only interested in the lowest order modes since the high-order modes occur close to the plasma frequency of the metal and are generally highly dissipative. This means that we do not need to compute modes that we have little interest in. A similar procedure can be applied to the excitation amplitudes in a coupled system [56].

~ k of a nanoparticle in the presence of N other nanoparticles is The excitation amplitude a p determined by the interaction of all the electric fields from the N nanoparticles, which leads to a modification of (6) N   a~pk  f pk   pk r p nˆ .E 0   E m r p  dS p , m 1  

(20)

 

where E m r p represents the electric field at position r p arising from particle m. This field is associated with the surface-charges  m rm  

 a~  r  of particle m according to j m

j m

m

j

E m r p    a~mj j

1 4 b

r

r

p

p

 rm   rm

3

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 mj rm dS m .

(21)

120

T. J. Davis When this is placed in (20) we obtain N

kj ~ j a~pk  a kp   C pm am , m 1

(22)

j

kj

where the coupling coefficient C pm is defined by

C

kj pm



f pk 4 b

k   p r p 

nˆ p .r p  rm  r p  rm

3

 mj rm dS m dS p ,

(23)

k

and the amplitude a p in (22) arises from substituting the integral over the applied field with (6). If we define C pp  0 , we can include all nanoparticles of the ensemble in the sum in kj

(22) which can then be written in the form

  N

m1

pm

kj ~ j am  a kp .  kj  C pm

(24)

j

By treating the terms in the sum as a matrix, the excitation amplitude of the coupled nanoparticle can be expressed in terms of the excitation amplitudes of the nanoparticles when they are uncoupled, or isolated from each other



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N

kj a~pk    pm kj  C pm m 1



1

amj ,

(25)

j

This equation is equivalent to that of Davis et al [56]. The term in brackets represents the inverse of a matrix involving a coupling coefficient that depends on the relative geometric arrangement between pairs of nanoparticles and on their resonant modes. We also note that the geometric term in (23) is the same as in (16)

C

kj pm



f pk 4 b

kj G pm ,

(26)

where we must impose G pp  0 . This condition is required to eliminate self-interactions, kj

which are already taken into account. Although these equations appear rather complicated, we will show that they can lead to simple expressions for the LSPR in nanoparticle ensembles. To understand the relationships between the particles in the ensembles and their resonant frequencies it is convenient, but not essential, to introduce a model for the electric permittivity of the metal. This model will allow us to make explicit the frequency-dependence

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121

of the resonance factor f p . For frequencies well-below the plasma frequency  P , the k

electric permittivity of a metal can be approximated by the Drude formula [64]

    1 

 P2 ,    i

(27)

which includes an imaginary term i to take account of losses in the metal. In many instances, we are only interested in the response of a nanoparticle in the vicinity of the resonance of one mode, say mode k. Since the denominator of (8) contains the resonance condition (4) then when (27) is substituted in (8) we find that the resonance factor depends on the difference between the applied frequency and frequency of the resonance  p . k

Furthermore, if we assume that the loss term is small and we only consider small deviations from resonance, it is straightforward to show that

 2 pk  b2 pk 2

f  k p



k p

3

 

 1  P2    pk  ipk 2 2





Apk

   pk  ipk 2

.

(28)

k

Here we represent the constants that multiply the resonance term by Ap . The coupling coefficient (26) then takes the form [66]

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C

kj pm



 Apk

kj G pm

4 b    pk  ipk 2

,

(29)

and similarly, the excitation amplitude (7) can be written as

a     k p

Apk p kp .E 0  

   pk  ipk 2

.

(30)

We will show in the following sections that much information about the LSPR of nanoparticle ensembles can be obtained from (25), (29) and (30) using only algebra. The resulting expressions allow us to interpret the mechanisms associated with the coupling and the resonance shifts. This is the key advantage of using this method. To simplify the analysis, it is useful to ignore the loss term i so that the expressions are real. jk

Although the geometric coupling term G pq can be determined numerically once the eigenfunctions are known, it is possible to write down an approximate form that is useful for estimating the relative magnitudes of the coupling. This is possible for the situation where the two coupled nanoparticles are far apart. We let the position of a point on the surface of nanoparticle q be given by rq  d q  x q where d q is the vector to the centre of the

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122

T. J. Davis

nanoparticle and x q is a vector from the centre to the surface. We assume that x q  d q . Likewise, for nanoparticle p, let rp  d p  x p . Then the first term in a series expansion can be taken, r p  rq

3





3 2 where d pq  d p  d q , d pq  d pq . The  d pq 1  3d pq .x p  x q  d pq

integrals of the surface-dipole and surface-charge distributions individually are zero, that is

  r dS k q

q

q

 0 and   pj r p dS p  0 since the nanoparticles are assumed to be uncharged

initially. Then the only non-zero terms are those for which the eigenfunctions are coupled with distances or position dependent terms. To lowest order in d pq , the geometric term takes the approximate form [66]

G pqjk 









1 3 p pj .dˆ pq p kq .dˆ pq  p pj .p kq . 3 d pq

(31)

This is the classical form of the dipole-dipole coupling that depends on the dipole

ˆ from nanoparticle q to moments of the modes of the nanoparticles and the unit vector d pq

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nanoparticle p. This expression represents the lowest-order multipole expansion of the geometric term and we have neglected the higher-order multipoles. It is useful for estimating the relative signs of coupling coefficients and for determining which couplings are approximately the same. In the following sections we apply the coupling method to a number of simple problems of nanoparticle coupling and show that a great deal of information can be gleaned out of the coupling problem using relatively simple algebra.

Figure 3. The surface-dipole modes corresponding to the interaction between two nanoparticle rods (rectangular prisms). The two modes form anti-symmetric and symmetric pairs. The top row shows identical rods where the symmetric mode has a zero net dipole moment. The bottom row has rods of different length and therefore different resonant frequencies. The anti-symmetric mode now has a nonzero net dipole moment.

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Evanescent Coupling between Resonant Plasmonic Nanoparticles…

123

TWO PARTICLE, SINGLE MODE COUPLING We first analyse a simple system consisting of two similar nanoparticles, labelled with subscript 1 and 2, each exhibiting one dominant mode, labelled with superscript 1. This system was analysed by Davis et al [56] and it highlights some of the key properties of coupled systems. For two nanoparticles with one resonant mode, (25) is written as

 a~11   1  1    a~    C 11 21  2 

 C1211   1 

1

 a11  1  1 C1211  a11   1    ,  a    C 11 1  a 1  2  21  2  2 

(32)

where the determinant of the matrix inverse is  2  1  C12 C 21 . We see that the amplitudes 11

11

a~pk of the LSPR of the coupled nanoparticles become large when the real part of the determinant  2 is zero. This is the condition for the resonance of the coupled system. The simplest case of two coupled nanoparticles is when both are identical and placed as shown in Figure 3. Then the coupling coefficients are identical, C12  C 21  C and we represent 11

11

them by a constant C. For simplicity, in this analysis we shall ignore the loss term i that appears in the coupling coefficients and amplitudes. Then the resonances of the coupled system occur when  2  1  C  0 . The two solutions are C  1 corresponding to two resonant frequencies. That is, if write out the frequency dependence of the coupling coefficient, as in (29), we would obtain a quadratic equation with two solutions (see below). This shows that the coupling of two nanoparticles with the same resonant frequency leads to two separate resonances. The shifts in the resonances are related to the strength of the coupling. Since the two nanoparticles are identical, then the uncoupled excitation amplitudes

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2

are the same a1  a 2  a , which we represent by a. Then the excitation amplitudes of the coupled nanoparticles are obtained from (32) as 1

1

a~11  a~21 

1  C a  a . 1  C 1  C  1  C 

(33)

It is interesting to note that, although there are two resonant frequencies associated with the two nanoparticles, there is only one resonance associated with their excitation amplitudes, namely when C  1 . The other resonance is unable to be excited in this system and it is associated with a dark mode [31, 33, 34]. The origin of this mode becomes apparent when we consider the case where the two nanoparticles are slightly different, with resonant frequencies 1 and  2 respectively. We will assume that the dipole moments are the same, so that the uncoupled excitation amplitudes are a1   a   1  and a 2   a    2  with a 1

1

some constant. The coupling coefficients are approximately C12  G   1  and 11

11 11 11 C 21  G    2  with G another constant. (Note from (31) that G12  G21  0 for this

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124

T. J. Davis 11

11

configuration. We have chosen the signs of C 12 and C 21 such that G  0 .) The resonances of 11 the coupled system occur when  2  1  C1211C 21  0 which leads to a quadratic equation for the resonant frequency with solutions



1   2  2



1 2

1   2 2  4G 2 .

(34)

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When the coupling is zero, such as when the nanoparticles are far apart, G  0 and the two resonant frequencies correspond to the individual resonances of the two nanoparticles when isolated. As the nanoparticles are brought closer together, the coupling increases and the resonances shift in frequency, appearing equally spaced about the average frequency 1   2  2 . The shift in resonant frequencies with changes to the nanopaticle coupling has been modelled previously using a number of methods. One method makes an analogy with molecular exciton coupling theory that describes the interactions between the dipole moments of molecules that aggregate under van der Waals attraction [29]. Another compares the plasmonic nanoparticle coupling with the hybridization of electron energy levels in molecular orbital theory. This latter method is known as the plasmon hybridization theory [51, 52]. Here we obtain the same results using the electrostatic eigenmode coupling method. The hybridized energy levels E   are exactly those obtained from the frequencies in Eq. (34). In this regard we may consider this method as an alternative route to determining the hybrid states of coupled plasmonic systems. The resonance shifts depend on the orientation of the nanoparticles since the geometric coupling, Eq. (31), depends on the relative orientation of the nanoparticle dipole moments and the vector separation between them. The excitation amplitudes are calculated from (32), yielding

 a  2  G  ,   2   1   G 2

a~11 

a~21 

 a  1  G  .   2   1   G 2

(35)

Again, when the coupling is zero, the excitation amplitudes equal those of the individual nanoparticles, as required. It is interesting to see what happens to the amplitudes if the two resonant frequencies deviate by a small amount,  2  1   . In this case the resonant frequencies of the coupled system to first order in

 R  1 

 are

 2

G.

(36)

The excitation amplitudes for the two resonant frequencies become

a~11 

a 2 ,   2  R  1  G 2



 R





a~21 

 a 2 ,   2  R  1  G 2



 R

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



(37)

Evanescent Coupling between Resonant Plasmonic Nanoparticles…

125

and

a~11 

 a  2  2G  ,   2  R  1  G 2



 R





a~21 

 a 2  2G  .   2  R  1  G 2



 R





(38)

The amplitudes of the higher-frequency resonance (38) have the same sign and are inphase. These amplitudes combine to form a large dipole moment for the coupled system.

~  a~ for the Using (10) for the scattering cross section and including both amplitudes a 1 2 combined dipole moments, we find there is strong scattering from the nanoparticle pair. However, the amplitudes of the lower-frequency resonance (37) have the opposite sign, which 1

1

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~ 1  a~ 1  0 so that, to first means that the resonances are in anti-phase. The sum leads to a 1 2 order in the deviation  of the resonant frequencies, this mode has a zero net dipole moment and does not scatter light. This mode is a quadrupole resonance and it would not normally be excited by the applied light field, as we discovered above. However, the small asymmetry in the resonances of the nanoparticles has introduced a small dipole moment through which the light can couple and excite this mode. In effect, we have broken the symmetry associated with the nanoparticle pair which has allowed the excitation of this mode. Symmetry is an important concept in nanoparticle systems [65]. As an example, we show in Figure 4 the scattering cross section of a pair of nanoparticle rods at different spacing which leads to different strength of the coupling. The nanoparticles have the permittivity of gold [62] and are assumed to be in air. As the coupling increases, the scattering cross section of the higher frequency mode becomes stronger whereas the cross section for the lower frequency mode becomes weaker and almost disappears. The resonance frequencies shift apart as the coupling increases. Also shown in the figure are the excitation amplitudes of each nanoparticle. The important feature is the relatively large excitation amplitudes indicating strong electric fields and the presence of localized surface plasmons even though the scattering from the dark mode is very small.

TWO PARTICLE, DOUBLE MODE COUPLING As a slightly more advanced problem, we consider a structure that has two modes coupled to a second structure with a single mode. Let the excitation amplitudes of the two 1

2

1

modes of the first structure be a1 and a1 , and the amplitude of the second structure be a 2 . The coupling matrix is found by noting that the two modes of the first structure do not couple to one another, so that C11  0 (see the comment after Eq. (26)). Then the excitation amplitudes of the coupled system are given by 12

 a~11   1  2   a~1    0  a~ 1    C 11 21  2 

0 1 12  C 21

11   C12  21  C12  1 

1

12 11 12 11  a11  1  C1221C 21  a11  C12 C 21 C12  2 1    11 11 11 1  C12 C 21 C1221  a12  ,  a1    C1221C 21  a 1   2  C 11 12 C 21 1  a12  21  2 

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(39)

126

T. J. Davis

where the two particle determinant is now  2  1  C12 C 21  C12 C 21 .

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11

11

21

12

Figure 4. The scattering cross section and the magnitudes of the excitation amplitudes of a pair of gold nanorods in air with slightly different resonant frequencies (shown as the bottom pair in Figure 3). As the separation between the rods decreases (in units of rod widths) the coupling increases. The symmetric mode increases in strength while the anti-symmetric mode diminishes becoming sub-radiant, or a dark mode. The excitation amplitudes remain strong, even for the dark mode, which suggests that even though the configuration scatters light very weakly, there are localized surface plasmons excited in the nanoparticles.

As a specific example, we consider the case shown in Figure 5. Taking account of the geometric coupling matrix using (31) we have the coupling coefficients with the form

11 11 12  C 21  G    2  with G  0 a C12  G   11  , C1221  G   12  and C 21

constant. The resonant frequencies of the two modes of nanoparticle 1 are 1 and 1 1

2

respectively, and the resonant frequency of nanoparticle 2 is  2 . The equation for  2  0 then becomes

           G 2   1 1

2 1

2

2

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1 1



 12  0 .

(40)

Evanescent Coupling between Resonant Plasmonic Nanoparticles…

127

To simplify the problem, we assume that the two resonant modes of nanoparticle 1 have the

same

frequency,

so

that

11  12  1 . Then the equation reduces to

  1   1    2   2G 2   0 with solutions   1 ,  

1   2 2



1 2

 2  1 2  8G 2 .

(41)

Since one of the solutions is the resonance of the uncoupled mode of nanoparticle 1,   1 , we would expect this to correspond to a situation where nanoparticle 2 is not excited and therefore it is not coupling to nanoparticle 1. The other two frequencies represent a splitting of modes equally about the average of the two resonances, with the degree of splitting depending on the strength of the geometric coupling. This is similar to the situation we found for the two parallel nanoparticles. The excitation amplitudes are obtained from (39). If the incident light field is polarized parallel to the long axis of particle 2, then we can write a 2   a    2  with a some constant. If we take the dipole moments of the two modes of particle 1 to be the same as particle 2 but note that they are at 45 degrees to the polarization of the incident field, then 1

a11  a12   a

2   1  . The excitation amplitudes are then given by





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 a    2  2  G a~11  a~12  ,   1    2   2G 2

a~21 





 a   1  2G .   1   2   2G 2

(42)

Figure 5. An example of the coupling of two nanoparticles. The top line shows the surface-dipole distributions of the two modes of nanoparticle 1 and the single mode of nanoparticle 2. The grey level represents the relative strength and the arrows indicate the direction of the dipole moments. The bottom line shows the three modes of the coupled nanoparticles. Mode 2 does not involve any coupling to the second nanoparticle, which has no LSP excitation.

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T. J. Davis

Figure 6. The real and imaginary components of the excitation amplitudes of the coupled nanoparticles under illumination by light polarized parallel to the long axis of nanoparticle 2 (see Figure 5). The numerical calculation is obtained from the complete numerical solution of the coupling problem including the calculated forms of the surface-charge and surface-dipole eigenfunctions. The nanoparticles were chosen to be gold embedded in a medium with  b  2.2 . The analytical model is

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based on Eqs. (42) with  2  0  0.2i , 1  1 and G  0.5 .

We immediately see that the resonance at   1 is unobtainable, since the amplitudes do not diverge at this frequency. This corresponds to mode 2 in Figure 5 which cannot be excited by light polarized parallel to the long axis of the second nanoparticle. The other two resonances exist for this polarization of light. The excitation amplitudes as functions of frequency are shown in Figure 6. The graph on the left was calculated using a numerical solution based on (39) with coupling coefficients given by (23). The surface-dipole and surface-charge distributions were calculated by using a surface mesh and converting the integral eigenvalue equations into matrix eigenvalue equations, as described by Mayergoyz et al [54]. The equations were solved numerically yielding the eigenfunctions and eigenvalues. The resonance factor was obtained from (8) using the electric permittivity of gold and taking the background permittivity as  b  2.2 . With these parameters the resonances of nanoparticle 1 occur at 1  1.575  10 radians.s1, which is equivalent to a wavelength of 1197 nm, whereas nanoparticle 2 has its resonance 15

at  2  1.628  10 radians.s-1 corresponding to 1158 nm. For comparison we also show the curves calculated from (42) using normalized frequencies and including a complex loss term to prevent zeroes in the denominator. The loss term and the coupling were chosen to give results qualitatively similar to the numerical solution. For this analytical model we used  2  0  0.2i , 1  1 and G  0.5 . Despite the number of simplifications that have been used in deriving Eqs. (42), the results show remarkable agreement. Having an analytical solution is of immense value when trying to understand the relationships between parameters and the associated LSPR. The numerator in (42) can be zero at particular frequencies which means that the real parts of the excitation amplitudes of the coupled nanoparticles are approximately zero at these frequencies (we need to keep in mind that there are imaginary terms that we are neglecting). 15

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129

This is in addition to the zeroes we would expect on resonance. For example, at

  2  2G the real component of the excitation amplitude of nanoparticle 1 is zero. This means that at this frequency the LSPR in nanoparticle 1 is out of phase with the incident field, since there is only an imaginary component remaining. As it is not at resonance, the amplitude will be small. Similarly at   1  2G the real part of the excitation amplitude of nanoparticle 2 is zero. To test this result with the numerical model, we can estimate G from (41) using the differences between the observed resonances. We note that the resonances occur in the numerical model at  R  1.503  10 radians.s-1 and  R  1.691  10 



15



radians.s-1 (see Figure 6). Rearranging (41) gives 8G 2   R   R

   2

2

15

2  1  so that

G  0.0638 1015 radians.s-1. Then from (42) we find that the amplitude of nanoparticle 1 is zero at   1.538 1015 radians.s-1 and the amplitude of nanoparticle 2 is zero at   1.485 1015 radians.s-1. These points agree well with the calculated data, as shown by the arrows in the graph. This shows that the analytical method provides relatively accurate predictions, despite the approximations associated with the use of the Drude model in deriving the frequency dependence.

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PLASMON-INDUCED TRANSPARENCY VIA THREE PARTICLE COUPLING In this section we consider the basic elements of a metamaterial proposed by Zhang et al [32] that exhibits what is known as plasmon-induced electromagnetic transparency. We will analyse this using the electrostatic coupling for three particles where each particle exhibits one dominant mode. The structure is shown in Figure 7. A variation on this structure was created using electron beam lithography by Liu et al [35] who demonstrated the plasmoninduced transparency effect. The coupling of three nanoparticles leads to a more complicated set of interactions with the coupling equation given by 11 11 11 11 11 11 11 11 a~11   1  C 23 C32 C12  C13 C32 C13  C12 C 23   a11   ~ 1  1  11 11 11 11 11 11 11 11   1  C 23  C 21 C13  a 2  a 2    C 21  C 23C31 1  C13 C31 3  11 11 11 11 11 11 11 11   1  a~31  1  C12 C 21   a3    C31  C32 C 21 C32  C31 C12 .

(43)

The three-particle determinant is 11 11 11 11 11 11 11 11 11 11 11 11  3  1  C12 C21  C23 C32  C31 C13  C12 C23C31  C13 C32C21 .

(44)

With three particles we note an interesting symmetry in the determinant. This consists of products of coupling coefficients that lead to closed cycles within the structure. For example, the term

11 C1211C 21 represents the coupling from particle 1 to particle 2 and then from particle 2

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T. J. Davis

C 11C 11C 11

back to particle 1. The three coupling coefficients 13 32 21 represent coupling from particle 1 to 2 to 3 and back to 1. In this context, it is perhaps not surprising that the determinant controls the resonances in the structure, since a resonance represents a condition where energy is trapped, or cycled around the components of the structure. As a general rule for ensembles where each nanoparticle has one dominant mode, the determinant is related to all the cyclic combinations of coupling coefficients. For the metamaterial, nanoparticles 1 and 2 are parallel to one another and have the same





resonant frequency p . Nanoparticle 3 with a resonant frequency s acts as an optical antenna lying across the ends of the other two nanoparticles. Using (31) to estimate the signs of the coupling coefficients, based on the directions of the dipole moments as in Figure 7, we have

11 C1211  C21  G p    p 

C  C  G 11 31

11 32

   

with

Gp  0

a constant,

11 C1311  C23  Gs    p 

G 0

s s s and with another constant. The three-particle determinant, when equated to zero, leads to the following equation

  s    p 2  G p2   s   2Gs2    p   2Gs2G p

0

,

(45)

with solutions

   p  Gp

 ,

 p  s  G p 2



1 G p2  8G s2   s   p 2G p   s   p  2 .

(46)

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As expected from a cubic equation, we have three resonant frequencies. For the metamaterial, the incident light is polarized parallel to the optical antenna, nanoparticle 3.

a11  a 12  0 since their dipole moments are perpendicular to the incident a1   a   s  with a constant. Then the excitation amplitudes light, and we can write 3 This means that

of the coupled nanoparticles are given by

aGs    p  G p   aGs    p  G p    p  G p  a~2  a~1  , a~3  . 2 2  3    s    p   3    s    p 

(47)

The excitation amplitudes of the two parallel nanoparticles have opposite signs and are therefore always out of phase. The net dipole moment associated with these particles is zero and there will be no dipole radiation scattering. Furthermore, the factor    p  G p in





the numerator is also one of the factors in the denominator, since it is a solution of

 3  0 (see (46)). This means that this resonance is not available for excitation by our light ~ of the third nanoparticle. For the field. This is also true for the excitation amplitude a 3

excitation amplitude of the optical antenna, nanoparticle 3, there is an additional factor in the

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numerator that causes it to be zero when    p  G p . However, the excitation amplitudes of the other two nanoparticles are not zero for this condition. This is quite an unusual

~  0 which will not scatter dipole situation. We have the main dipole mode given by a 3 ~  a~  0 so that the two parallel nanoparticles will not emit dipole radiation and we have a 1 2 radiation either. This means that the structure will not scatter light and will appear transparent, except for a small amount of absorption. The coupling of the three nanoparticles, that would otherwise scatter light strongly, has led to a situation where they scatter light very poorly. This has been referred to as plasmon-induced electromagnetic transparency [32, 35]. The interesting feature of this effect is that, even though collectively there is no light ~  0 and a~  0 , which means that there are LSPR scattering, individually we have a 1 2

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oscillations induced in these nanoparticles.

Figure 7. Three coupled plasmonic nanoparticles that exhibit plasmon-induced electromagnetic transparency. The structures (top-left) consist of two parallel nanoparticles (1 & 2) of thickness 1.0 units, width 1.5, length 5.0 and 1.5 units apart. The third nanoparticle is 1.0 unit thick, 2.5 wide and 7.0 long. As the separation is reduced, the scattering cross section develops a minimum representing a

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T. J. Davis

transparent region (the material used here was silver). The graphs on the right show the excitation amplitudes of the three nanoparticles, which gives an indication of the strength of the LSP induced in them.

In the design of Zhang et al, the frequency of the dipole resonance of the single nanoparticle was matched to the quadrupole mode of the parallel pair of nanoparticles. The quadrupole resonance is obtained from (34) yielding sign for the quadrupole resonance and set

   p  Gp

s   p  G p

. We take the negative

. This is one of the conditions for

~  0 . The resonance condition (46) simplifies to    p  G p  2Gs which which a 3 shows a splitting that is symmetric about the resonance of the single dipole. The splitting depends directly on the coupling which can be altered by changing the separation between nanoparticle 3 and the parallel pair. As a demonstration of this effect, we model the configuration given by Zhang et al and plot the scattering cross section at three different separations (Figure 7). We also plot the excitation amplitudes at the smallest separation. The calculations are again based on the full numerical solution for the electrostatic eigenvalue problem assuming the nanoparticles are made from silver [67] and are in air,  b  1 . Since the electrostatic calculation is dimensionless, we only give the relative dimensions of the structure. As expected, the resonance at 478 THz splits into two as the separation decreases

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( G s increasing) leaving a minimum that represents low scattering. The real parts of the excitation amplitudes of nanoparticles 1 and 2, however, remain relatively large, indicating they have significant excitation of localized surface plasmons. Because the amplitudes have the opposite sign, the LSPR resonate out of phase with one another and prevent dipole radiation. Furthermore, these modes are relatively lossless because the imaginary terms are almost zero. A similar effect is observed with the two asymmetric nanoparticles shown in Figure 4. In that example the amplitude of nanoparticle 2 behaves in a similar fashion to the optical antenna in the metamaterial and the amplitude of nanoparticle 1 behaves like the pair of nanoparticles. Comparing Eqs. (35) and (47) shows that they have the same form and therefore we would expect similar behaviour. The induced-transparency effect is a consequence of the relative phases of the resonating nanoparticles that results from their mutual interaction. In general, when we couple plasmonic nanoparticles we are altering the resonant frequencies and creating interference between them. In the following section we show how to use this effect to make a highly sensitive plasmonic circuit that provides an output relating to optical phase differences.

A PLASMONIC CIRCUIT – THE WHEATSTONE BRIDGE When taking accurate measurements of a quantity, very sensitive methods can be made using the principle of a null measurement. In this situation, the measuring apparatus is contrived to produce zero output when the measurement reaches a desired value. The idea is that small signals about zero are much easier to measure than small signals about a large value. In electrical measurements, a special circuit is often constructed to provide this null measurement. An example is the Wheatstone bridge circuit that was popularized by Sir

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Charles Wheatstone in 1843 (Figure 8). The circuit consists of a resistance bridge with an unknown component and a voltmeter for measuring the potential difference between two arms of the bridge [68, 69]. Under the null or balanced condition (zero voltage) the value of the unknown resistance is given by the ratio R x  Rc Rb Ra . More importantly, small changes in the resistance can induce large changes in the measured voltage which provides a sensitive measure of the small changes. If the applied voltage is replaced by a signal generator to provide an alternating voltage source, then the voltage measurement could, in principle, be measured across two capacitances. Instead of this arrangement, we can consider two parallel plasmonic nanoparticles as arms of the bridge and a third nanoparticle as an optical antenna to out-couple difference signals, thereby playing the role of the voltmeter (Figure 8). Such a configuration was described by Davis et al [40]. Although superficially it resembles the threeparticle metamaterial discussed in the last section, its operation is fundamentally different. The analysis of the plasmonic Wheatstone bridge follows the same procedures as with all the other configurations. We assume that the two parallel nanoparticles are far enough apart that we can ignore their mutual coupling, so that C12  C 21  0 . Furthermore, we assume 11

11

that nanoparticles 1 and 2 are slightly different with resonances at  1 and at  2 respectively. Then the coupling coefficients are

11 11 C13   G   2  , and  G   1  , C23

11 11 C31  C32  G   3  with G  0 a constant. In this configuration, we illuminate the a1  0 since its structure with light polarized parallel to nanoparticles 1 and 2. Then we have 3

  dipole moment is perpendicular to the incident light, and we can write a1   a   1 1

a 1   a   



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2 and 2 with a a constant. The excitation amplitudes of the coupled nanoparticles are obtained from (43) and are given by

a~1 

 a  1   3   a  2   3  ~  , a , 2  3   1   2   3   3   1   2   3  Ga~1  a~2   aG1  2  , a~3     3  3   1   2   3 

(48)

(49)

and the resonances are found from the solution of  3  0 and are given by

  1   2   3   G 2 2  1  2   0 .

(50)

The main feature of this configuration is that the excitation amplitude of nanoparticle 3 depends on the difference between the excitation amplitudes of the other two nanoparticles which, in this example, depends on the difference between their resonant frequencies, as shown in Eq. (49). If the two nanoparticles have the same resonant frequency then there is no LSPR excited in nanoparticle 3. On the other hand, if one of the resonances changes then a

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T. J. Davis

surface plasmon is excited and nanoparticle 3 will emit dipole radiation. This radiation is polarized perpendicular to the incident radiation and can, therefore, be distinguished from it [40]. It is useful in our analysis to include the damping term, as in (28), so we will write the resonant frequencies as 1   R  i 2 and  2   R   R  i 2 where  R is a small difference between the resonances in nanopartcles 1 and 2. To obtain maximum sensitivity, the amplitude of the LSPR in the optical antenna should be at resonance. For  R  0 we find that a resonance of the antenna occurs when 3   R . Then for    R the amplitude in the optical antenna is

a~3 

aGR iaGR ,  2 2 R G   4  i G   8  G 2   2 8



2

2











(51)

which scales with the coupling and the damping for small  R . If the coupling is very strong, so that G   , Eq. (51) has the LSPR amplitude in the antenna scaling as

a~3  ia R G which shows low sensitivity with large coupling. In the regime, where G   , we have a~  ia8G 3 and again the sensitivity is low because the coupling 3

R

is small. We find from (51) that the optimum sensitivity occurs where the coupling is matched to the damping according to

G   2 2 for which the LSPR amplitude is

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a~3  ia 2R 2 . This analysis shows that we need to match the coupling to the damping in the metal to optimise the sensitivity of the plasmonic bridge circuit. To demonstrate this result, we calculate the imaginary part of the excitation amplitude of the antenna as a function of distance from the parallel nanoparticles. The distance is measured from the centre line of nanoparticle 3 to the top of nanoparticle 1 (Figure 8). The calculation was done using the full numerical solution for three nanoparticles of thickness 1.0 units, width 2.5 and length 6.4. Nanoparticle 2 was 3% longer than the others in order to shift its resonance frequency slightly so that  R  0 . From the numerical calculation, the eigenvalues were found to be   1.15 . With the nanoparticles made from gold and the permittivity of the background medium set to  b  1.77 , the resonances occur at 834 nm and at 846 nm for the longer nanoparticle. The results in Figure 9 demonstrate that the excitation amplitude has a maximum at a distance of about 5.2 units. The coupling constant, also shown in Figure 9, has a value of G  1.3  10  R at this distance. To compare this result with our analytical model, we estimate the Drude damping factor which can be written in terms of the real and imaginary parts of the electric permittivity,    Im    1  Re    . Using 2

the known values for the permittivity of gold we estimate that   3 10  R which 2

predicts an optimum coupling of G   2 2  1.1102 R . This value compares well with the results of the numerical calculation.

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Figure 8. Left: the Wheatstone bridge circuit from electronics; right: the plasmonic equivalent of the Wheatstone bridge. The circuit consists of three metallic nanoparticles in the form of rectangular prisms. Nanoparticles 1 and 2 form the arms of the bridge and 3 is an optical antenna for coupling out the difference signal. Molecules binding to one of the arms of the bridge can unbalance the circuit causing surface plasmon resonances in the antenna.

Figure 9. The coupling constant (solid line) and the imaginary component of the excitation amplitude of the optical antenna (dashed line) as a function of the distance of the optical antenna from the top of the parallel pair of nanoparticles. The data were calculated using the numerically determined eigenfunctions and taking the nanoparticle material as gold in a background medium with  b  1.77

As we have discussed, the plasmonic Wheatstone bridge is sensitive to the resonance changes in the bridge circuit. For sensing applications, the resonance will shift due to the presence of molecules, such as proteins, adsorbing onto one of the nanoparticles. In effect, the adsorption causes a small shift in the electric permittivity about the nanoparticle which changes the resonant frequency. The effects of the interactions of molecules on the LSPR of nanoparticle structures were analysed by Davis et al. [63]

CONCLUSION The reader may now be finding that the analysis of complicated plasmonic structures is relatively trivial, despite the complicated mathematical machinery developed at the beginning

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of this chapter. The mathematics is there to provide the theoretical background required to understand the method, to highlight the assumptions that underlie it and its limitations. The key point, however, is that the method is easy to implement and does not require the use of complex numerical methods. Although we have only considered the coupling of up to three nanoparticles, it is possible to represent the coupling of any number of nanoparticles, although the inversion of the matrix in (25) becomes quite complicated. Even so, for systems that display a high degree of symmetry, the matrix and the determinant can be relatively simple. Moreover, it is possible to deduce the form of the eigenfunctions using group theory, as discussed by Gómez et al [65]. The coupling method has also been applied to a number of other problems relating to coupled plasmonic systems. As we have already discussed, the method has been used to model the interactions of molecules with metallic nanoparticles supporting LSPR [63]. Another important problem is the effect of a substrate on the resonances. In most experimental situations, the nanoparticle rests on a substrate and the evanescent electric fields induce polarization charges at the substrate surface that also affect the resonant frequencies. If the substrate is a plane, which is usually the case, it is straightforward to represent the effects of the induced charges by “image” charges [70]. In this situation, the image charge appears like another nanoparticle in the substrate (that is, a pseudoparticle) and its effects can be modelled using the coupling theory [71]. While numerical methods are useful, and perhaps essential for accurate predications of resonances (particularly when retardation is important) the underlying physical mechanisms are contained in the electrostatic eigenmode method. It should be noted that the method does not take into account magnetic effects. It is purely electrostatic in nature, modelling the effects of the evanescent electric fields. However, it is possible to calculate the magnetic fields produced by the LSPR from the time-dependent oscillations of the dipole moments, even though the theory assumes that these magnetic fields do not react back on the LSPR. The back-reaction is a second order effect [54] and arises over distances such that the condition

 b kd2  1

fails. Even with this limitation, the electrostatic eigenmode method

provides useful analytical expressions that enable an analysis of coupled plasmonic systems and provides a framework by which systems can be designed. As the use of nanoparticle systems becomes more wide spread, it is likely that the effects of coupling will become important for creating different optical effects or for manipulating light at the nanoscale. For these applications, the coupling method presented here will be of value in providing a means for estimating the effects of the coupling, independent of the shapes of the nanoparticles.

ACKNOWLEDGMENTS The author would like to thank Daniel Gómez and Kristy Vernon for their reading of the manuscript and for providing helpful suggestions. This work was sponsored by the CSIRO Future Manufacturing Flagship.

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[17] Barbillon, G., Bijeon, J. L., Plain, J., Chapelle, M. L. d. l., Adam, P. M. & Royer, P. (2007). "Electron beam lithography designed chemical nanosensors based on localized surface plasmon resonance," Surface Science, 601, 5057-5061. [18] Lucas, B. D., Kim, J. S., Chin, C. & Guo, L. J. (2008). "Nanoimprint Lithography Based Approach for the Fabrication of Large-Area, Uniformly Oriented Plasmonic," Advanced Materials, 20, 1129-1134. [19] Song, Y., Yanga, W. & King, M. (2008). "Shape controlled synthesis of sub-3 nm Ag nanoparticles and their localized surface plasmonic properties," Chemical Physics Letters, 455, 218 -224. [20] Jana, N. R., Gearheart, L. & Murphy, C. J. (2001). "Seed-Mediated Growth Approach for Shape-Controlled Synthesis of Spheroidal and Rod-like Gold Nanoparticles Using a Surfactant Template," Advanced Materials, 13, 1389-1393. [21] Nikoobakht, B. & El-Sayed, M. A. (2003). "Preparation and Growth Mechanism of Gold Nanorods (NRs) Using Seed-Mediated Growth Method," Chemistry of Materials, 15, 1957-1962. [22] Haynes, C. L. & Van Duyne, R. P. (2001). "Nanosphere Lithography: A Versatile Nanofabrication Tool for Studies of Size-Dependent Nanoparticle Optics," The Journal of Physical Chemistry B, 105, 5599-5611. [23] Rechberger, W., Hohenau, A., Leitner, A., Krenn, J. R., Lamprecht, B. & Aussenegg, F. R. (2003). "Optical properties of two interacting gold nanoparticles," Optics Communications, 220, 137 -141. [24] Nordlander, P., Oubre, C., Prodan, E., Li, K. & Stockman, M. I. (2004). "Plasmon Hybridization in Nanoparticle Dimers," Nano Letters, 4, 899-903. [25] Jain, P. K., Eustis, S. & El-Sayed, M. A. (2006). "Plasmon Coupling in Nanorod Assemblies: Optical Absorption, Discrete Dipole Approximation Simulation, and Exciton-Coupling Model," The Journal of Physical Chemistry B, 110, 18243-18253. [26] Thaxton, C. S. & Mirkin, C. A. (2005). "Plasmon coupling measures up," Nature Biology, 23, 681-682. [27] Tabor, C., Murali, R., Mahmoud, M. & El-Sayed, M. A. (2008). "On the Use of Plasmonic Nanoparticle Pairs As a Plasmon Ruler: The Dependence of the Near-Field Dipole Plasmon Coupling on Nanoparticle Size and Shape " The Journal of Physical Chemistry A, 113, 1946-1953. [28] Funston, A. M., Novo, C., Davis, T. J. & Mulvaney, P. (2009). "Plasmon Coupling of Gold Nanorods at Short Distances and in Different Geometries," Nano Letters, 9, 16511658. [29] Tabor, C., Van Haute, D. & El-Sayed, M. A. (2009). "Effect of Orientation on Plasmonic Coupling between Gold Nanorods," ACS Nano, 3, 3670-3678. [30] Verellen, N., Sonnefraud, Y., Sobhani, H., Hao, F., Moshchalkov, V. V., Dorpe, P. V., Nordlander, P. & Maier, S. A. (2009). "Fano Resonances in Individual Coherent Plasmonic Nanocavities," Nano Letters, 9, 1663-1667. [31] Stockman, M. I., Faleev, S. V. & Bergman, D. J. (2001). "Localization versus Delocalization of Surface Plasmons in Nanosystems: Can One State Have Both Characteristics?," Physical Review Letters, 87, 167401 -167401. [32] Zhang, S., Genov, D. A., Wang, Y., Liu, M. & Zhang, X. (2008). "Plasmon-Induced Transparency in Metamaterials," Physical Review Letters, 101, 47401-47401.

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[33] Davis, T. J., Vernon, K. C. & Gómez, D. E. (2009). "Designing plasmonic systems: applications to dark modes in nanoparticle pairs and triplets," Proceedings of the SPIE, 7394, 739423. [34] Yang, S. C., Kobori, H., He, C. L., Lin, M. H., Chen, H. Y., Li, C., Kanehara, M., Teranishi, T. & Gwo, S. (2010). "Plasmon Hybridization in Individual Gold Nanocrystal Dimers: Direct Observation of Bright and Dark Modes," Nano Letters, 10, 632-637. [35] Liu, N., Langguth, L., Weiss, T., Kästel, J., Fleischhauer, M., Pfau, T. & Giessen, H. (2009). "Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit," Nature Materials, 8, 758-762. [36] Hao, F., Nordlander, P., Sonnefraud, Y., Dorpe, P. V. & Maier, S. A. (2009). "Tunability of Subradiant Dipolar and Fano-Type Plasmon Resonances in Metallic Ring/Disk Cavities: Implications for Nanoscale Optical Sensing," ACS Nano, 3, 643652. [37] Mirin, N. A., Bao, K. & Nordlander, P. (2009). "Fano Resonances in Plasmonic Nanoparticle Aggregates," The Journal of Physical Chemistry A 113, 4028-4034. [38] Hao, F., Sonnefraud, Y., Dorpe, P. V., Maier, S. A., Halas, N. J. & Nordlander, P. (2008). "Symmetry Breaking in Plasmonic Nanocavities: Subradiant LSPR Sensing and a Tunable Fano Resonance," Nano Letters, 8, 3983-3988. [39] Ac´imovic´, S. S., Kreuzer, M. P., González, M. U. & Quidant, R. (2009). "Plasmon Near-Field Coupling in Metal Dimers as a Step toward Single-Molecule Sensing," Nano, 3, 1231 -1237. [40] Davis, T. J., Vernon, K. C. & Gomez, D. E. (2009). "A plasmonic ``ac Wheatstone bridge'' circuit for high-sensitivity phase measurement and single-molecule detection," Journal of Applied Physics, 106, 043502. [41] Yee, K. (1966). "Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media," Antennas and Propagation, IEEE Transactions on, 14, 302-307. [42] Taflove, A. & Brodwin, M. E. (1975). "Numerical Solution of Steady-State Electromagnetic Scattering Problems Using the Time-Dependent Maxwell's Equations," Microwave Theory and Techniques, IEEE Transactions on, 23, 623-630. [43] García de Abajo, F. J. & Howie, A. (1998). "Relativistic Electron Energy Loss and Electron-Induced Photon Emission in Inhomogeneous Dielectrics," Physical Review Letters, 80, 5180. [44] García de Abajo, F. J. & Howie, A. (2002). "Retarded field calculation of electron energy loss in inhomogeneous dielectrics," Physical Review B, 65, 115418. [45] Draine, B. T. & Flatau, P. J. (1994). "Discrete-dipole approximation for scattering calculations," J. Opt. Soc. Am. A, 11, 1491-1499. [46] Noponen, E. & Turunen, J. (1944). "Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles," J. Opt. Soc. Am. A, 11, 24942502. [47] Moharam, M. G., Grann, E. B., Pommet, D. A. & Gaylord, T. K. (1995). "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A, 12, 1068-1076.

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[48] Moharam, M. G., Pommet, D. A., Grann, E. B. & Gaylord, T. K. (1995). "Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach," J. Opt. Soc. Am. A, 12, 1077-1086. [49] Li, L. (1997). "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am., A 14, 2758-2767. [50] Prodan, E. & Nordlander, P. (2003). "Structural Tunability of the Plasmon Resonances in Metallic Nanoshells," Nano Letters, 3, 543-547. [51] Prodan, E. & Nordlander, P. (2004). "Plasmon hybridization in spherical nanoparticles," Journal of Chemical Physics, 120, 5444 -5454. [52] Prodan, E., Radloff, C., Halas, N. J. & Nordlander, P. (2003). "A Hybridization Model for the Plasmon Response of Complex Nanostructures," Science, 302, 419 -422. [53] Ouyang, F. & Isaacson, M. (1989). "Surface plasmon excitation of objects with arbitrary shape and dielectric constant," Philosophical Magazine B, 60, 481-492. [54] Mayergoyz, I. D., Fredkin, D. R. & Zhang, Z. (2005). "Electrostatic (plasmon) resonances in nanoparticles," Physical Review B, 72, 155412. [55] Mayergoyz, I. D., Zhang, Z. & Miano, G. (2007). "Analysis of Dynamics of Excitation and Dephasing of Plasmon Resonance Modes in Nanoparticles," Physical Review Letters, 98, 147401. [56] Davis, T. J., Vernon, K. C. & Gomez, D. E. (2009). "Designing plasmonic systems using optical coupling between nanoparticles," Physical Review B (Condensed Matter and Materials Physics), 79, 155423. [57] Davis, T. J., Vernon, K. C. & Gómez, D. E. (2009). "Effect of retardation on localized surface plasmon resonances in a metallic nanorod," Opt. Express 17, 23655-23663. [58] Kellog, O. D. (1929). Foundations of Potential Theory (Frederick Ungar Publishing Co, New York,). [59] Courant, R. & Hilbert, D. (1953). Methods of Mathematical Physics (Interscience Publishers, New York). [60] Bergman, D. J. (1978). "The dielectric constant of a composite material--A problem in classical physics," Physics Reports, 43, 377-407. [61] Bergman, D. J. (1979). "Dielectric constant of a two-component granular composite: a practical scheme for calculating the pole spectrum," Physical Review B 19, 2359-2368. [62] Johnson, P. N. & Christy, R. W. (1972). "Optical Constants of Noble Metals," Physical Review B, 6, 4370-4379. [63] Davis, T. J., Gómez, D. E. & Vernon, K. C. (2010). "Interaction of molecules with localized surface plasmons in metallic nanoparticles," Physical Review B., 81, 045432. [64] Bohren, C. F. & Huffman, D. R. (1983). Absorption and scattering of light by small particles (John Wiley and Sons, New York). [65] Gómez, D. E., Vernon, K. C. & Davis, T. J. (2010). "Symmetry effects on the optical coupling between plasmonic nanoparticles with applications to hierarchical structures," Physical Review B, 81, 075414. [66] Davis, T. J., Gomez, D. E. & Vernon, K. C. (2010). "A simple model for the hybridization of surface plasmon resonances in metallic nanoparticles," In press. [67] Ordal, M. A., Long, L. L., Bell, R. J., Bell, S. E., Bell, R. R., Alexander, J. R. W. & Ward, C. A. (1983). "Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared," Appl. Opt., 22, 1099-1119.

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[68] Brophy, J. J. (1977). Basic electronics for scientists (McGraw Hill Kogakusha Ltd, Tokyo). [69] Diefenderfer, A. J. (1972). Principles of electronic instrumentation (W. B. Saunders Co. London,). [70] Jackson, J. D. (1975). Classical electrodynamics (John Wiley & Sons Inc, New York). [71] Vernon, K. C., Funston, A. M., Novo, C., Gómez, D. E., Mulvaney, P. & Davis, T. J. (2010). "Influence of particle-substrate interaction on localised plasmon resonances," to be published.

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In: Plasmons: Theory and Applications Editor: Kristina N. Helsey, pp.143-160

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Chapter 5

SPR-FACILITATED INTERROGATION OF ENZYMATIC REACTIONS INVOLVING NUCLEIC ACIDS Wendi M. David* Department of Chemistry & Biochemistry, Texas State University-San Marcos, San Marcos, Texas

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ABSTRACT Processing of nucleic acids during transcription, translation, replication, and repair must proceed smoothly for survival. Numerous proteins interact with DNA and RNA to mediate these processes, ranging from those that allow access to the genome and participate in transmission of coded information to those that directly repair damaged areas of the genome. Interactions between proteins and nucleic acids are often investigated using polyacrylamide gel electrophoresis (PAGE) analysis, which necessarily yields information concerning the end result of a particular interaction. More recent analytical techniques to monitor intermediate or real-time interactions include fluorescence resonance energy transfer (FRET) spectroscopy and surface plasmon resonance (SPR). SPR is label-free, an advantage for interrogating many biomolecular interaction events. Although SPR has been used extensively to determine small molecule/DNA and protein/DNA binding interactions, SPR analysis of enzymatic reactions involving nucleic acids has lagged behind. However, several recent studies of enzymatic reactions that extend nucleic acids or unwind different topological forms of DNA demonstrate significant advantages of SPR for discerning aspects of these interactions that are difficult to “see” using other techniques. This chapter focuses on the emerging application of SPR technology for elucidating enzymatic reactions involving nucleic acids.

* Corresponding

author: 512-245-4637 (ph), [email protected] (email)

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INTRODUCTION Nucleic acids are one of the most important biomolecules, serving as the repository of genetic information and means of perpetuating life. DNA is composed of the four nucleobases -adenine, guanine, cytosine, and thymine - connected by glycosidic bonds to a deoxyribosephosphate backbone. Uracil substitutes for thymine in RNA molecules. DNA and RNA are helical molecules, with DNA normally found as a right-handed duplex of anti-parallel selfcomplementary strands held together by hydrogen bonds between complementary nucleobases on opposite strands (Figure 1). DNA triplex and quadruplex structures also form. RNA adopts a number of conformations, from single-stranded or duplex, to hairpin and other higher order structures. The terms ‘oligonucleotide’ or ‘oligoribonucleotide’ typically refer to short, synthetic DNA or RNA polymers, respectively but will be used interchangeably with ‘DNA’ and ‘RNA’ in this chapter. In cells, DNA is compacted into nucleolids (prokaryotic cells) or chromosomes (eukaryotic cells) through interaction with specific proteins, by which access to the genetic information is controlled. In order for efficient cellular functioning to occur, DNA must be repaired, copied, and transcribed. The transmission of genetic information from DNA to protein is mediated by RNA, and the number of processes manipulating both DNA and RNA is vast. This chapter will review the application of surface plasmon resonance (SPR) technology towards the investigation of enzymatic processes involving nucleic acids. Consequently, these enzymatic processes will be grouped into the following broad categories: polymerization (synthesis of DNA or RNA), repair (excision of damage, ligation), topological manipulation (unwinding), and cleavage. Many other enzymatic manipulations of nucleic acids exist but have not been directly investigated using SPR. Traditional techniques for monitoring nucleic acid processing often rely on the detection of radioactively- or fluorescently-labeled oligonucleotides. The most common technique is polyacrylamide gel electrophoresis (PAGE) to detect the length of radiolabeled oligonucleotides. Although recent, miniaturized versions of PAGE minimize the amount of material required [1], some types of processes (such as unwinding of unimolecular quadruplex DNA) are not easily visualized. Fluorescence resonance energy transfer (FRET) methods are also commonly employed for monitoring changes in nucleic acid substrates but similarly require labeling of oligonucleotide substrates [2]. SPR techniques are widely used for monitoring and quantifying binding interactions between biomolecules and the number of yearly journal articles covering the wide range of binding interactions is increasing rapidly [3-5]. A major advantage of SPR-based biosensing is the ability to determine real-time binding interactions and obtain kinetic information in a label-free manner with very little sample required. Complicated binding interactions can often be deconstructed, yielding biologically relevant information. Exploitation of surface enzymatic processes coupled with SPR is not new, but has primarily focused on protein transformations [6, 7]. The use of SPR for directly investigating enzymatic processing of nucleic acids has lagged behind, but is potentially transformative for obtaining information about individual steps in these processes.

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Figure 1. Nucleic acids and several topological forms. Uracil and thymine nucleobases are shown linked to a sugar-phosphate molecule through an N-glycosidic bond to form a ribonucleotide (UTP) and deoxyribonucleotide (dTTP), respectively. Phosphodiester linkage of successive (deoxy)ribonucleotides forms single-stranded oligonucleotides. Hydrogen bonding between cytosine and thymine nucleobases is denoted with dashed lines to indicate normal base pairing found in DNA. Schematic representation of B-form helical duplex DNA and various topological types of G-quadruplex DNA displays the sugarphosphate backbone as a ribbon or solid line, respectively.

This chapter will focus on the emerging SPR-based strategies for monitoring real-time enzymatic transformations of nucleic acids, especially those strategies designed to gain mechanistic insights. A brief overview of enzyme-based processes for increasing sensitivity of nucleic acid detection by SPR will also be given. Methods reviewed below employ prism coupling-based SPR measurements, often determined with commercially available biosensors. Experiments utilizing alternative types of SPR measurements [8, 9] are not considered in this chapter.

DIRECT MEASUREMENT OF ENZYMATIC ACTIVITY INVOLVING OLIGONUCLEOTIDES DNA and RNA Polymerization Polymerization (or synthesis) of DNA and RNA fundamentally involves the catalyzed formation of phosphodiester bonds to link deoxyribonucleotides or ribonucleotides together in a growing chain. A transesterification reaction between a 5’-deoxyribonucleoside triphosphate (5’-dNTP) and the 3’-hydroxyl end of the growing chain is catalyzed by DNA

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polymerases; RNA polymerases add a 5’-ribonucleotide triphosphate (5’-NTP) to the growing RNA chain. Polymerization occurs in a number of different processes, including replication of DNA, some types of repair, and RNA synthesis during transcription. Polymerization of DNA always requires a primer sequence (a ‘starter’ nucleic acid chain), whereas RNA polymerization does not require a primer. Both processes require a template strand to direct the polymerization of the correct nucleotides in the strand being synthesized. Reverse transcription involves synthesis of DNA from a RNA template. One of the earliest SPR assays for monitoring nucleic acid template-primer elongation was developed in 1996 [10]. A Biacore instrument was employed for the study. Templateprimer substrates for the Moloney murine leukemia virus reverse transcriptase (MoMLV-RT) were formed in situ (Figure 2). First, short oligonucleotide sequences were immobilized to streptavidin-coated (SA) sensor chips through a 3’-biotin linker. Next, a short primer was hybridized to the immobilized strand through a partially complementary region. The template was formed by adding a third oligonucleotide such that a single-stranded template region extended past the primer. Binding of MoMLV-RT to several different template-primers caused an increase in the observed SPR signal and was quantified; subsequent nucleotide polymerization was observed to be dependent upon addition of the correct nucleotide (Figure 2). In the example shown in Figure 2, MoMLV-RT added a nucleotide stretch containing 25 cytosine residues complementary to the template-primer G25 region upon the addition of 5’dCTP. Subsequent addition of adenine triphosphate (5’-dATP) and RT generated an A25 extension complementary to the template-primer T25 region. Furthermore, the addition of a RT inhibitor drug, 3’-deoxy-3’-azidothymidine triphosphate (AZT), abolished elongation and increased the observed dissociation of MoMLV-RT from the template-primer.

5’-(T)25 (G)25

5’-(T)25 (G)25

5’

+ RT +dCTP

3’-(C)25

5’

Figure 2. Reverse transcriptase polymerization assay. DNA oligonucleotides were immobilized to the streptavidin-coated sensor surface through a 3’-biotin linker. A primer was hybridized to the immobilized oligo, followed by a third oligonucleotide template to form the gapped duplex shown on the left. The template is denoted with (G)25 and (T)25 sequences. Following injection of the reverse transcriptase (RT) and the correct nucleotide dCTP, a (C)25 sequence was synthesized on the primer. Further injection of RT and dATP added an (A)25 stretch.

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A later SPR investigation of RNA polymerase activity demonstrated formation of an E. coli RNA polymerase (RNAP) initiation complex on an immobilized DNA substrate, as well as synthesis of RNA from the DNA template [11]. Binding of RNAP to a DNA promoter sequence and a subsequent conformational change to form the processive elongation enzyme occurs prior to elongation of a nascent RNA strand along the template [12]. Each of these successive steps in RNAP activity was distinctly observed by SPR. A 5’-biotinylated 203base pair DNA fragment representing the lac UV5 promoter was immobilized on SA sensor chips; SPR detection was performed with a Biacore instrument. Binding of RNAP to the promoter sequence (lac UV5) was determined to have a stoichiometry of 1:1 and multiphasic dissociation of the complex was observed. Addition of more RNAP to the sensor surface along with the appropriate ribonucleotide triphosphates necessary for RNA polymerization led to a dramatic increase in signal and an apparent stoichiometry of a maximum of 2 RNAP bound per immobilized DNA template during the elongation phase. Synthesis of the RNA transcripts was confirmed by using 32P-labeled UTP for elongation and gel electrophoretic analysis of transcripts recovered from the flow channel after the reaction. Interestingly, PCR analysis of transcripts indicated that RNAP transcribed one strand of the immobilized DNA template, moved over and around the biotin-streptavidin linkage of the immobilized DNA, and transcribed the other strand. A corresponding experiment with DNA immobilized on magnetic beads produced a similar result. Recently, von Hippel and coworkers have extended the initial SPR studies of transcription to more effectively monitor E. coli RNA transcription and directly distinguish between paused and terminated states [13]. SPR was used to quantitatively analyze initiation, elongation, and termination events on immobilized DNA substrates. RNA transcription is tightly regulated to control gene expression; control is often exerted at the point of termination [14]. Elongation of a transcript competes with arrested, editing, or termination states, which are often modulated by protein cofactors and sequence-specific signals. Estimation of termination rates using traditional methods is often indirect, especially since discrimination between paused RNAP complexes and release of RNAP is difficult. Initiation of RNA transcription begins when the holoenzyme plus a protein called sigma bind to the DNA promoter region. Initiation and escape from the promoter with concomitant loss of sigma leads to the RNA transcription elongation complex (TEC), the core of which is conserved in all organisms. The TEC from E. coli undergoes intrinsic termination after an RNA terminator hairpin has been formed, which effectively causes dissociation of RNAP [15]. Elongation rates, as well as the discrimination between stalled and terminated TECs can be determined using the SPR method. Formation of the RNAP initiation complex was monitored in real-time. Over time the SPR signal gradually decreased, reflecting a loss of non-specifically bound components. Addition of only three of the four possible nucleotides started transcription, and formation of the TEC was reflected by a decrease in signal corresponding to the mass change from loss of sigma, indicating an initiation-elongation transition (Figure 3). The mass change was confirmed to be due to sigma by mass spectrometric analysis of the flow-through recovered from the sensor surface. A lower limit for an elongation rate of ~10 nucleotides per second was measured. With only three NTPs added, a stable stalled transcription complex was formed with a half-life of ~3h at 25 °C. This stalled complex was resistant to dissociation even under high salt. Further injection of all four NTPs led to RNA elongation until intrinsic termination occurred. The decrease in signal upon TEC dissociation at termination must be

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due to loss of TECs at terminator regions since stalled complexes are resistant to dissociation. Dissociation curves for release of the TEC at the terminator region are multiphasic, probably reflecting a statistically variable population of TECs along the template. Pseudo-first order analysis of the leading edge of the dissociation curves gave an apparent dissociation rate of 23 per second, in range of values reported in other studies [16]. It is difficult to explore the rate between transcription pause, formation of a pretermination complex, and dissociation of the TEC. But by varying the template, and investigating the consequence for pausing or TEC dissociation, SPR can be utilized to obtain quantitative information for these different reaction pathways since stalled and terminated complexes are easily distinguished. An interesting application of SPR technology towards the analysis of nucleic acid elongation is the telomeric repeat elongation (TRE) assay developed by Maesawa and coworkers [17]. Telomerase, a ribonucleoprotein complex, includes a unique reverse transcriptase responsible for elongating telomeric DNA [18]. Telomeric DNA in normal somatic cells progressively shortens with each replicative cycle due to the DNA endreplication problem [19]. In contrast, germ cells and immortalized cancer cells overexpress telomerase, thereby maintaining telomeric lengths. Consequently, telomerase is an attractive target for chemotherapeutic agents. Traditionally, the effectiveness of telomerase inhibitors is assessed using the telomerase repeat amplification protocol (TRAP) assay, a PCR-based assay that amplifies extensions, followed by PAGE to analyze the extension products [20]. Disadvantages of the TRAP assay include PCR-derived artifacts and the requirements of postTRAP workup protocols. The TRE was performed with the commercially available Biacore 3000 instrument. Oligonucleotides (5’-biotinylated) were immobilized to SA sensor chips and exposed in situ to telomerase extracts from eighteen different human cancer cell lines, and three normal fibroblast cell lines. Elongation of the immobilized DNA occurred after injection of telomerase extracts in the presence of dNTPs (Figure 4). The elongation was dependent upon the concentration of telomerase extract and time-dependent, as expected from normal enzymatic activity. The number of bases added to the growing oligonucleotide was calculated from the observed increase in response using the empirical relationship of 1500 RU = 1.8 ng DNA. Elongation rates of 2.4– 3.9 nucleotides per minute were determined for the fibroblast cell lines, with the cancer cell line elongation rates 2-10 times higher. Comparison to rates determined by the TRAP assay for the cancer cell lines was reasonable; elongation rates for the fibroblast cell lines were presumably too low to be amplified in the TRAP assay, but are detectable in the TRE assay. Furthermore, the assay is specific for telomerase extension of DNA containing the telomeric repeat [human (TTA GGG)n]. Non-telomeric oligonucleotides were not elongated, indicating any additional polymerases were not contained in the telomerase extracts. In addition, heat-inactivated telomerase did not bind to the immobilized oligonucleotides or extend them. Binding, but not elongation, did occur when injected cellular telomerase extracts did not contain dNTPs. The RT inhibitor AZT inhibited telomerase activity in two breast cancer cell lines to the same extent in both the TRE and TRAP assays. The benefits of this SPR-based TRE assay include no labeling requirements, rapid quantitative measurement of activity, the ability to measure telomerase activity of somatic cells, and the potentially rapid screening of inhibitors.

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Relative RU

Addition of 3 NTPs

 -120 RU

Time

Figure 3. Sensorgram representing the transition from RNA polymerization initiation to elongation. Addition of three ribonucleotides to a preformed RNAP/immobilized template complex was followed by a decrease in response proportional to the loss of sigma protein. A stalled complex resistant to further dissociation formed until addition of all four possible NTPs restarted elongation.

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G G G A T T

DNA elongation by telomerase

T T G G G A T T

A

G G GT

Figure 4. Telomerase repeat elongation (TRE) assay. Extension of an immobilized template with the human telomeric sequence, [TTA GGG], occurred after addition of cellular telomerase extracts and dNTPs. The increased SPR response observed in this assay is directly proportional to the amount of DNA added to the sensor surface.

Excision of DNA Damage DNA is under constant repair to correct for errors in replication, spontaneous damage and damage caused by environmental factors. Many types of DNA repair pathways for mitigating genomic damage exist. Base excision repair (BER) is responsible for repairing damage resulting from alkylation and oxidation of nucleobases. The first step in the pathway is accomplished by different DNA N-glycosylases that excise a specific damaged base. Cleavage of the N-glycosidic bond between the nucleobase and the DNA sugar-phosphate backbone leaves an apurinic/apyrimidinic (AP) site. Further AP-endonuclease or AP-lyase activity removes the AP site, leaving a gap in the DNA duplex. The gap is filled by a DNA polymerase and ligated by a DNA ligase to reform the continuous DNA strand. A hint that DNA repair could be observed in real-time using SPR occurred during an investigation of uracil DNA glycosylase (UDG) substrate preference [21]. UDG’s are highly

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conserved enzymes that excise uracil residues from DNA with high precision [22]. Uracil is a RNA base not normally found in DNA. The incorporation of uracil into DNA can occur either through misincorporation of dUTP instead of dTTP during replication or through deamination of cytosine to uracil. The shift from cytosine to uracil is pro-mutagenic, leading to mutations in the genome. The primary focus of the work by Pearl and coworkers was to investigate UDG substrate recognition, in particular whether UDG actively ‘flips’ out a uracil base from the oligonucleotide strand for excision, or simply recognizes spontaneous extrahelical uracil residues [21]. Binding of UDG to immobilized oligonucleotides (single-stranded and doublestranded) containing uracil bases (U-DNA) was quantified for several UDG mutants that were catalytically inactive but capable of binding U-DNA. Experiments were conducted with a Biacore Biosensor instrument using 5’-biotinylated oligonucleotides immobilized on SA chips. The affinity between U-DNA and UDG mutants depended on the structural context of the uracil base, with single-stranded U-DNA binding to UDG mutants with greater affinity than duplex DNA containing a uracil substitution. However, if the immobilized U-DNA was exposed to wild-type UDG, further binding between the mutant UDG’s and the immobilized DNA did not occur to any great extent, indicating that the uracil bases had been excised by wild-type UDG. Neither the wild-type nor mutant UDG’s showed any detectable affinity for the abasic site resulting from uracil excision. Other studies using SPR to quantify binding between repair-associated proteins and damaged DNA sites similarly focused on characterizing binding affinities [23, 24]. More recently, however, Gasparutto and coworkers have used SPR imaging to observe both the binding of the repair protein E. coli Fapy DNA N-glycosylase (Fpg or mutM) to damaged bases, and to observe repair activity in real-time [25, 26]. The natural substrate of Fpg is 8oxo-7,8-dihydro-2’-deoxyguanosine (8-oxoG) resulting from oxidative damage to guanine. For measurement of Fpg binding and activity, a GenOptics SPR instrument was employed, in which reflectivity changes on an array slide were monitored by CCD camera. DNA array slides were prepared by first coating the gold layer of a glass prism slide with a hydrophobic thiol monolayer and then grafting oligonucleotides containing polypyrrole linkers onto the monolayer using a pyrrole copolymerization process [27]. Hybridization to form duplex substrates was then performed in situ. Injection of CytC was used to block any underivatized gold surface and prevent non-specific binding to the surface of the slide. Strong binding of Fpg to the 8-oxoG damaged site of a duplex substrate was observed, in comparison to weak binding with undamaged duplex DNA and no binding with singlestranded DNA, in agreement with known behavior of the enzyme. However, successive injections of Fpg over damaged DNA previously bound by Fpg showed progressively weaker binding, indicating that excision of the damaged base had occurred and that binding to the duplex now resembled that for the non-specific duplex/Fpg complex. A dissociation constant for the non-specific duplex/Fpg complex was determined to be ~100 nM, lower than the value determined by stopped-flow techniques [28], perhaps due to the increased length of the DNA substrate used in this study or to influence of the surface. Fpg was also demonstrated to bind to a second type of damaged base, 5’,8-cyclo-2’deoxyribosyladenine [(5’S)-5’,8-CyclodA] but enzymatic activity did not occur in this case. Intriguingly, although this is not the primary substrate of Fpg, the authors suggest that it may serve as a competitive inhibiter in vivo, perhaps compromising repair efficiency [26].

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Damaged base

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45 degrees C Gapped duplex

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Figure 5. Excision repair assay. Immobilized oligonucleotides were hybridized to strands containing the damaged base 8-oxoG or (5’S)-5’,8-CyclodA. Exposure of the E. coli DNA repair enzyme Fpg to the 8oxoG-containing duplexes resulted in excision of the damaged guanine and a gapped duplex. The extent of repair was determined by heat-denaturing gapped duplexes, leading to a decrease in observed SPR response.

To confirm and quantify enzymatic activity, a temperature-dependent assay was developed to selectively denature any gapped duplex regions formed in the DNA substrates due to repair activity (Figure 5). DNA denatures, or melts, in a length-dependent fashion since longer duplexes are stabilized by hydrogen bonding between more base pairs than shorter duplexes. The gapped duplex is composed of two shorter duplex regions, with correspondingly lower melting temperatures than the original DNA substrate. Only DNA that had been enzymatically processed would be expected to undergo denaturation at 45 °C since the length of the substrate DNA is otherwise sufficient to prohibit duplex melting at this temperature. After reaction of Fpg with 8-oxoG duplex substrates and application of increased temperature, dissociation of the resulting gapped duplex occurred and was confirmed by reinjection of an oligonucleotide complementary to the specific region. Importantly, no dissociation occurred after injection of Fpg over (5’S)-5’,8-CyclodA duplexes, confirming that Fpg binds this particular lesion but does not excise it from the duplex. In the context of an array slide, this clever assay quickly determines the extent of repair activity and could be easily modified to investigate a number of different parameters simultaneously.

DNA Ligation Ligation of DNA strands occurs to seal a ‘nick’ in a DNA strand. DNA ligase transfers an adenine monophosphate (AMP) residue onto a 5’-phosphate group at the nick of the DNA. The 3’-hydroxyl on the other side of the nick then attacks the newly formed phosphodiester bond between AMP and the 5’-end. A new phosphodiester bond ligating the two proximal ends of the nick is generated, releasing AMP. The reaction is similar to polymerization of DNA. As for many other types of DNA processing reactions, DNA ligase assays are typically performed using PAGE to visualize radiolabeled oligonucleotide products. Although several

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recent electrochemical and fluorescent based detection methods have been developed for sensitivity [29-30] they still require some type of labeling and are time-consuming or costly. A very recent SPR-based method employing a home-built high-resolution SPR instrument with a hairpin DNA probe was developed to assess E. coli ligase activity [31]. This method exploits a conformational change that occurs in an immobilized DNA hairpin probe as a consequence of ligation activity (Figure 6). The immobilized hairpin was first hybridized to two oligonucleotide fragments complementary to the open loop: the first contained a 5’-phosphate end and the second fragment presented a proximal 3’-hydroxyl forming a ‘nicked’ oligonucleotide substrate in the hairpin probe. After ligase joined the nick, the loop region of the original hairpin probe was base paired to form a continuous duplex segment that is longer the original self-paired region of the hairpin. This resulting longer duplex segment is more stable than the short self-complementary duplex stem region of the hairpin, leading to ‘melting’ of the hairpin stem. The conformational change resulting after ligation activity led to a decrease in the observed SPR signal. A lower limit of detection was determined (0.6 nM E. coli ligase) that is comparable to detection limits obtained by fluorescence resonance energy transfer (FRET) methods [30]. Regeneration of the hairpin probe was easily accomplished by removal of the non-immobilized complement strand. No corresponding signal shift was observed when the two oligonucleotides hybridized to the loop portion of the hairpin contained a 5’-hydroxyl end and a proximal 3’-hydroxyl end. Since ligation requires a 5’-phosphate end and 3’-hydroxyl end for joining, the lack of response observed indicates that the enzymatic reaction must occur for the corresponding SPR shift to be observed. Furthermore, the assay is sensitive to mismatches that affect ligation efficiency, especially mismatches near the site of ligation. In addition to the demonstrated specificity of the assay and low detection limit, inhibition with the small molecule Quinacrine was also characterized. Inhibition of ligase activity was dose dependent and reproducible (0.58 M caused 50% inhibition with 60 M ligase).

Figure 6. DNA ligation assay. A DNA hairpin probe was immobilized to the sensor chip surface. Hybridization of two oligonucleotides complementary to the loop region of the hairpin generated the substrate, shown at left, with the 5’-phosphate and 3’-hydroxyl termini indicated. Ligation of the two oligonucleotides by E. coli ligase formed a continuous duplex in the loop region, causing a conformational change to the form shown at right.

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The changes in mass observed in the ligase assay were very small and the key in this study seems to be the home-built high resolution instrument for detecting a conformational change in the hairpin. Restricted access to such an instrument may mitigate the applicability of the assay but modifications could presumably be made such that the detection of the ligated strand could be otherwise accomplished. For instance, perhaps a reverse strategy of the temperature-dependent assay described above for detection of a gapped duplex could be employed to quantitate ligation efficiency. Another important SPR-based ligation assay was developed to investigate HIV integrase [32]. Incorporation of HIV genetic material into host DNA involves DNA ligation and is accomplished by HIV integrase (IN) in two steps. First, IN processes HIV viral DNA. It forms 3’-hydroxyl termini on both ends of the HIV viral long terminal repeat (LTR) DNA segment by cleaving a 5’-phosphorylated dimer (pGT) from each end. During the second step, strand transfer of the LTR to the host DNA is accomplished by a ligation reaction between the 3’-hydroxyl viral DNA end and a 5’-phosphate end of host DNA. Both catalytic activities of IN have been demonstrated in vitro using either radio- or fluorescently-labeled oligonucleotide LTR mimics. However fluorescent labels may exhibit non-specific binding to IN [33]. Additionally, the interaction of IN with host DNA is mediated by different divalent metal ions in a manner that is not well understood. SPR was used to monitor binding and strand transfer activity of IN [32]. The effect of four different divalent cations on the ability of IN to bind LTR mimics as well as reference oligonucleotide targets was determined. Label-free direct monitoring of the IN strand transfer reaction was accomplished with high sensitivity using a supercoiled plasmid DNA target (pUC 19) and a short oligonucleotide LTR mimic. These experiments were conducted with a SPR sensor developed at Institute of Photonics and Electronics (Prague, Czech Republic). The instrumental set-up is similar to that of Biacore instruments – a 4- or 8- channel sensor using the change in refractive index at the surface of a gold film (50 nm) under temperature controlled conditions. Sensor chips contained a self-assembled monolayer (SAM) coating covalently attached to streptavidin. Biotinylated oligonucleotides were captured directly by streptavidin. IN was observed to bind to duplex-LTR mimics and reference triplex substrates to the same extent but not to the underivatized streptavidin sensor surface. Binding was most effective in the presence of Mg2+ or Mn2+, as opposed to Co2+ and Ni2+. After rigorous washing and correction for nonspecific binding, strand transfer was directly observed only for the target duplex-LTR mimics and not for reference surfaces (Figure 7). The pUC 19 plasmid was covalently attached to the duplex-LTR mimic at a low concentration, perhaps due to steric hindrance of the large plasmid, but only ~10% of the immobilized duplex-LTR was removed by washing, leaving the rest effectively ligated through strand transfer. Strand transfer was most efficient with Mn2+, as opposed to Mg2+. Strand transfer of a much shorter target oligonucleotide (12-mer) was accomplished to demonstrate that the actual covalent transesterification process did occur, as opposed to simply tight binding of the IN/immobilized DNA complex to the long pUC 19 plasmid. The authors reported kinetic constants for binding of IN to the duplex-LTR mimics and suggested that modification of the duplex-LTR mimics may lead to an increase in sensitivity. The assay is potentially useful for identifying inhibitors of HIV-IN activity, especially if used in an array-type format.

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Figure 7. Strand transfer assay for HIV integrase (IN). Immobilization of a short oligonucleotide LTR mimic substrate, followed by binding of IN led to excision of a 5’-dinucleotide (dGT), leaving a 3’hydroxyl end. Subsequent injection of the pUC 19 plasmid DNA over the LTR mimic/IN complex led to ligation of pUC 19 to the duplex LTR mimic.

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DNA Topological Manipulation Helicase proteins function to unwind DNA and RNA in a specific direction – either 5’-3’ or 3’-5’. Translocation along the nucleic acids requires ATP hydrolysis to power the helicase. Numerous helicases are known and are involved in maintenance of DNA and RNA topological forms. While unwinding of duplex DNA and bimolecular or tetramolecular Gquadruplex substrates (G2 or G4, respectively) can be monitored using traditional gel shift assays, unwinding of unimolecular G-quadruplex structures (G4’) is not easily visualized. However, formation of G-quadruplex DNA is increasingly implicated in genome maintenance and function [34-36] and efforts to develop and characterize helicase inhibitors are ongoing. The ability to monitor unwinding of G4’ structures is important in this endeavor. We recently developed the first SPR-based assay of helicase activity for investigation of duplex and G-quadruplex DNA unwinding [37, 38]. The helicase employed in these studies was large T-antigen (T-ag), from simian virus 40 (SV40). T-ag, a member of the helicase superfamily III (SF3), is a hexameric helicase with duplex and G-quadruplex unwinding activity [39-40]. Prior to our work, unwinding of G4’ structures by T-ag had not been demonstrated, although at least one important potential G-quadruplex-forming region of the SV40 genome is unimolecular in structure. Binding of T-ag to single-stranded, duplex, and unimolecular G4’ oligonucleotide substrates was characterized using a Biacore X SPR instrument. DNA substrates were immobilized through 5’- or 3’-biotinylated linkers on SA sensor chips. The binding was ATPdependent as expected and the response was much greater for oligonucleotides immobilized through 5’-biotin linkers. Since T-ag is a 3’-5’ helicase (it moves along the DNA from the 3’ to 5’ direction) this binding behavior is in agreement with the mechanism of T-ag [39]. Association curves for binding to the 5’-immobilized oligonucleotides indicated multiple hexamers bound per strand, indicating processive loading of T-ag onto the oligonucleotides from the free 3’-end in solution. A dissociation constant for the binding of T-ag to singlestranded DNA (assuming 1:1 stoichiometry) was determined to be 0.54 M, consistent with the non-specific binding of T-ag to random oligonucleotides. Conversely, T-ag possesses high affinity for the SV40 genome origin sequence and the observed binding between T-ag and

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immobilized oligonucleotides containing part of the origin sequence was very strong and resistant to dissociation [38].

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Figure 8. T-ag helicase assays. Duplex (top sequence) or G4’ quadruplex (bottom sequence) substrates were formed on the sensor chip such that helicase activity would release a non-immobilized DNA strand, leading to a decrease in observed SPR response. Binding of T-ag and unwinding occurred in the 3’-5’ direction as expected and was ATP-dependent.

Figure 9. Sensorgram of duplex DNA unwinding. T-ag unwinding of an immobilized duplex substrate (schematically depicted in Figure 8) led to a decrease in response due to release of the duplex complementary strand. The decrease was exactly compensated by rehybridization to reform the duplex substrate, indicating that the response is a consequence of helicase activity.

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Unwinding of a duplex and G4’ substrate formed from the human telomeric repeat sequence (TTA GGG)4 was observed (Figure 8). Unwinding of duplex led to an immediate drop in the observed response since helicase activity removed the non-immobilized complementary strand from the sensor chip (Figure 9). The observed drop in response was exactly compensated through rehybridization of the complement strand to the immobilized oligonucleotide. Unwinding of the G4’ substrate was initially monitored indirectly using E. coli single-stranded binding protein as a probe of DNA conformation [37]; however, in an effort to simplify the assay the G4’ substrate was constructed such that the G4’ element was contained on an oligonucleotide partially complementary to an immobilized strand [38] (Figure 8). Unwinding of the short duplex stem of the DNA substrate would only occur if Tag first unwound the G4’ region. The observed kinetics of unwinding were too fast to reflect assembly of an active T-ag hexamer past the G4’ region of the substrate that would only unwind the short duplex region. Therefore, a loss of the complementary strand from the sensor chip surface represented unwinding of the G4’ element. Formation of the quadruplex structure on the sensor chip was confirmed by challenging the substrate with a third oligonucleotide strand complementary to the sequence of the quadruplex (Figure 10). If the quadruplex were unfolded, it would be free to bind this complementary single-stranded oligonucleotide to form a duplex region. The percent folded substrate was thus calculated to be >85% in 150 mM K+-containing buffer. In fact, quadruplex structures are in equilibrium with the unfolded structure and the equilibrium rate on sensor surfaces has been calculated [41]. The percent folded G-quadruplex calculated in our study thus represents a lower limit for the amount of folded substrate, indicating a high degree of folding occurs on the chip under the appropriate conditions.

Figure 10. Immobilized G-quadruplex substrate folding. A G4’-quadruplex containing substrate was formed on the sensor chip, as depicted in Figure 8. If the G4’ element was unfolded, it hybridized to an oligonucleotide complementary to the [TTA GGG] 4 sequence. The extent of hybrization determined the amount of unfolded substrate.

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Besides the ability to monitor unwinding of a G4’ quadruplex structure, the SPR-based assay of helicase activity is also amenable to determining inhibition by small molecules. The effect of several different small molecules, including known G-quadruplex binding agents, was determined [37, 38]. Interestingly, no inhibition occurred unless the small molecules were pre-equilibrated with the DNA substrates, indicating that the DNA/small molecule complex is necessary for inhibition and that the small molecules do not directly bind to T-ag, thus inhibiting its activity. This type of information is important for determining what aspects of different inhibitors are important for the observed inhibition. Techniques that monitor only the end product of unwinding are unable to “see” this aspect of inhibition.

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ENZYME-BASED STRATEGIES FOR INCREASING NUCLEIC ACID DETECTION SENSITIVITY As mentioned previously, SPR is widely used for the detection of biomolecular binding interactions but is also becoming an important technique for gaining insight into enzymatic reactions involving nucleic acids. Recently, reports in the literature have also coupled surface enzyme reactions with SPR detection for the purpose of increasing sensitivity for detection of nucleic acids as well as engineering surfaces that are specific for nucleic acids [42]. These types of methods have been most often used with array-type sensor surfaces (Figure 11) such that enzymatic reaction and detection of product can be accomplished in microarray format. For instance, if an enzymatic reaction occurred on the black spots in the example array sensor shown in Figure 11, the product of that reaction could be detected by some designed specific interaction in the clear spots of the array. Experiments discussed below were conducted with an SPR imaging instrument in which reflectivity changes at the surface of the prism coated sensor chip were recorded by a CCD camera. The formation of a RNA microarray was accomplished using the strategy depicted in Figure 11 [42]. A single-stranded oligonucleotide containing a T7 polymerase promoter sequence was first immobilized on the sensor chip (black spots, for instance). Subsequent hybridization to a single-stranded DNA template formed a duplex array element on those spots. Then, exposure of the surface to T7 polymerase in the presence of the correct ribonucleotides allowed synthesis of a RNA transcript that was complementary to the DNA template. After formation of the RNA transcript, T7 polymerase was released into solution and was then free to bind again to the immobilized duplex substrates, allowing amplified production of RNA. The single-stranded RNA produced was then captured on the clear areas of the array (Figure 11) through sequence-specific hybridization to a short complementary oligonucleotide. Since RNA is not very stable, this method of producing the RNA just previous to binding studies is advantageous. If protein, peptide, or small molecule targets were immobilized (on the clear-only spots, for instance) then binding between the RNA generated in the first step could be determined with these immobilized targets immediately after synthesis. A kinetic model for the interaction of RNaseH and Exo III exonuclease at the surface of these array-type sensor surfaces was also developed [42, 43].

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Figure 11. Formation of a RNA microarray sensor chip. The type of array sensor chip is shown schematically on the left with spatially separate areas designated by black and clear boxes. For construction of a RNA microarray chip, RNA was synthesized by T7 polymerase from an immobilized DNA template. The synthesized RNA can then be captured or investigated for different interactions on separate array areas.

Amplified SPR-based detection of target DNA was reported in 2004 [44]. A RNA microarray was constructed in which the RNA strands were immobilized to the sensor surface. Subsequent hybridization of a particular target DNA formed a RNA-DNA heteroduplex, which is a substrate for cleavage by RNaseH. After exposure of the heteroduplex-derivatized sensor surface to RNaseH, the RNA strand was cleaved and both the DNA strand and enzyme were released back into solution. The DNA was then free to bind to remaining immobilized RNA strands, and the process repeated. Amplification of the signal resulting from the target DNA, and not other DNA strands, thus resulted.

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CONCLUSION The advent of SPR-based analysis of nucleic-acid associated enzymatic processes has contributed to a number of advances, not the least of which is the ability to discriminate between different stages of dynamic processes in a real-time, label-free manner. Rapid analysis of nucleic acid elongation, repair rates, topological manipulation, and inhibition of enzymatic activity has been demonstrated. In many cases, traditional methods of analyzing nucleic acid processing reactions do not provide the level of specificity possible with SPR measurements without costly and time-consuming labeling procedures. Perhaps especially promising is the potential to determine binding interactions and enzymatic processes simultaneously by SPR detection in a multiplexed format. Insights into specific mechanisms of transcription termination, DNA repair binding interactions, and enzyme inhibition discussed in this chapter are only the beginning of those sure to follow from increased use of SPR-based methods.

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Willander, M; Al-Hilli, S. Methods Molecular Biol., 2009, 544, 201-229. Navratilova, I; Myszka, DB. Springer Series Chemical Sensors Biosensors, 2006, 4, 155-176. Rich, R; Myszka, DG. J. Mol. Recognit., 2010, 23, 1-64. Laurent, N; Voglmeir, J; Flitsch, SL. Chem. Comm., 2008, 37, 4400, 4412. Krenkova, J; Foret, F. Electrophoresis, 2004, 25, 3550-3563. Homola, J. Chem. Rev., 2008, 108, 462-493. Fan, X; White, IM; Shopova, SI; Zhu, H; Suter, JD; Sun, Y. Anal. Chim. Acta, 2008, 620, 8-26. Buckle, M; Williams, RM; Negroni, M; Buc, H. Proc. Natl. Acad., Sci. U.S.A., 1996, 93, 889-894. Pemberton, IK; Buckle, M. J. Mol. Recognit., 1999, 12, 322-327. Gill, SC; Weitzel, SE; von Hippel, PH. J. Mol. Biol., 1991, 220, 307-324. Grieve, SJ; Weitzel, SE; Goodarzi, JP; Main, LJ; Pasman, Z; von Hippel, PH. Proc. Natl. Acad. Sci., U. S. A. 2008, 105, 3315-3320. Gomak, N; West, S; Proudfoot, NJ. Mol. Cell. Biol., 2006, 26, 3986-3996. Grieve, SJ; von Hippel, PH. Nat. Rev. Mol. Cell. Biol., 2005, 6, 221-232. Yin, H; Artsimovitch, I; Landick, R; Gelles, J. Proc. Natl. Acad. Sci., USA, 1999, 96, 13124-13129. Maesawa, C; Inaba, T; Sato, H; Iijima, S; Ishida, K; Terashima, M; Sato, R; Suzuki, M; Yahima, A; Ogasawara, S; Oikawa, H; Sato, N; Saito, K; Masuda, T. Nucleic Acids Res., 2003, 31, e4. Greider, CW. Annu. Rev. Biochem., 1996, 65, 337-365. Hayflic, L; Moorhead, PS. Exp. Cell Res., 1961, 25, 585-621. Kim, NW; Piatyszek, MA; Prowse, KR; Harley, CB; West, MD; Ho, PL; Coviello, GM; Wright, WE; Weinrich, SL; Shay, JW. Science, 1994, 266, 2011-2015. Panayotou, G; Brown, T; Barlow, T; Pearl, L. H; Savva, R. J. Biol. Chem., 1998, 273, 45-50. Lindahl, T. Nature, 1993, 362, 709-715. I. Hegde, V; Wang, M; Deutsch, WA. DNA Repair, 2004, 3, 121-126. Wang, M; Mahrenholz, A; Lee, SH. Biochem., 2000, 39, 6443-6439. Corne, C; Fiche, JB; Cunin, V; Buhot, A; Fuchs, J; Calemczuk, R; Favier, A; Livache, T; Gasparutto, D. Nucleic Acids Symposium., 2008, Series 52, 249-250. Corne, C; Fiche, JB; Gasparutto, D; Cunin, V; Suraniti, E; Buhot, A; Fuchs, J; Calemczuk, R; Livache, T; Favier, A. Analyst., 2008, 133, 1036-1045. Livache, T; Roget, A; Dejean, E; Barthet, C; Bidan, G; Teoule, R. Nucleic Acids Res., 1994, 22, 2915-2921. Zharkov, DO; Ishchenko, AA; Douglas, KT; Nevinsky, GA. Mutat. Res., 2003, 531, 141-156. Zauner, G; Wang, YT; Lavesa-Curto, M; MacDonald, A; Mayes, AG; Bowater, RP; Butt, JN. Analyst, 2005, 130, 345. Tang, ZW; Wang, KM; Tan, WH; Li, J; Liu, LF; Guo, QP; Meng, XX; Ma, CB; Huang, SS. Nucleic Acids Res., 2003, 31, e148. Luan, Q; Xue, Y; Yao, X; Lu, W. Analyst., 2010, 135, 414-418. Vaisocherová, H; Snášel, J; Špringer, T; Šipová, H; Rosenberg, I; Štěpánek, J; Homola, J. Anal. Bioanal. Chem., 2009, 393, 1165-1172.

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[33] Esposito, D; Craigie, R. EMBO J., 1998, 17, 5832. [34] Dexheimer, TS; Fry, M; Hurley, LH. In Quadruplex Nucleic Acids; Neidle, S. Ed; Royal Society of Chemistry: Cambridge, UK, 2006, 180-207. [35] Eddy, J; Maizels, N. Nucleic Acids Res., 2006, 34, 3887-3896. [36] Wu, Y; Shin-ya, K; Brosh, R. M. Jr. Mol. Cell. Biol., 2008, 28, 4116-4128. [37] Plyler, JR; Jasheway, K; Tuesuwan, B; Karr, J; Brennan, JS; Kerwin, SM; David, WM. Cell Biochem. Biophys., 2009, 53, 43-52. [38] Plyler, JR; Sanjar, F; Howard, R; Araki, N; David, WM. J. Biotech Res., 2010, 2, 5666. [39] Simmons, DT. SV40 large T-antigen functions in DNA replication and transformation. Adv. Virus Res., 2000, 55, 75-134. [40] Tuesuwan, B; Kern, JT; Thomas, P. W; Rodriguez, M; Li, J; David, W. M; Kerwin, S. M. Biochem., 2008, 47, 1896-1909. [41] Halder, K; Chowdhury, S. Biochem., 2007, 46, 14762-14770. [42] Lee, HJ; Wark, AW; Corn, RM. Langmuir, 2006, 22, 5241-5250. [43] Lee, HJ; Wark, A; Goodrich, TT; Fang, S; Corn, RM. Langmuir, 2005, 21, 4050-4057. [44] Goodrich, TT; Lee, HJ; Corn, RM. J. Am. Chem. Soc., 2004, 126, 4086-4087.

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Chapter 6

SURFACE PLASMON ASSISTED MICROSCOPY: REVERSE KRETSCHMANN FLUORESCENCE ANALYSIS OF KINETICS OF HYPERTROPHIC CARDIOMYOPATHY HEART J. Borejdo*, P. Mettikolla, N. Calander, R. Luchowski, I. Gryczynski and Z. Gryczynski Dept of Molecular Biology & Immunology and Center for Commercialization of Fluorescence Technology, the University of North Texas HSC, Fort Worth, Texas

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ABSTRACT It is believed that the alteration of the kinetics of interaction between actin and myosin causes a serious heart disease called Familial Hypertrophic Cardiomyopathy (FHC) by making a heart pump blood inefficiently. To check this hypothesis in ex-vivo heart, we constructed Surface Plasmon Assisted Microscope (SPAM) and used it in a reverse Kretschmann (RK) configuration. In SPAM fluorescence is the result of nearfield coupling of fluorophores excited in the vicinity of the metal coated surface of a coverslip with the surface plasmons propagating in the metal. Surface plasmons decouple on opposite side of the metal film and emit in directional manner as a far-field ppolarized radiation. In RK-SPAM a sample is illuminated directly by the laser beam. During contraction of heart muscle a myosin cross-bridge imparts periodic force impulses to actin. The impulses were analyzed by RK-SPAM by Fluorescence Correlation Spectroscopy (FCS) of fluorescently labeled actin. The rate of changes of orientation were significantly faster in contracting cardiac myofibrils of transgenic (R58Q) mice than of wild type (WT). These results suggest a way to rapidly diagnose this disease.

*

Corresponding author: Department of Molecular Biology, University of North Texas Health Science Center, 3500 Camp Bowie Blvd, Fort Worth, TX 76107, tel:817 735-2106; fax: 817 735 2118; email: [email protected]

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INTRODUCTION Contraction of every muscle, including a heart, is a result of periodic interaction of myosin cross-bridges with actin. Interaction involves binding of myosin to actin, delivery of force impulse and dissociation of myosin from actin. Familial Hypertrophic Cardiomyopathy (FHC) is an autosomal dominant disease characterized by left ventricular and septal hypertrophy and by myofibrillar disarray [1]. Single-point-mutation R58Qin the myosin regulatory light chain (RLC) is associated with a malignant FHC disease phenotype [2]. It has been suggested that FHC mutations cause a heart to pump blood inefficiently by altering kinetics of acto-myosin interaction, [3]. However, a suggestion based on measurements of macroscopic quantity (e.g. muscle force) can never be completely proved because it is impossible to extraction of kinetic information from steady-state ensemble measurements is unreliable. Isometric force is a temporal average of trillions of individual force impulses [4] and averaging over many molecules will always mask individual impulses. In contrast, true kinetic information can be extracted from stochastic fluctuations originating from small ensemble of molecules. Fluorescence Correlation Spectroscopy (FCS) [5-7] is a method of choice in such measurements. The method is based on the ability to observe few molecules of actin in an ex-vivo heart muscle. It is illustrated in Figure 1. Muscle actin is labeled with extremely diluted dye. A small volume within labeled heart (10-16L), containing only a few actin molecules, is observed by a confocal microscope. In rigor muscle (left panel), the thin filaments are stationary. There are no fluctuations in orientation of the transition dipole (red arrow). The situation is quite different during isometric contraction (right panel). Now myosin cross-bridges deliver impulses to actin [4]. The force impulses deform actin filament, changing the orientation of a transition dipole of rhodamine. The orientation of rhodamine dipole fluctuates in time. In addition, fluctuations arise because rhodamine may leave and reenter observational volume, as cross-bridges pull filaments to the left, and filaments recoil during isometric contraction to the right. We measure fluctuations by recording parallel (║) and perpendicular () components of fluorescent light emitted by an actin-bound fluorophore. The ratio of these components, called Polarized Fluorescence, is a sensitive indicator of the orientation of transition dipole of the fluorophore [8-15]. However, we report fluctuations only in one of the orthogonal components of fluorescent light. The fluctuations of the polarization of fluorescence are very noisy (because they are the ratio of two noisy signals). However, Fortunately, it has been shown experimentally [16] and theoretically [17] that the individual components can also serve as indicators of orientation change. FCS is typically used together with confocal detectionIn order to satisfy FCS requirement that only a small ensemble of molecules be examined., FCS is typically used together with confocal detection. However, if a sample is ex-vivo tissue, a significant background is contributed by the reflection and scattering of the excitation light from the surface/interface. Moreover, in confocal detection the axial dimension of the detection volume is of the order of few microns, large compared to the axial extent of the attached biomolecule. Total Internal Reflection Fluorescence (TIRF) [18] decreases the thickness of confocal section and thus allows measurement of single molecule of myosin II and V in vitro [19, 20] [21-24]. But scattering and reflections still make it difficult to use in tissues.

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Figure 1. Origin of fluctuations. Actin is labeled with diluted fluorescent dye (orange red star) and is placed in the center of confocal volume of a confocal microscope. The detection volume (DV) is an ellipsoid of revolution with major diameter (1.2 μm) equal to the diameter of the confocal pinhole (2ωo=50 µm) divided by the magnification of the objective (40x). The waist of the ellipsoid is ωo and its height, zo, is equal to the thickness of a typical myofibril. Taking this thickness as 1 µm, DV=4/3πω o2zo is ~1 µm3. This is approximately equal to the volume of a half-sarcomere (typical length, width and height of 1, 1, and 1 µm, respectively). Therefore the signal detected by the instrument is contributed by the fluorescent molecules in one half-sarcomere. In rigor muscle (left panel), the thin filaments are stationary. There are no fluctuations in orientation of the transition dipole (red arrow). During isometric contraction (right panel) myosin cross-bridges deliver impulses to actin which deform actin filament, changing the orientation of a transition dipole of rhodamine. The orientation of rhodamine dipole fluctuates in time. In addition, fluctuations arise because rhodamine may leave and re-enter observational volume, as cross-bridges pull filaments to the left, and filaments recoil during isometric contraction to the right. We measure fluctuations by recording parallel (║) and perpendicular () components of fluorescent light emitted by an actin-bound fluorophore

The introduction of Surface Plasmon Assisted Microscope (SPAM) allowed furthersignificant decrease of axial dimension. This decrease is a consequence of the fact that the Surface Plasmon Coupled Emission (SPCE) restricts the range of near field interactions due to restricted range of near-field interactions of excited fluorophores with surface plasmon polaritons in thin metal film [25-27]. The coupling distance is further reduced at a close proximity (below 10 nm) to a surface by fluorescence quenching by a metal. As a result the axial dimension of SPCE is ~50 nm. SPAM avoids reflections and scattering because of reflection of fluorescent light by metal film (see below). SPAM produces extremely thin optical sectioning and provides excellent background rejection. SPAM typically uses two types of excitation modes. Sample can be excited directly with the laser beam (reverse Kretschmann configuration (RK)) or through surface plasmons evanescent field (Kretschmann configuration (KR)). KR mode requires very precise incident angle adjustment and polarization of the exciting light, but the evanescent wave excitation further reduces the depth of the observed layer. RK configuration is very simple to implement and the exciting light need not to be angle adjusted or polarized. In this work we used RK configuration. In both configurations, a sample is observed through the metal film. This assures excellent background rejection because scattered excitation light is unable to penetrate the metal layer and enter the objective. An additional benefit is the fact that coupling of fluorophores to surface plasmons strongly depends on fluorophore orientation. The dipoles perpendicular to the surface couple very efficiently, resulting in p-polarized emission on the other side of the

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film. The dipoles parallel to the surface do not couple (s-polarized modes are not supported by the surface plasmons). We show here that coupling to surface plasmons is more sensitive to changes of dipole orientation than the classically used photoselection and fluorescence polarization measurements. In this work we utilized FCS detection to follow kinetics of subtle conformational changes occurring upon FHC point mutation. This is a simple extension of FCS technology that takes advantage of the fact that orientational fluctuations are strongly reflected in the observed SPCE signal. A muscle axis was always vertical (V) and the direction of polarization of the laser was also always vertical (V) in the laboratory frame of reference. In this work we focused on actin since it can be labeled with very low concentration of fluorescent phalloidin. Such labeling preserves the regular structure of a myofibril. Moreover, phalloidin does not alter enzymatic properties of muscle [28, 29] and labels actin stoichiometrically, which allows strict control of the degree of labeling. Finally, attachment of phalloidin to actin is non-covalent but strong. We show here and have shown previously that in such an attachment the fluorescent moiety rotates little with respect to protein, i.e. change of orientation of rhodamine parallels change of orientation of protein [30]. Observing orientation of actin is a valid way of observing interaction with cross-bridges, because actin changes orientation in response to cross-bridge binding [29, 31-33] and those changes parallel changes of orientation of a cross-bridge [34]. In this report we a present application of SPCS to show that actin in transgenic hearts that are affected by R58Q mutation in RLC responds to cross-bridge impulse faster than actin in WT hearts.

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MATERIALS AND METHODS Chemicals, solutions, microscope slides. Rhodamine-labeled microspheres (100 nm diameter) were from Molecular Probes (MP, Eugene, OR). Spheres were supplied at 3.6 x 1013 /mL and used at 100x dilution. Rhodamine-phalloidin (RP) and Alexa647-phalloidin were was also from MP. All other chemicals including 1-ethyl-3-(3’-dimethylaminopropyl) carbodiimide (EDC), dithiotreitol (DTT), were from Sigma. EDTA-rigor solution, Ca-rigor solution, Mg-rigor solution and contracting solution were as in [35]. Microscope slides were glass or sapphire. They were covered with metal by vapor deposition by EMF Corp. (Ithaca, NY). A 48-nm layer of gold was deposited on the slides. A 2-nm chromium undercoat was used as an adhesive background. NA=1.45 objective was used with gold coated glass slides, and NA=1.65 objective with gold coated sapphire slides. Preparation of myofibrils, labeling and cross-linking. Mouse hearts (a gift from Dr D. Szczesna-Cordary from Miami Univ of Miami) were from 6-7 month old transgenic (Tg) R58Q and Tg-WT mice. They were quickly removed and glycerinated after euthanasia as previously described [36, 37]. Myofibrils from right ventricles were prepared from glycerinated fiber bundles stored at -20°C in glycerinating solution as described in Mettikolla et al. [36]. After application to a coated coverslip, myofibrils were washed with 5 volumes of the Ca2+-rigor solution [36]. Myofibrils were freshly prepared for each experiment. Labeled myofibrils (25 μl) were applied to a gold-coated coverslip. The sample was left on a coverslip for 3 minutes to allow the myofibrils to adhere. The gold-coated cover slip was covered with

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a 10x10 mm coverslip to prevent drying. For labeling, 1 mg/mL myofibrils (~ 4 μM actin) were mixed with 10 nM rhodamine-phalloidin+10 μM unlabeled-phalloidin. Unlabeled phalloidin was necessary to prevent uneven labeling. In its absence, the sarcomeres closest to the tip of the pipette used to add the label, would have contained more chromophores than sarcomeres further away from the tip. The degree of labeling was 10 μM/10 nM = 1,000, i.e. on the average 1 actin protomer in 103 was fluorescently labeled. Labeled myofibrils were washed with 5 volumes of the Ca2+-rigor solution by applying the solution to one end of the channel and absorbing with #1 filter paper at the other end. To prevent shortening of muscle in contracting solution myofibrils (1 mg/mL) were incubated with 20 mM EDC for 10 min at room temperature according to procedure of Herrmann [38]. The reaction was stopped by 20 mM DTT. The lack of shortening was checked in a DIC microscope by comparing the length of a myofibril before and 100 sec after inducing contraction. Within the limits of measuring accuracy on the computer screen (~1%), the length always remained unchanged.

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Preparation of fluospheres. Orange fluorescent (540/560 nm) carboxylate modified microspheres 0.1μm in diameter (cat # F-8800, lot 91B2-3) were from Invitrogen (Carlsbad, Ca). Spheres (2% solids) were diluted 100 times with water to 3.6 x 1011 spheres /mL. Data collection and analysis. ISS Alba FCS (ISS, Urbana, IL) confocal system coupled to Olympus IX 71 microscope was used to collect the data. A sample was illuminated with water immersion objective (Zeiss Planapochromat NA=1.2, x40) and collected with either high aperture objective (Olympus Plan Apo NA=1.45, 60x) through oil with refractive index 1.512, or with very high aperture objective (Olympus Apo, NA=1.65, 100x) through high refractive index (1.78) liquid (Cargill) (Figure 1D). The excitation was by a 532 nm CW laser. Confocal pinhole was 50 μm. Fluorescence was collected every 10 μs for 20 s, and signal was smoothed by binning 1000 points together (final frequency response = 100 Hz). Orthogonally linearly polarized analyzers were placed before Avalanche PhotoDiodes (APD’s). The laser was polarized vertically (on the microscope stage). Myofibril was also vertical. Correlation functions were computed and fitted using ISS Vista ver. 4.0.29 software. Rotation of rhodamine-phalloidin on f-actin. We checked whether rhodamine moiety of fluorescent label was immobile on the surface of actin. It is important that it be so, because otherwise measurements would reflect motion of the fluorophore, not of protein. This was done by measuring fluorescence anisotropy by the time-domain technique using FluoTime 200 fluorometer (PicoQuant, Inc.). Actin (1 μM) was labeled by 0.2 μM Alexa488 phalloidin. The excitation was by a 475-nm laser pulsed diode, and the observation was through a monochromator at 590 nm with a supporting 590-nm long wave pass filter. FWHM of pulse response function was 68 ps (measured by PicoQuant, Inc.). Time resolution was better than 10 ps. The intensity decays were analyzed in terms of a multi-exponential model using FluoFit software (PicoQuant, Inc.). 100% of the intensity of Alexa-phalloidin alone (in the absence of actin) decayed with the time constant of 0.519 ns, consistent with rotation of a molecule with Mw=1,250. No independent rotation of rhodamine moiety was observed. The decay of anisotropy of Alexa-phalloidin coupled to F-actin was best fitted by the two exponents with correlation times of 0.665 and 36.8 ns with the relative contributions of 13.7 and 86.3%, respectively. The short correlation time is due to the rotation of rhodaminephalloidin and the long one to the rotation of F-actin oligomers. We conclude that over 86%

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of fluorescent phalloidin was immobilized by F-actin, i.e. that fluorescent phalloidin is a suitable dye for monitoring rotational motion of acrtin. The anisotropy probably reflects change in the number of the bending modes of actin filament rather than the reorientation of the actin monomer, because the work of Yangida and collaborators implied the lack of any gross rotational motion of a monomer [31, 32].

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OVERVIEW Consider a conventional case where muscle is examined in an upright microscope (Figure 2A). The red fluorescence of the labeled molecules is excited by light incident on a sample from above (green) through an objective (acting as a condenser). Fluorescence is collected by high numerical aperture (NA) objective through the immersion oil (yellow). The light reflected from the surface of a coverslip or scattered by a sample (marked scattered excitation light) excites some fluorophores indirectly. Unfortunately, this light (far field fluorescence) is collected by an objective together with the main fluorescent light. The effect is best illustrated by creating artificial background and collecting fluorescence from a sample plus background. Skeletal muscle myofibrils were labeled with 100 nM Alexa647-phalloidin (+10 µM unlabeled phalloidin acting as competition for actin). in the presence of Tthe background was in the form of 0.5 mM of Rhodamine 800. Since the concentration of the background dye was 5,000 times higher than a sample, the fluorescence was completely dominated by the background and no image of myofibrils was discernible (Figure 2B). For comparison, we also show the Back Focal Plane (BFP) image (Figure 2C). At the BFP, the microscope objective maps rays emanating from in focus source into off-axis radial positions. For spherical refracting surface, a ray originating from a source on-axis at the focal plane and propagating at an angle θ with respect to the axis will cross the objective’s BFP at an off-axis radial distance r = nf sinθ where n is the refractive index of the medium (1.515) and f is the focal length of the objective [39]. Therefore strong central spot in Figure 2C arises from the illuminating laser, because a significant amount of exciting light is able to pass to the detector (the detector is looking directly at the laser; all the light impinging on a sample at 0o angle is passing through the center at BFP). The weak peripheral ring is due to weak coupling to a high refractive index glass even in the absence of metal [40]. The diffuse interior is the image. In the RKproposed configuration, the situation is quite different. Here a sample rests on a coverslip coated with a thin layer of noble metal (Figure 2D). The incident light coming from above through an objective (acting as condenser) excites the fluorophores in the whole sample volume. About 95% of beam is reflected by the metal surface and never gets to the detection system. Fluorophore that are excited in the close proximity to the surface (below 50 nm) couple via near-field interactions to induce surface plasmons. Surface plasmons decouple on the other side of metal film as a directional emission. Other fluorophores (further than 100 nm from the metal surface) are able to emit light as a far-field radiation, but the metallic surface does not transmit it to the detector. Fluorophores closer than 10 nm to the surface are quenched by the metal. Thus, even though the exciting light does not produce evanescent wave, the RK-SPAM produces the effect similar to TIRF because only the fluorescence from molecules within 10 to 50 nm of the metal layer (indicated by a dashed line) can penetrate the metal layer via plasmon resonance. The scattered light emitted as far field fluorescence is

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reflected by the metal and is not collected by the objective. The image of the sample observed by SPAM in RK configurations was no longer dominated by the background (Figure 2E). Figure 2F shows the BFP image of the rhodamine fluorophore on gold coated glass. Most of the light is now emitted at well defined angle, so the BFP image is now doughnut-shaped. Figure 2B & E demonstrates that RK-SPAM provides excellent background rejection because the scattered light is unable to penetrate the metal [41]. A significant advantage of the method is that the excitation volume is comparable to TIRF, without the need to adjust incidence angle.

Figure 2. A concept of SPAM microscope. A cardiac myofibril is illuminated from above. In conventional microscope (A) all light, including scattered (background) light, is able to penetrate the coverslip. Green dots represent fluorophores that are out of the field of excitation. Red dots represent fluorophores that are in the path of direct or scattered excitation light. In RK/SPAM (D) sample is placed on a metal coated coverslip and excited with green light (right). The excitation energy from the excited fluorphore couples to the surface plasmons and radiates through the metal film (red) to the objective as a surface of a cone with half angle equal to the SPCE angle. Metal can be a thin layer of Al (20 nm thick) or Ag or Au (50 nm thick). The scattered light is unable to penetrate the coverslip and is radiated into free space. B & E: The background rejection by SPAM. 0.5 mM Rhodamine 800 added as background obscures the image in ordinary TIRF (B). SPAM in RK configuration eliminates much of the background contribution (E). Myofibrils (0.1 mg/mL) were labeled with 100 nM Alexa647phalloidin + 10 μM unlabeled phalloidin for 5 min at room temperature, then extensively washed with rigor buffer containing 50 mM KCl, 2 mM MgCl2, 1 mM DTT, 10 mM TRIS pH 7.0. 633 nm excitation, 1.65 NA x100 Olympus objective, sapphire substrate, 1.78 Refractive Index immersion oil. The bars are 5 µm in B and 10 μm in E. C & F: The Back Focal Plane image of a sample in a microscope in conventional configuration (C) consists of weak outer ring and a diffuse interior with s strong center corresponding to imperfect blockage of exciting light. The BFP image of a sample in a microscope in RK configuration (F) consists of a strong ring corresponding to emission into free space in a cone.

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Figure 3. A: Power entering the objective at various distances from the surface of a coverslip. B: the SPCE (solid line) and TIRF (broken line) power entering objective at various polar angles. Curves normalized to power at θ=0o

Characteristic feature of SPAM is that fluorescence coupling to surface plasmons dramatically depends on the orientation of the transition moment of the dye.. This is illustrated in Figure 3A which compares the power entering the microscope objective for SPCE and TIRF mode of detection. For different polar angles of the transition dipole, the power is calculated for different distances from the coverslip. More fluorescence is contributed by SPCE than TIRF at distances larger than 10 nm. The method is also more

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sensitive than TIRF to changes in polar angle. Figure 3B compares the sensitivity of SPCE vs. TIRF to change in angle. This makes the SPCE method well suited to the measurements of orientation changes. Therefore in the experiments reported here, the polarized intensity fluctuations arise from changes in the degree of coupling as the dipole changes orientation, in addition to differential absorption of light at different angles like in classical polarization of fluorescence measurements[42].

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RESULTS Polystyrene spheres. In order to show that the method is superior to conventional confocal technique of measuring motions of protein-sized particles, we compared Brownian motion of 100 nm polystyrene spheres. Figure 4 compares typical RK-SPAM FCS experiments of 0.1 μm fluoro-spheres diffusing on glass, gold-coated-glass and gold-coatedsapphire coverslips. The average counts were IH= 1746±694 and IV= 2189±785 photons/10ms, respectively. The total photon rate was ITot=IV +2*G*IH = 5681 photons/10ms ≈ 600 Kcounts/s, where G is the correction factor (0.98). Figure 4A compares correlation function of spheres diffusing on glass (red), gold-coated-glass (green) and gold-coatedsapphire (blue). For spheres diffusing on glass, the magnitude of the correlation function at zero delay time is equal to the inverse of the number of particles in the detection volume and gives ~ 22 spheres in the DV. Since the confocal volume is ~3 fL the concentration of spheres is 7 x 1012/mL, close to the number estimated by Molecular Probes (100x dilution of the stock concentration of 3.6 x 1013 /mL). Figure 4A shows also that the magnitude of correlation function at zero delay time of spheres diffusing on gold-coated glass was nearly 4 times larger than for spheres diffusing on glass. This is consistent with the fact that gold coating decreases DV by quenching fluorescence near the metal surface [43]. Figure 4B shows that the decay of the correlation function is the fastest for spheres diffusing on gold-coated-sapphire and the slowest for glass. This is due to the fact that experiments on sapphire were carried out using high aperture objective (NA=1.65) which has the thinnest depth-of-focus, and the fact that gold coating decreases DV. Since the correlation function was noisy for gold-coated-sapphire, we used gold-coated-glass for the rest of the experiments. The decay of correlation occurs between 10-3 and 10-2 s, consistent with earlier results obtained for a 100 nm diameter sphere with a diffusion coefficient of 4.12x10-12 m2/s [44] diffusing along Z-axis through a distance of 36 nm [41]. Contracting heart muscle. Figure 5A shows a typical RK-SPAM signal from contracting R58Q mutated muscle. Figure 5B is the control intensity trace of the same muscle in rigor, when no mechanical activity occurs. The average counts on gold of contracting heart was IV= 28±8 and IH=72 ±17 photons/10ms for the total photon rate ITot = 174 photons/10ms. In all the RK experiments, the myofibril was labeled such that only one in 1,000 actin monomers carried fluorescent phalloidin (muscle was irrigated with 10 nM RP + 10 µM UP). The volume of a typical half-sarcomere (HS) is ~ 5 x 10-16 L (the length, width and height of a typical HS are 1, 1, and 0.5 µm, respectively). Since the concentration of actin in muscle is 0.6 mM [45], this volume contains ~200,000 actin monomers and ~ 200 fluorophores . The objective captures the fluorescence from the entire HS, because the diameter of the detection volume is equal to the diameter of the confocal pinhole (50 µm) divided by the magnification

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of the objective (60x). Since the signal from a freely diffusing chromophore is 200 photons/10 ms , the fact that the observed signal was very much smaller suggests that photobleaching is a significant factor in our experiments. This is not surprising, because in free diffusion, fluorophores spend only μseconds in the DV. In our experiments , in contrast, muscle is immobilized on a coverslip and each fluorophore is exposed to light for a few minutes required to focus and measure.

Figure 4. RK-SPAM FCS experiments using fluorescent spheres diffusing on glass, gold-coated-glass and gold-coated-sapphire coverslips. A - correlation function of spheres diffusing on glass (red), goldcoated-glass (green) and gold-coated-sapphire (blue). The average number of spheres in DV was 6, 8 and 22, respectively. B - the decay of the correlation function is the fastest for spheres diffusing on gold-coated-sapphire and the slowest for glass. The total photon rate was ≈ 600 Kcounts/s. The incident power 150 μW.

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Figure 5.The time traces from the left ventricle on gold-coated glass labeled with 10 nM rhodamine phalloidin + 10 μM unlabeled phalloidin .Contracting (A) and rigor (B) muscle. Parallel (I H) and perpendicular (IV) intensities shown in the blue and red, respectively. The incident laser power 150 μW. Insets show histograms of counts during contraction (A) and rigor (B). Note that the Gaussian curve used to fit the contraction data is asymmetrical.

Insets to Figure 5A and B show histograms of photocounts during contraction and rigor. In all experiments the Gaussian curve used to fit the data was highly asymmetrical during contraction and symmetrical during rigor. In the example shown in inset to Figure 5, the fit to the function f=aexp(-.5((x-xo)/b)2) gave R2=0.985 for ch2 during rigor and R2=0.922 for ch2 during contraction. We carried out the Normality test on the data. The normality test, which checks whether the data passed or failed the test of the assumption that the source population is normally distributed around the regression f=aexp(-.5((x-xo)/b)2). It also computes the P value. The contraction data failed normality test (Test Failed with P = 0.0007). Rigor data, in contrast, passed the test with P = 0.3755. Failure of the normality test can indicate the

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presence of outlying influential points or an incorrect regression model. This finding is consistent with the original suggestion of Huxley & Simmons [46] that muscle contraction maybe a multi-step process. The analytical expression for rotational autocorrelation function is very complex. The correlation functions computed from the signal shown in Figure 5 is plotted in Figure 6. Even in the simplest case, where the transition dipole rotates on glass (i.e. not on the metal), the shape of correlation function depends on the angle between the excitation and emission transition dipole moments, the polarization of the excitation, and the presence of emission polarizers. In the simplest case, where the molecule does not rotate during the excited state lifetime, the excitation and emission transition dipole moments are parallel, muscle is excited

Figure 6. Correlation function of Tg-R58Q myofibrils. A – contracting myofibrils, B – rigor myofibrils.  -autocorrelation function of ch1 (IH),  - ch2 (IV),  - cross-correlation ch1 x ch2

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with polarized light and the emission is observed without polarizers, the complex expression for the rotational correlation function assumes exponential form G(τ)=Bexp(-τ/τr) where B is a constant and τr is the rotational correlation time [47]. Since the excited state lifetime of rhodamine is 4.08 ns [48] and actin filament does not rotate to any significant extent during such a short time and the excitation and emission transition dipole moments of rhodamine are almost parallel above 470 nm, the decay of correlation function is well described by a simple exponential. Pre-exponential constant B is taken as a measure reflecting kinetics of contracting heart muscle. While this scheme is an oversimplification of the actual events, mostly because rotation occurs on metal, (see Discussion), constant B reflects a clear difference between WT and mutated muscle. The experiment was repeated 4 times for TgWT and 5 times for Tg-mutated hearts. Figure 7 summarizes the results and shows that there is statistically significant (t=-3.16, P=0.034, paired-test) difference between WT and mutated myofibrils. Correlation function of the rigor control shows no decay at all (Figure 6B).

Figure 7. Pre-exponential constant B from a fit of WT (left) and mutated (right) myofibrils. The error bar is SD. N=5 for R58Q, 4 for WT

DISCUSSION RK-SPAM is well suited for the measurements of rotational motion from live tissue. Firstly, it has an excellent background rejection, made possible by the fact that scattered excitation light is unable to penetrate the metal layer and enter the objective (Figure 2). In contrast, the scattered light has no difficulty penetrating the coverslip and entering the objective in the conventional detection. Secondly, the coupling of the fluorescence is strongly distance dependent and extends only to about fifty nanometers into a sample, smaller than 100-200 nm characteristic of TIRF (Figure 3). It is further reduced at close proximity (below 10 nm) to a surface by quenching by a metal. Thus using SPAM, detection of signal is confined to a window of 10 nm above the surface to 50 nm above the surface of the metal plate. This property of only detecting light that originates from this 40 nm wide window is a

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main factor in reducing the background signal. In effect, RK-SPAM is equivalent to TIRF excitation without the need to produce evanescent wave. Finally, fluorescence coupling to surface plasmons dramatically depends on the orientation of the molecule transition moment, i.e. the method is particularly suited to measurements of macromolecular orientation changes. This feature has been exploited here. The above characteristic of RK-SPAM convinced us that RK-SPAM can be conveniently applied to the study a Familial Hypertrophic Cardiomyopathy. This serious heart disease is associated with single point mutations which occur in sarcomeric proteins. In particular we studied R58Q mutation in a genes that encodes for the ventricular myosin regulatory light chain (RLC) [2, 49-54]. The contraction of heart, like the contraction of every muscle, results from periodic interaction of myosin cross-bridges with actin. During this interaction myosin delivers periodic force impulses to the thin filament. Integration of these impulses is a gross contractile force delivered to actin by myosin. It is commonly believed that alteration of kinetics of this interaction, which is causing a heart to pump blood inefficiently [3] is a cause of FHC. We compared RK-SPCE spectra of myofibrils prepared from Tg-WT and Tg-R58Q hearts. In spite of the fact that signals were noisy, because they were contributed by a small fraction of ~200,000 actins present in a half-sarcomere, we managed to compare correlation functions of healthy (WT) and diseased (R58Q) hearts. The correlation function of R58Q muscles was approximated by a single exponential function. We recognize that fitting our data to simple exponential decay is a gross oversimplification of the actual events. The orientation change of myosin cross-bridges, and therefore of actin, does not occur in a single reversible step, but as suggested by asymmetry of histogram shown in inset to Figure 5A, but probably occurs in a series of smaller steps. Nevertheless, the kinetic parameter extracted from the correlation spectra (the amplitude of correlation function) showed statistically significant difference between Tg-WT and Tg-mutated heart muscle. The fact that the amplitude is increased in mutated muscle is consistent with the fact that ATPase rate and the maximal velocity of actin translation in the in-vitro motility assay was larger for R58Q myosin in comparison with WT myosin [55]. At the same time, maximal force developed by mutated muscle was not increased, suggesting a decrease in the efficiency of ATP utilization. Thus the present work is consistent with the idea that FHC results from decreased efficiency.

ACKNOWLEDGMENTS Supported by NIH grants R01AR048622 and R01 HL090786 to JB and by Texas ETF grant (CCFT).

ABBREVIATIONS FHC FCS RLC RP DV

Familial Hypertrophic Cardiomyopathy Fluorescence Correlation Spectroscopy Regulatory Light Chain Rhodamine-Phalloidin Detection Volume

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Tin, L. L., Beevers, D. G. & Lip, G. Y. (2002). Hypertension, left ventricular hypertrophy, and sudden death. Curr Cardiol Rep., 4(6), 449-57. Richard, P. et al. (2003). Hypertrophic Cardiomyopathy: Distribution of Disease Genes, Spectrum of Mutations, and Implications for a Molecular Diagnosis Strategy. Circulation, 107, 2227-2232. Kerrick, W. G. et al. (2009). Malignant familial hypertrophic cardiomyopathy D166V mutation in the ventricular myosin regulatory light chain causes profound effects in skinned and intact papillary muscle fibers from transgenic mice. Faseb J., 23(3), 85565. Epub 2008 Nov 5. Oplatka, A. (1972). On the mechanochemistry of muscular contraction. J Theor Biol, 34(2), 379-403. Elson, E. L. & Magde, D. (1974). Fluorescence Correlation Spectroscopy: Coceptual Basis and Theory. Biopolymers, 13, 1-28. Elson, E. L. (1985). Fluorescence correlataion spectroscopy and photobleaching recovery. Annu. Rev. Phys. Chem., 36, 379-406. Elson, E. L. (2004). Quick tour of fluorescence correlation spectroscopy from its inception. J Biomed Opt., 9(5), 857-64. Dos Remedios, C. G., Millikan, R. G. & Morales, M. F. (1972). Polarization of tryptophan fluorescence from single striated muscle fibers. A molecular probe of contractile state. J. Gen. Physiol., 59, 103-120. Dos Remedios, C. G., Yount, R. G. & Morales, M. F. (1972). Individual states in the cycle of muscle contraction. Proc Natl Acad Sci U S A, 69, 2542-2546. Nihei, T., Mendelson, R. A. & Botts, J. (1974). Use of fluorescence polarization to observe changes in attitude of S1 moieties in muscle fibers. Biophys. J. , 14, 236-242. Tregear, R. T. & Mendelson, R. A. (1975). Polarization from a helix of fluorophores and its relation to that obtained from muscle. Biophys. J., 15, 455-467. Morales, M. F. (1984). Calculation of the polarized fluorescence from a labeled muscle fiber. Proc Nat Acad Sci USA, 81, 145-9. Sabido-David, C. et al. (1998). Orientation changes of fluorescent probes at five sites on the myosin regulatory light chain during contraction of single skeletal muscle fibres. J Mol Biol, 279(2), 387-402. Hopkins, S. C. et al. (1998). Fluorescence polarization transients from rhodamine isomers on the myosin regulatory light chain in skeletal muscle fibers. Biophys J, 74(6), 3093-110. Hopkins, S. C. et al. (2002). Orientation changes of the myosin light chain domain during filament sliding in active and rigor muscle. J Mol Biol., 318(5), 1275-91.

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[16] Muthu, P. et al. (2008). Cross-bridge duty cycle in isometric contraction of skeletal myofibrils. Biochemistry, 47, 5657-5667. [17] Barcellona, M. L. et al. (2004). Polarized fluorescence correlation spectroscopy of DNA-DAPI complexes. Microsc Res Tech, 65(4-5), 205-17. [18] Axelrod, D. (2001). Total internal reflection fluorescence microscopy in cell biology. Traffic, 2, 764-74. [19] Warshaw, D. M. et al. (1998). Myosin conformational states determined by single fluorophore polarization. Proc Natl Acad Sci, U S A, 95(14), 8034-8039. [20] Quinlan, M. E., Forkey, J. N. & Goldman, Y. E. (2005). Orientation of the myosin light chain region by single molecule total internal reflection fluorescence polarization microscopy. Biophys J., 89(2), 1132-42. Epub May 13. [21] Enderlein, J. & Ambrose, W. P. (1997). Optical collection efficiency function in singlemolecule detection experiments. Appl Opt., 36(22), 5298-302. [22] Willets, K. A. et al. (2003). Novel fluorophores for single-molecule imaging. J Am Chem Soc., 125(5), 1174-5. [23] Wang, Y. et al. (2007). Single-molecule structural dynamics of EF-G--ribosome interaction during translocation. Biochemistry., 46(38), 10767-75. Epub Aug 30. [24] Taniguchi, Y. et al. (2007). Single molecule thermodynamics in biological motors. Biosystems., 88(3), 283-92. Epub 2006 Nov 10. [25] Gryczynski, I. et al. (2004). Surface Plasmon-Coupled Emission with Gold Films. J. Phys. Chem., 108, 12568-12574. [26] Gryczynski, I. et al. (2005). Two-photon induced surface plasmon-coupled emission. Thin Solid Films, 491, 173-176. [27] Borejdo, J., et al. (2006). Application of Surface Plasmon Coupled Emission to Study of Muscle. Biophys. J., 91, 2626-2635. [28] Bukatina, A. E., Fuchs, F. & Watkins, S. C. (1996). A study on the mechanism of phalloidin-induced tension changes in skinned rabbit psoas muscle fibres. J Muscle Res Cell Motil, 17(3), 365-71. [29] Prochniewicz-Nakayama, E., Yanagida, T. & Oosawa, F. (1983). Studies on conformation of F-actin in muscle fibers in the relaxed state, rigor, and during contraction using fluorescent phalloidin. J. Cell Biol., 97, 1663-1667. [30] Mettikolla, P., et al. (2010). Kinetics of a Single Cross-Bridge in a Familial Hypertrophic Cardiomyopathy Heart Muscle Measured by Reverse Kretschmann Fluorescence. J. Biomed. Optics, 15(1):017011in press. [31] Yanagida, T. & Oosawa, F. (1980). Conformational changes of F-actin-epsilon-ADP in thin filaments in myosin-free muscle fibers induced by Ca2+. J Mol Biol., 140(2), 31320. [32] Yanagida, T. & Oosawa, F. (1978). Polarized fluorescence from epsilon-ADP incorporated into F-actin in a myosin-free single fiber: conformation of F-actin and changes induced in it by heavy meromyosin. J Mol Biol., 126(3), 507-24. [33] Borovikov Yu, S., Kuleva, N. V. & Khoroshev, M. I. (1991). Polarization microfluorimetry study of interaction between myosin head and F-actin in muscle fibers. Gen Physiol Biophys., 10(5), 441-59. [34] Borejdo, J., et al. (2004). Changes in orientation of actin during contraction of muscle. Biophys. J., 86, 2308-2317.

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[35] Muthu, P., et al. (2009). Single Molecule Kinetics in the Familial Hypertrophic Cardiomyopathy D166V Mutant Mouse Heart. JMCC, in press. [36] Mettikolla, P., et al. (2009). Fluorescence Lifetime of Actin in the FHC Transgenic Heart. Biochemistry, 48(6), 1264-1271. [37] Szczesna-Cordary, D., et al. (2005). The E22K mutation in myosin RLC that causes familial hypertrophic cardiomyopathy increases calcium sensitivity of force and ATPase in transgenic mice. J. Cell Science, 118. [38] Herrmann, C., et al. (1993). A structural and kinetic study on myofibrils prevented from shortening by chemical cross-linking. Biochemistry, 32(28), 7255-63. [39] Mattheyses, A. L. & Axelrod, D. (2005). Fluorescence emission patterns near glass and metal-coated surfaces investigated with back focal plane imaging. Journal of Biomedical Optics, 10(5), 054007. [40] Enderlein, J. & Ruckstuhl, T. (2005). The efficiency of surface-plasmon coupled emission for sensitive fluorescence detection. Optics Express, 13(22), 8855-8865. [41] Borejdo, J., et al. (2006). Fluorescence correlation spectroscopy in surface plasmon coupled emission microscope. Optics Express, 14(17), 7878-7888. [42] Pesce, A. J., Rosen, C. G. & Pasby, T. L. (1971). In: "Fluorescence Spectroscopy". Marcel Dekker, New York. [43] Calander, N., et al. (2008). Fluorescence correlation spectroscopy in a reverse Kretchmann surface plasmon assisted microscope. Opt Express., 16(17), 13381-90. [44] Tanford, C. (1963). Physical Chemistry of Macromolecules., New York: John Wiley & Sons. [45] Bagshaw, C. R. (1982). "Muscle Contraction". Chapman & Hall, London. [46] Huxley, A. F. & Simmons, R. M. (1971). Proposed mechanism of force generation in striated muscle. Nature, 233, 533-538. [47] Ehrenberg, M. & Rigler, R. (1976). Fluorescence correlation spectroscopy applied to rotational diffusion of macromolecules. Q Rev Biophys, 9(1), 69-81. [48] ISS, Lifetime Data of Selected Fluorophores. http://www.iss.com/resources/ fluorophores.html, 2008. [49] Poetter, K,. et al. (1996). Mutations in either the essential or regulatory light chains of myosin are associated with a rare myopathy in human heart and skeletal muscle. Nat Genet, 13(1), 63-9. [50] Flavigny, J., et al. (1998). Identification of two novel mutations in the ventricular regulatory myosin light chain gene (MYL2) associated with familial and classical forms of hypertrophic cardiomyopathy. J Mol Med, 76(3-4), 208-14. [51] Andersen, P. S., et al. (2001). Myosin light chain mutations in familial hypertrophic cardiomyopathy: phenotypic presentation and frequency in Danish and South African populations. J Med Genet., 38(12), E43. [52] Kabaeva, Z. T., et al. (2002). Systematic analysis of the regulatory and essential myosin light chain genes: genetic variants and mutations in hypertrophic cardiomyopathy. Eur J Hum Genet., 10(11), 741-8. [53] Morner, S., et al. (2003). Identification of the genotypes causing hypertrophic cardiomyopathy in northern Sweden. J Mol Cell Cardiol., 35(7), 841-9. [54] Hougs, L., et al. (2005). One third of Danish hypertrophic cardiomyopathy patients have mutations in MYH7 rod region. Eur J Hum Genet., 13, 161-165.

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[55] Greenberg, M. J., et al. (2008). Regulatory light chain mutations associated with cardiomyopathy affect myosin mechanics and kinetics. J Mol Cell Cardiol., 2009. 46(1), 108-15. Epub Sep 27.

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Chapter 7

TWO-DIMENSIONAL PLASMON POLARITON NANOOPTICS BY IMAGING IN FAR-FIELD Andrey L. Stepanov1,* and Joachim R. Krenn2 1

Laser Zentrum Hannover, 30419 Hannover, Germany; Kazan Physical-Technical Institute, Russian Academy of Sciences, 420029 Kazan, Russian Federation, Kazan Federal University, 420008 Kazan, Russian Federation 2 Institute of Physics, Karl-Franzens University, 8010 Graz, Austria

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ABSTRACT A review of the experimental realization of key high efficiency two-dimensional optical elements, built up from metal nanostructures, such as nanoparticles and nanowires to manipulate plasmon polaritons propagating on metal surfaces is reported. Beamsplitters, Bragg mirrors and interferometers designed and produced by elelectronbeam lithography are investigated. The plasmon field profiles are imaged in the optical far-field by leakage radiation microscopy or by detecting the fluorescence of an organic film deposited on the metal structures. It is demonstrated that these optical far-field methods are effectively suited for direct observation and quantitative analysis of plasmon polariton wave propagation and interaction with nanostructures on thin metal films. Several examples of two-dimensional nanooptical devices fabricated and studied in recent years are presented.

INTRODUCTION The ongoing miniaturization in today’s innovative technology has triggered the emergence of nanotechnology. Within this field intense research effort in, e.g., material science, chemistry, electronics or microscopy is conducted, often combined to interdisciplinary research. When it comes to optics, however, we find the nanoscale to be out * Corresponding

author: E-mail: [email protected]

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of reach for conventional optics: diffraction limits the spatial resolution to a value given by about half the light wavelength. To overcome this limitation is of great interest for both basic research and technological applications. The investigation of the wealth of optical phenomena by the broad range of readily available radiation sources and detection devices would clearly profit from an extension of the spatial resolution beyond the diffraction limit. Furthermore the continuous miniaturization of existing electronic devices is expected to meet fundamental limitations in the near future. In particular, the metallic interconnects used to link individual electronic elements within data-processing units as well as interfacing these units with their surrounding electronic infrastructure will not be able to deal with much higher computation frequencies than are used in existing devices. One potential solution to this problem is to substitute conventional electronic devices with integrated optical elements working in nanoscale dimensions. Nowadays, the drive towards highly integrated optical devices and circuits for use in high-speed communication technologies and in future all-optical photonic chips has generated considerable interest in the field of optical nanotechnology [1,2]. Since the beginning of the 1990's new routes towards the control of light wave propagation at the micro- and nano-scale are explored. Relying on dielectric structures, photonic band gap materials [3, 4] and high dielectric contrast materials have been explored. For the latter a strong light field confinement permits to reduce the cross section of waveguides and to decrease radiative losses in bends. Recent results have demonstrated that visible light from an evanescent local source can be efficiently propagated through a TiO2 waveguide only 200 nm wide [5 - 7]. Besides dielectric materials, metals have been investigated for their potential in downsizing optics beyond the diffraction limit. Recently surface plasmon polaritons (SPPs) excited in metal nanostructures were identified as promising candidates to serve that need. SPPs are resonant electromagnetic surface modes constituted by a light field coupled to a collective oscillation of conduction electrons at the interface of a metal and a dielectric. While SPPs propagate along the metal-dielectric interface, their perpendicular field components decay exponentially into both neighboring media [8]. The corresponding electromagnetic fields are thus strongly localized at the interface. The value of the according wave vectors of optical SPPs is larger than the light wave vector at a given frequency, however, SPP waves can propagate on a two-dimensional surface with almost the speed of light. Increased interest in optical SPPs comes from recent advances that allow metals to be structured and characterized on the nanometer scale. If metal-dielectric interfaces or surface elements are formed on the nanoscale, it is the spatial dimension of the nanostructures rather than the light wavelength that determines the spatial extension of the SPP field, thus rendering feasible optics beyond the diffraction limit. Therefore, SPPs could allow the realization of novel photonic devices that meet the need of manipulating or guiding electromagnetic fields in nanoscale dimensions. The principal feasibility of nano-optics based on metal nanostructures is well documented by recent experiments [9-11]. The existing nanooptical technology and know-how seem complete enough to allow the realization of functional SPP based nanoscale devices. As a propagating wave, a SPP in an extented metal structure can be used for the direction of light fields, corresponding to signal transfer or optical addressing. Indeed, recent results on SPP propagation in m-wide metal stripes and nanowires [12, 13] demonstrate the feasibility of SPP based subwavelength light field transport. Moreover, when a thin metal film acts as a waveguide for SPPs the same structure can be used to carry electrical signals.

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Propagating SPPs are characterized by a dispersion relation defining the plasmon wavelength to be smaller than the vacuum light wavelength for any given light frequency. Consequently prism, grating or edge coupling [8] have to be applied for optical SPP excitation. The metals of choice are usually gold or silver, as these metals show SPP modes in the visible or nearinfrared spectral range, besides rather low ohmic damping [1]. The SPP propagation distance, while depending on material and wavelength is in the order of 10 μm in visible spectral range. On the other hand, noble metal nanoparticles are known for a long time for spectrally selective absorption and scattering due to particle plasmon excitation [14-16]. The plasmon resonance frequency is a function of particle geometry, the polarization direction of the exciting light wave with respect to the particle axes and the dielectric functions of the particle metal and the surrounding medium. The excitation of particle plasmons gives rise to a spectrally narrow extinction band and to an electromagnetic field strongly enhanced around the nanoparticle with respect to the exciting light field. As in many applications ensembles rather than individual particles are applied, the mutual interaction of particle plasmons is of high interest. This interaction was recently identified to be particularly strong for regularly arranged nanostructures due to grating effects and can be used for SPP excitation on a surface. Further recent results revealing the properties of particle plasmons indicate that this phenomenon can be efficiently exploited for the local manipulation of light fields on the nanoscale. The findings include light field squeezing [11], local plasmon excitation [10], the controlled design of near-field enhancement [17] and spectral selectivity [18], all of which can be applied to manipulate with SPPs on a surface. In summary, propagating SPPs and particle plasmons can be used for light signal propagation and nanoscale light field manipulation, respectively. A combination of both could obviously allow the realization of nanoscale plasmon-based (plasmonic) optical devices. Besides connecting nanoscale metal structures, propagating SPPs can also provide the interface between classical optics (freely propagating or dielectric waveguide bound light) and nanoscale optical and the plasmonic devices.

LITHOGRAPHIC NANOSCALE SAMPLE PREPARATION Metal nanostructures are available for a long time in the form of island films and colloidal nanoparticles [14-16]. Nevertheless, the detailed investigation of particle plasmon effects proved to be rather difficult as tailoring the geometry and spatial position of the particles within such samples is virtually impossible. These drawbacks can be eliminated by lithographic techniques such as electron beam lithography (EBL), which was recently adapted for the production of metal nanoparticle ensembles well controlled in geometry and position [19]. EBL relies on the local exposure of an electron sensitive resist by a focused electron beam [20]. Usually a standard scanning electron microscope is equipped with commercial hard- and software to extend it to a lithography device. A resist layer (thickness typically 100 nm) can be formed by spin coating substrates such as glass plates with, e.g., polymethylmethacrylate (PMMA). The glass plates are often covered with a few nm of indium tin oxide, which combines the weak electric conductivity needed for EBL with high optical transparency adequate for optical experiments. Electron beam exposure (electron

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energy ≈ 25 keV) breaks the PMMA molecules into fragments that can be preferentially chemically dissolved. Therewith a PMMA mask for a subsequent deposition process of the material constituting the nanostructures is produced. Deposition is carried out by thermal or electron beam evaporation of materials such as silica, silver or gold in high vacuum. The height of the deposited material, typically 60 nm is monitored with a quartz microbalance. In a final step, the PMMA mask is removed chemically, leaving the dielectric or metal nanostructures on the substrate. Residual PMMA is ashed by exposing the sample to an argon plasma. When aiming at propagating SPPs, the fabricated discrete nanostructures can be covered by a thin continuous metal film with a typical thickness of 60 nm. A typical structure fabricated by this approach is illustrated in Figure 1 [21]. EBL provides the means to produce metal nanostructures with smallest feature sizes of about 20 nm over areas of typically 100×100 μm.

Figure 1. Exemplary plasmonic sample structure: Sketch of sample geometry combined with atomic force microscopy image of the surface [21]. The area of the upper image is 25×25 μm.

LITHOGRAPHIC FLUORESCENCE IMAGING To couple light to SPPs the light wave vector has to be increased as for a given frequency it is always smaller than the SPP wave vector due to the SPP dispersion relation [8]. By EBL it is possible to create suitable nanoscale surface protrusions on the metal film with geometries giving rise to scattering that provides the Fourier components necessary to match the light and SPP wave vectors. As an example, we launch SPPs by focusing a laser beam onto nanostructure on a gold thin film through a optical microscope objective. A direct way to the SPP field profile is to monitor the fluorescence of a thin molecule layer close to the metal surface by a charge-coupled device (CCD) camera (Figure 2) coupled to an optical microscope. This technique relies on the fluorescence of organic molecules placed in the vicinity of the SPP-carrying metal surface. A 10-4 molar solution of DiR or Rhodamine 6G was prepared by dispersing the molecules in a solution of 1 % PMMA in chlorobenzene. The resulting solution was spin cast on the samples yielding a film 30 nm thick when dried. Based on conventional optical microscopy, fluorescence imaging is a quite quick and reliable

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technique for probing SPP fields. As a further example, Figure 3 depicts the fluorescence images of SPPs launched from differently shaped nanostructures, leading to distinctly different plasmon profiles [21]. For comparison, the SPP propagating along the nanowire in Figure 4 has been imaged with a near-field optical microcsope [11].

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Figure 2. Sketch of the experimental setup for fluorescence imaging of SPPs launched from a nanoscale surface protrusion, combined with an actual SPP fluorescence image. OBJ optical microscope objective (100, numerical aperture 0.75); BS beam splitter; CCD charge-coupled device camera [22].

Figure 3. Fluorescence images of SPPs launched from lithographically designed surface features, as sketched in the insets, light wavelength 515nm [21].

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Figure 4. SPP propagation along a gold nanowire (a) topography, (b) near-field optical image acquired with a near field scanning microscopy, light wavelength 800nm [13].

The present know-how and technology with regard to plasmon effects on both the microand nanoscale seem sufficiently advanced to allow the development of functional SPP based optical devices. Therefore basic elements in analogy to conventional optics, as mirrors, beamsplitters, filters, polarizers or resonators have to be realized. Precisely tailored metal nanostructures can serve as the elementary building blocks for these elements. The practical aim of this line of research is the experimental realization and optimization of plasmonic components. For example, together with the local launch of directed SPPs via a metal nanostructure, SPP mirrors and beamsplitters allow the realization of a SPP interferometer. Additionally, a question of fundamental interest concerns the detailed understanding of how the nanoparticles and -structures that make these SPP optical elements interact with propagating SPPs, i.e., what is the role of their geometries, resonance frequencies, etc. Directed SPPs can be launched as outlined above (Figure 3, [21]). These SPPs propagating in the flat metal/air interface can be reflected and/or transmitted in-plane by suitable mirrors and beamsplitters. These elements can be built up and optimized from metal nanostructures. The principle feasibility of this approach to form functional optical elements for a 2D-dimensional SPP optics was recently demonstrated. Figures 5 and 6 display examples of SPPs locally launched on a silver surface and manipulated by silver nanoparticles forming Bragg mirrors and, finally, an interferometer [23]. Based on earlier work [21] we identified in detail the interference conditions for different optical pathlengths in the interferometer arms, based on both lateral position changes of the SPP mirrors and changes in their angular orientation (Figure 7). SPP interferometers constitute thus the first two-dimensional SPP device that is thoroughly characterized.

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Figure 5. SPP Bragg mirror: (a) Scanning electron microscope image, the inset shows a 6x-magnified view of the area marked by the white box. The white circle and the arrows indicate the laser focus position and the SPP propagation directions, respectively. (b) Corresponding fluorescence image [23].

Figure 6. SPP interferometer: (a) Scanning electron microscope image, the white circle and the arrows indicate the laser focus position and the SPP propagation directions, respectively. (b) and (c) show two corresponding fluorescence images with different lateral positions of the right Bragg mirror [23].

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Figure 7. SPP interferometer: (bottom) Scanning electron microscopy image. The white circle and the arrows mark the laser focus position and the SPP propagation directions, respectively, the angle  defines the SPP mirror orientation. (top) Fluorescence images for different mirror positions.

Besides the realization of a SPP interferometer consisting of mirrors and beamsplitters, the feasibility of a SPP based spectral filter was investigated. Recent results have demonstrated that light extinction from a periodic array of gold nanoparticles deposited onto a planar dielectric waveguide can be controlled by adjusting the array periodicity [24, 25]. Selective suppression of light extinction has been attributed to the coupling of light scattered by the particles with a surface mode of the underneath surface. Despite of the advantages of fluorescence imaging for the visualization of SPP propagation on metal surfaces and their interaction with surface nanostructures we identified restrictions for the practical application of this approach: First, fluorescence images have to be recorded within a very limited time (typically a few seconds) after the beginning of laser irradiation because of molecule bleaching. This often restricts the possibilities for a precise optical adjustment and the acquisition of images. Second, the fluorescence intensity is in general not proportional to the local SPP field intensity, making quantitative measurements impossible. Furthermore, the efficiency of light/SPP coupling is difficult to assess in quantitative terms, as usually no control over the relative contributions of absorbed, transmitted and scattered light field intensities is achieved. In the following, we discuss and demonstrate that the imaging of SPP fields by means of leakage radiation (LR) allows to overcome the drawbacks of fluorescence imaging. We will especially focus on the fact that this approach allows for quantitative measurements of the spatial SPP field profile.

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LEAKAGE RADIATION IMAGING The intensity decay of a plane SPP wave in a perfectly planar metal film between two dielectric media (Figure 8, [26]) defines its intrinsic decay length Lint = 1/2k, which is a measure of the “ideality” of the electron gas. k is defined as the imaginary part of the complex surface plasmon wave vector kSPP = k + ik. Intrinsic losses are caused by inelastic scattering of conduction electrons, scattering of electrons at interfaces and LR [8, 27, 28]. LR is emitted from the interface between the metal thin film and a higher-refractiveindex dielectric medium (substrate), for example, glass (Figure 8). When the electromagnetic plasmon field crosses the metal film and reaches the substrate, LR appears at a characteristic angle of inclination LR [8] with respect to the interface normal. At this angle the LR wave satisfies kSPP = nk0sinLR where kSPP and nk0 are the wave vectors of the SPP (real part) and the LR, respectively, with n being the refractive index of glass. For glass with n = 1.5 it follows that LR  44 fulfills the phase matching condition, which is larger than the critical angle of total internal reflection, CRITIC = 41.8 [8, 27]. It should be mentioned that LR can as well be observed experimentally when SPPs are excited at the metal/air interface by electrons. Although LR contributes to SPP damping, it permits the direct linearly proportional detection in the far-field of the SPP spatial intensity distribution at the metal/air interface. Indeed, the intensity distribution at the metal/glass interface is proportional to that of the SPPs at the same lateral position and the azimuthal intensity profile equals that of the SPP profile at each point of the interface [8, 27, 28]. A typical experimental setup used for LR imaging is shown in Figure 9 [29]. As mentioned above, LR arises at the interface between the thin metal film and the glass substrate (Figure 8). To avoid total reflection inside the glass, an immersion objective in contact with the bottom part of the sample is required to collect LR images [30]. Note that LR is generated as well in the so-called Kretschmann configuration with a prism [31], where it interferes with the incoming light.

Figure 8. Leakage radiation from SPPs. The SPP field is maximum in the interface and decays exponentially in the perpendicular directions SPPs propagating along at the interface metal/air will couple and transform their energy to electromagnetic fields in the glass substrate [26]. Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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Figure 9. Experimental scheme for LR imaging (bottom). SPPs are exited by laser light focused onto a structured gold film on a glass substrate. LR is emitted into the glass substrate at an angle LR (compare Figure 8). F (gray) filter, P polarizer, BB beam block. A picture of the setup is shown on top [29].

The conditions of local excitation have an important impact on the SPP properties, as we now show by applying linearly polarized light from a Titan:Sapphire laser at a wavelength of between 780 and 900 nm focused onto a gold ridge (200 nm wide, 60 nm high) on a 60 nm thick gold film through two different microscope objectives (Zeiss 50, numerical aperture 0.7 and Zeiss Achroplan 10, numerical aperture 0.25). With different objectives and thus focus diameters, SPP waves of different divergence angles can be launched as illustrated in Figure 10. Thus, conventional microscopy based LR imaging proves to be a quick and reliable technique for probing the spatial profile of SPP fields [29]. In the meantime, LR imaging has become a standard technique used by many scientific groups [32 -36]. The quality of LR images is significantly enhanced when the directly transmitted part of exiting laser light is blocked from contributing to the LR image [37]. Therefore, a reverse diaphragm can be introduced as a central beam block in the first image plane of the sample with a diameter chosen such that the LR appearing under the specific angle LR remains unaffected while the directly transmitted laser light is blocked (see Figure 9, „Beam Block“). For illustration, in Figure 11, LR images with and without beam blocked configuration are compared. Large improvements in signal constant and reduced background in the LR image acquired with a beam block are clearly seen.

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Figure 10. LR images of propagating (and reflected) SPPs acquired with different objectives for focusing the laser beam: Zeiss Achroplan 10x, numerical aperture 0.25 (left) and Zeiss 50, numerical aperture 0.7 (right). P indicates the laser polarization direction.

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Figure 11. LR images acquired in the configuration with a beam block for the directly transmitted laser beam (left) and without blocking (right). P indicates the laser polarization direction.

INTERACTION OF SPPS WITH BEAMSPLITTERS By applying LR imaging we performed the quantitative analysis of the efficiencies of nanoparticle chains on metal thin films that can be used as SPP beamsplitter in terms of SPP transmittance and reflectance as a function of laser wavelength applied for SPP excitation [26, 29]. Figure 12 depicts the LR image of a structured silver film shown in the scanning electron microscope (SEM) micrograph in the inset. The SPP propagation length is found to be of similar value as that reported from fluorescence imaging [22]. The center of the excitation at the position of the focused laser spot is overexposed due to the limited dynamic range of the CCD camera. Actually the SPP excitation intensity was adjusted to a level beyond the maximum CCD response near the source point in order to image the decay of the SPP over a decently wide area. As shown in Figure 12, two-counter propagating directed SPPs are locally exited by focusing the laser beam onto a nanowire (200 nm wide, 60 nm high), similar to the previously discussed fluorescence imaging studies [21 - 23]. The LR image reveals two highly directed SPP beams propagating to the right and left with a quite small divergence angle. The right-bound SPP beam interacts with a beamsplitter consisting of a chain of individual nanoparticles (60 nm high, diameter of 260 nm, and separated by a

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distance of 400 nm), see Figure 12. The incidence angle of the SPP is 45 with regard to the particle chain direction. The SPP is both partly reflected and transmitted by the the beamsplitter, as depicted by the arrows in the figure. In this particular case the intensity of the transmitted beam is comparably high, with only small contributions from reflectance and scattering. An intensity cross-cut through the intensity distribution along the arrows in Figure 12 is shown in Figure 13. The lateral variations of the SPPs intensity are fit by the expression given in same figure. This behavior is characteristic for the damped radiation from a dipole in two dimensions [27, 28]. To obtain a correct fit, the center of the fit functionas plotted in Figure 13 together with the experimental data has to be accordingly corrected for background ligtht. The quantitative analysis of the fitting curves yields then all information on transmittance, reflectance and scattering of SPPs at the beamsplitter, which is summarized in Figure 14 [29].

Figure 12. LR image of SPPs interacting with a nanoparticle beamsplitter. The arrows define the directions of SPP propagation. P indicates the polarization direction of the exciting laser beam. The inset shows a magnified SEM image [29].

Figure 13. Cross-section intensity profiles along the arrows in Figure 12 together with fitting curves calculated by the inset equation. Y is the intensity; Y0 is the value of the background intensity in the image; t is the decay constant of SPPs; x is the distance from the focused laser spot and l is the selected distance between laser spot and starting point for profile measurements in the images.

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In the near-infrared spectral region for wavelengths longer than ≈ 875 nm the beam splitter is almost transparent for SPPs (T ≈ 85 %), while the according reflectance is very weak and cannot be measured with the present technique. Therefore ≈ 15 % of the incident SPP intensity has to be attributed to SPP scattering to light and absorption by the metal. When moving towards shorter laser wavelengths, the transmittance monotonically decreases and has a minimum at ≈ 800 nm, whereas the reflectance simultaneously increases to a maximum value at the same wavelength. Proceeding further towards lower wavelengths restores the high transmittance and almost negligible reflectance values (Figure 14). SPP transmittance and reflectance thus shows a resonance-like behavior with a resonance wavelength of approximately 800 nm. Although such resonances are well known from metal nanoparticles in dielectric matrices [8], our results show that nanoparticle-on-film systems maintain a resonant behavior in qualitative accordance with theoretical work [38].

Figure 14. Transmission (T) and reflection (R) efficiencies of the nanoparticle beamsplitter versus laser wavelength. The solid curves are guides to the eye [29].

In Figure 15 we plot the dependence on the direction of the SPP beam reflected from the nanoparticle beamsplitter (Figure 12) in dependence on the laser wavelength. As parameter for the measurements the angle  between the nanoparticle chain and the direction of the reflectance was selected, as depicted in Figure 12. We find that an increase in wavelength leads a decrease in angle , illustrating the interplay of beamsplitter parameters and the SPP wavelength. For further analysis we now turn to a nanoparticle beamsplitter with a similar silver nanostructure geometry as the one presented in Figure 12 but with a modified parameters of the nanoparticle chain (height of 70 nm, diameter of 220 nm and separated by a distance of 300 nm), see Figure 16. Again, we analyze SPP transmission and reflection as a function of the incidence angle of the SPP beam. Using the analysis method described above the

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corresponding values for transmittance, reflectance and scattering were derived and summarized in the Table 1. We find that increasing the incidence angle of the SPP beam with respect to the particle chain direction increases the SPP reflectance and decreases the according transmittance values. All these results form a solid base for the application-oriented design of components as mirrors, beamsplitters and interferometers [39, 40].

Figure 15. Angle () reflectance dependence on the laser wavelength as indicated in Figure 12. The solid curve is a guide to the eye.

Table 1. Beamsplitter efficiency for different incidence angles of SPP incidence. Angle

Transmittance

Reflectance

Scattering

30

92 %

2%

6%

45

86 %

9%

5%

60

65 %

24 %

11 %

75

14 %

80 %

6%

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Figure 16. LR images of SPP nanoparticle beamsplitters for different SPP incidence angles. A nanowire as the SPP launching structure and SPP beamsplitters built up from chains of individual nanoparticles are imaged by SEM to the panels on the left hand side. The corresponding LR images are shown on the rigth hand side.

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ELLIPTICAL BRAGG REFLECTORS AND INTERFEROMETER With a specific pattering of, e.g., gold nanoparticles on a thin gold film an elliptical Bragg reflector for SPPs can be realized (Figure 17). In such an elliptical structure, SPPs are generated by laser excitation on a single particle in one focal point F1. Following reflection from the elliptical Bragg mirror, SPP focusing takes place in the second ellipse focus F2 [41, 42]. In order to achieve an effective SPP reflectance from the elliptical mirror we used a system of five confocal ellipses arranged by nanoparticles (Figure 17). The distances between the elliptically shaped nanoparticle chains were selected to enable Bragg reflection at a wavelength of 750 nm, just as it was demonstrated before for Bragg mirror constructed by nanoparticles arranged in parallel lines [22].

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Figure 17. Elliptical SPP nanoparticle reflector. Sketches of the reflector geometry and the cross-section of an individual nanoparticle are combined with a SEM image of a part of the reflector [41].

Figure 18. Experimental LR images of locally exited SPP beams upon reflection from elliptical Bragg nanoparticle reflectors. SPP are exited locally in the left focus for (a) vertical and (b) horizontal polarization of the exciting laser beam. Images (c) and (d) show the corresponding theoretical modeling [41]. The black arrows depict the laser polarization direction, the dashed arrows sketch the SPP propagation direction towards the right hand focus after reflection.

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LR images of the SPP field distributions corresponding to two polarization orientations of the SPP-exciting laser beam are shown in Figures 18a and b. We find that SPP focusing in F2 is indeed working while the SPP intensity profile within the ellipse depends on the polarization conditions of the exciting laser beam. In Figures 18c and d depict numerical simulations for comparison with the experimental data. Therefore, we characterized the SPP waves by a cos2() angular intensity distribution  being the azimuthal angle between the polarization axis of the laser beam and the direction to the observation point. For modeling we considered the SPP waves from F1 and the contribution due to SPP reflection from the elliptical mirror. The excellent agreement of measured and calculated images confirms that 2D-elliptical mirrors for SPPs can indeed be used for SPP focussing. The successful experimental realization of the elliptical Bragg reflectors and the effective focalization of SPPs as dicussed in the previous paragraph can be used as the starting point to construct a SPP interferometer based on elliptical Bragg mirrors. The idea of this interferometer is the dependence of the SPP intensity in the focal point F2 on the polarization angle of the exciting laser beam which launches SPPs in F1. The according experimental images are shown in Figure 19 and are found again in excellent agreement with the theoretical modeling for different polarization angles (here defined as the angle between the exciting laser beam polarization and the horizontal), as shown in the same figure [43]. The values of the SPP intensity in F2 extracted from the experimental measurements are presented in Figure 20 as a function of the angle . This figure quantitatively illustrates the interference condition around F2 due to SPP reflected from the elliptical Bragg mirror and how this intensity is changed in dependence on laser polarization. Figure 20 is a clear illustration that an efficient angular polarization SPP interferometer can be realized.

Figure 19. Experimental LR images of SPPs propagating from F1 to F2 for different polarization angles , defined as the angle between the exciting laser beam polarization and the horizontal (left set of images). Corresponding theoretical modeling of the LR images (right set of images). The polarization directions are indicated by the white arrows [43].

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Figure 20. Experimental data (dots) and theoretical model (solid line) for the dependence of the SPP intensity at F2 on the polarization angle  [41].

Figure 21. SPP excitation and reflection from Bragg mirrors constituted by gold nanowires. The SEM images (left) show the Bragg mirrors consisting of 10 lines, fabricated for various SPP beam incidence angles. The LR images (right) were acquired with different divergence angles of the SPP beam.

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LINEAR BRAGG REFLECTOR Besides being built from nanoparticles SPP Bragg mirrors can be constituted as well by arrays of gold nanowires on a gold thin film, which we now apply to demonstrate that LR imaging provides quantitative data on the reflection efficiencies of SPPs interacting with surface nanostructures. In Figure 21 the SEM and LR images for different structural parameters of the Bragg mirrors (interline distance) and incidence angles are presented. The images were acquired with different divergence angles of the SPP beam by laser excitation through different microscopy objectives. The quantitative analysis of SPP reflectance was done by the methods described in [29]. Summarizing the main aspects revealed by Figure 21, we conclude: The reflectance efficiency is higher for the SPP beam with smaller divergence angle. This finding is readily explained by the fact that the SPP intensity is distributed over a larger angular sector for the case wider divergence angles. Using a higher divergence angle allows us however to directly assess the acceptance angle of a Bragg mirror. The increase of incidence angle of the SPP beam leads to a decrease of the acceptance angle. This is due a feature of the interference between the incident SPP beam with the SPP beam reflected from individual lines in the Bragg mirror, corrsponding to a respectively wider bandgap for the Bragg mirrors optimized for smaller incidence angles.

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CONCLUSION Allowing for quantitative analysis leakage radiation imaging has significantly expanded the experimental analysis toolbox with regard to SPP effects on the micro- and nanoscale. It is an important tool for investigating and further optimizing functional SPP based optical components and devices, aiming at basic elements in analogy to conventional optics, as mirrors, beamsplitters, frequency selective filters etc. On the fabrication side these developments can rely on advanced lithographic techniques for the precise fabrication of metal nanostructures as elementary building blocks for SPP photonics. For large scale fabrication, nanoimprining lithography migth be a technique enabling both high-resolution and high-throughput production of plasmonic devices.

ACKNOWLEDGMENTS We wish to thank our partners and co-authors F.R. Aussenegg, A. Drezet, H. Ditlbacher, A. Hohenau, A. Leitner, B. Steinberger, N. Galler from the Karl-Franzens University in Graz (Austria) and A. Dereux, J.-C. Weeber, M.U. Gonzalez, A.-L. Baudrion from University of Dijon in France. Also, A.L.S. gratefully acknowledges the Alexander von Humboldt Foundation in Germany and the Austrian Scientific Foundation in the frame of a Lise Meitner Fellowship.

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[12] [13]

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[14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

Barnes, WL. Dereux, A. Ebbesen, TW. Nature, 2003, 424, 824-830. Krenn, JR; Weeber, JR; Phil. JC. Trans. Roy. Soc. A 2004, 362, 739-756 Yablonovitch, E. Phys. Rev. Lett. 1987, 58, 2059-2062. Joannopoulos, JD; Villeneuve, PR; Fan, S. Nature, 1997, 386, 143-149. Weeber, JC; Dereux, A; Girard, Ch; Colas des Francs, G; Krenn, JR. Phys. Rev. E, 2000, 62, 7381-7388. Quidant, R; Weeber, JC; Dereux, A; Peyrade, D; Colas des Francs, G; Girard, Ch. Chen, Y. Phys. Rev. E, 2001, 64, 066607-1-066607-6. Quidant, R; Weeber, JC; Dereux, A; Peyrade, D; Chen, Y; Girard, Ch. Europhys. Lett., 2002, 57, 191-197. Raether, H. In Surface Plasmons; Höhler, G. Ed., Springer: Berlin, 1988. Smolyaninov, II; Mazzoni, DL; Mait, J; Davi, CC. Phys. Rev. B, 1997, 56, 1601-1611. Krenn, JR; Weeber, JC; Dereux, A; Bourillot, E; Goudonnet, JP; Schider, B; Leitner, A; Aussenegg, FR; Girard, C. Phys. Rev., B 1999, 60, 5029-5033. Krenn, JR; Dereux, A; Weeber, JC; Bourillot, E; Lacroute, Y; Goudonnet, JP; Schider, B; Gotschy, W; Leitner, A; Aussenegg, FR; Girard, C. Phys. Rev. Lett., 1999, 82, 25902593. Lamprecht, B; Krenn, JR; Schider, G; Ditlbacher, H; Salerno, M; Felidj, N; Leitner, A; Aussenegg, FR. Appl. Phys. Lett. 2001, 79, 51-53. Krenn, JR; Lamprecht, B; Ditlbacher, H; Schider, G; Salerno, M; Leitner, A; Aussenegg, FR. Eutophys. Lett., 2002, 60, 663-669. Kreibig, U; Vollmer, M. Optical properties of Metal Clusters, Springer: Berlin, 1995. Stepanov, AL. In Metal-Polymer Nanocomposites; Nicolais, L. Carotenuto, G. Eds., John Wiley & Sons Publ: London, 2004, pp. 241-263. Stepanov, AL. In Silver nanoparticles; Perez, DP., Ed., In-Tech: Vukovar, 2010. Lamprecht, B; Schider, G; Lechner, RT; Ditlbacher, H; Krenn, JR; Leitner, A; Aussenegg, FR. Phys. Rev. Lett., 2001, 84, 4721-4724. Salerno, M; Felidj, N; Krenn, JR; Leitner, A; Aussenegg, FR. Phys. Rev., B 2001, 63, 165422-1 - 165422-6. Gotschy, W; Vonmetz, K; Leitner, A; Aussenegg, FR. Appl. Phys., B 1996, 63, 381384. Handbook of microlithography, micromachining and microfabrication, Vol. 12 of IEE Materials and Devices Series, Rai-Choudhury, P. Ed. SPIE: Washington, 1997. Ditlbacher, H; Krenn, JR; Schider, G; Leitner, A; Aussenegg, FR. Appl. Phys. Lett., 2002, 81, 1762-1764. Ditlbacher, H; Krenn, JR; Felidj, N; Lamprecht, B; Schider, G; Salerno, M; Leitner, A; Aussenegg, FR. Appl. Phys. Lett., 2002, 80, 404-406. Krenn, JR; Ditlbacher, H; Schider, G; Hohenau, A; Leitner, A; Aussenegg, FR. J. Microsc., 2003, 209, 167-172. Linden, S; Kuhl, J; Giessen, H. Phys. Rev. Lett., 2001, 86, 4688-4691. Linden, S; Christ, A; Kuhl, J; Giessen, H. Appl. Phys. B 2001, 73, 311-317.

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[26] Stepanov, AL; Kiyan, R; Reinhardt, C; Chichkov, BN. In Laser Beams: Theory, Properties and Applications; Thys, M. Desmet, E. Eds., NOVA Sci. Publ.: New York, 2009. [27] Hecht, B; Bielefeldt, H; Novotny, L; Inouye, Y; Pohl, DW. Phys. Rev. Lett., 1996, 77, 1889-1892. [28] Bouhelier, A; Husere, Th. Tamaru, H; Güntherodt, HJ; Pohl, DW; Baida, FI; van Labeke, D. Phys. Rev., B 2001, 63, 155404-1 -155404-9. [29] Stepanov, AL; Krenn, JR; Ditlbacher, H; Hohenau, A; Drezet, A; Steinberger, B; Leitner, A; Aussenegg, FR. Opt. Lett., 2005, 30, 1524-1526. [30] Hohenau, A; Krenn, JR; Stepanov, AL; Drezet, A; Ditlbacher, H; Steinberger, B; Leitner, A; Aussenegg, FR. Opt. Lett., 2005, 30, 893-895. [31] Kretschmann, E; Raether, HZ. Naturforsch., A 1968, 23, 2135-2136. [32] Kiyan, R; Reinhardt, C; Passinger, S; Stepanov, AL; Hohenau, A; Krenn, JR; Chichkov, BN. Opt. Express 2007, 15, 4205-4215. [33] Radko, IP; Bozhevolhyi, SI; Brucoli, G; Martin-Moreno, L; Garcia-Vidal, FJ; Boltasseva, A. Phys. Rev. B 2008, 78, 115115-1-7. [34] Zhang, D; Yuan, X; Bouhelier, A. Appl. Opt. 2010, 49, 875-879. [35] Grandidier, J; Colas, G; Francs, D; Massenot, S; Bouheier, A; Markey, L.; Weeber, J-C; Dereux, A. J. Microsc. 2010, 239,167-172. [36] Wang, J; Zhao, C; Zhang, J. Opt. Lett. 2010, 35, 1944-1946. [37] Drezet, A; Hohenau, A; Stepanov, AL; Ditlbacher, H; Steinberger, B; Galler, N; Aussenegg, FR; Leitner, A; Krenn, JR. Appl. Phys. Lett., 2006, 89, 91117-1 - 91117-3. [38] Ruppin, R. Solid State Commun., 1981, 39, 903-906. [39] Gonzalez, MU; Weber, J-C; Baudrion, A-L; Dereux, A; Stepanov, AL; Krenn, JR; Devaux, E; Ebbesen, TW. Phys. Rev. B 2006, 73, 155416-1-13. [40] Gonzalez, MU; Stepanov, AL; Weber, J-C; Hohenau, A.; Dereux, A; Quidant, R; Krenn, JR. Opt. Lett. 2007, 32, 2704-2706. [41] Drezet, A; Stepanov, AL; Ditlbacher, H; Hohenau, A; Steinberger, B; Aussenegg, FR; Leitner, A; Krenn, JR. Appl. Phys. Lett., 2005, 86, 74104-1-74104-3. [42] Drezet, A; Hohenau, A; Stepanov, AL; Ditlbacher, H; Steinberger, B; Aussenegg, FR; Leitner, A; Krenn, JR. Plasmonics, 2006, 1, 141-145. [43] Drezet, A; Hohenau, A; Koller, D; Stepanov, AL; Ditlbacher, H; Steinberger, B; Aussenegg, FR; Leitner, A; Krenn, JR. Mater. Sci. Eng., B 2008, 149, 220-229.

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Chapter 8

SURFACE PLASMON RESONANCE SPECTROSCOPY FOR BIOMIMETIC MEMBRANE ASSEMBLY AND PROTEIN-MEMBRANE INTERACTIONS STUDIES Joël Chopineau1*, Laure Beven2** Daniel Ladant3*** and Claire Rossi4****

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1

Institut Charles Gerhardt Institut Charles Gerhardt Montpellier, UMR 5253 CNRS-ENSCM-UM2-UM1, Ecole Nationale Supérieure deChimie, 8 rue de l’Ecole Normale –34296 Montpellier-France 2 umr Gdpp Inra, Université de Bordeaux 2 – INRA, Centre INRA de Bordeaux, BP 81, 33883 Villenave d'Ornon Cedex –France, 3 Unite de Biochimie des Interactions Macromoléculaires, Departement de Biologie Structurale et Chimie - CNRS URA 2185, Institut Pasteur, 28 rue du Dr Roux, 75724 Paris CEDEX 15, France 4 Université de Technologie de Compiègne, UMR CNRS 6022 – Génie Enzymatique et Cellulaire, Centre de Recherche de Royallieu, BP 20529 - 60205 COMPIEGNE Cedex, France

ABSTRACT Surface Plasmon Resonance (SPR) spectroscopy is a powerful technique for monitoring in real time interactions at surface/liquid or air interfaces. This detection method is widely used in the area of biomimetic membrane sensors. Biomimetic membranes are adaptable lipid supported structures which can be used to study protein or ligand association with cellular membrane. Different lipidic environments and membrane architectures have been described: vesicles, hybrid bilayer membranes, supported lipid bilayers, tethered lipid bilayers and vesicle layers. The formation of these architectures and their complete characterization involved complementary techniques such as fluorescence recovery after photobleaching, atomic force microscopy, quartz crystal microbalance and surface plasmon resonance spectroscopy. SPR spectroscopy technique has been found as a method of choice for studying the interaction of protein or peptides

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Joël Chopineau, Laure Beven, Daniel Ladant et al. and membranes. The membrane binding properties of extrinsic proteins such as myristoylated proteins was followed on gold supported biomimetic membranes. SPR spectroscopy allowed the determination of the kinetic binding parameters and equilibrium constant of these interactions. The reconstitution of integral membrane proteins in solid supported membranes represents a major goal for the researchers specialized in biomimetic membranes. The supported tethered lipid bilayers, in which the bilayer delimits two distinct compartments, is considered to be the preferred architecture to mimic an authentic cell membrane. SPR spectroscopy was used to monitor the insertion and/or reconstitution of proteins in such supported lipid environments. It was demonstrated to be very appropriate for the study of peptides or proteins insertion into or translocation across lipid bilayers.

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INTRODUCTION Surface plasmon resonance (SPR) is recognized as an important biochemical technique for the study of molecular interactions [1,2]. The main advantages of this technique are: (1) it allows a direct determination of the association and dissociation rate constants for binding events between two partners without requiring prior labeling of the components and (ii) it is sensitive enough to be carried out with only limited amounts of molecules. Protein-protein interactions are widely studied using the SPR technique in number of laboratories. However, application of this technique for the study of protein interactions with biological membranes requires the development of dedicated biomimetic membrane structures. Different lipidic structures on flat surfaces can be developed for the construction of membrane mimics [3]. Gold surfaces are chosen for the construction of supported biomimetic membranes because of the reactivity of thiol towards gold and the property of gold to generate plasmons. The membrane binding properties of extrinsic proteins such as acylated proteins can be followed on simple platforms such as hybrid bilayer membranes. These structures are obtained in two steps; alkyl thiols which self assemble as a monolayer on the gold surface to form the distal layer and the hydrophobic adsorption of liposomes followed by their rupture to obtain the proximal layer [4]. SPR spectroscopy is useful to monitor both the process of membrane formation and subsequently to visualize the interactions of proteins or ligands with the reconstituted artificial membrane. The kinetic parameters and equilibrium binding constants governing the interactions can be easily and accurately determined from the SPR data [5]. Yet, the hybrid bilayer structures are only poor equivalent of authentic biological membranes. Indeed, integral membrane proteins can hardly be inserted in such bilayers in a functional state, mainly because of the rigidity of the covalently attached alkyl under-layer. The reconstitution and the study of integral membrane proteins into biomimetic membranes represent a challenging task which requires the design of specific supported membranes that is at the forefront of the research in the membrane field. Promising models have been developed in the past two decades and the tethered lipid bilayer design represents a significant advance toward a more biological relevant membrane environment [6]. A tethered lipid bilayer model based on a fast and reliable procedure is of utmost interest for the study of transport process across membranes. Indeed, an authentic biomimetic membrane should delimit two distinct “cis” and “trans” compartments that are characteristics of native biological barriers. This is an essential experimental set up for studying the passage of

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Surface Plasmon Resonance (SPR) Technique and Apparatus SPR spectroscopy is a well-known and powerful technique which was developed for measurements of thin films deposited on a metallic surface [7]. The method is based on excitation of the surface plasmons by p-polarized light at the noble metal-dielectric interface. Surface plasmons are described as electromagnetic evanescent waves propagating along the surface with a typical penetration depth of 200-300 nm. The requirement of a noble metal surface and the property of gold to form covalent linkage with thiol groups favor the choice of gold surfaces for the assembly of biomimetic membranes [8]. An optical home made set-up (Figure 1) in the Kretschmann configuration is used to create an evanescent field on the gold surface [9,10]. The gold-coated glass slide is vertically assembled with a PTFE sample cell (0.1-1 mL) and a 90° high refractive index LaSFN9 prism (n = 1.85) is mounted with a glass slide using a matching fluid (n = 1.656). Monochromatic ppolarized light from a He-Ne laser beam ( = 633 nm) is reflected from the backside of the gold-coated glass slide. Gold-coated glass slides are obtained by thermal evaporation under vacuum; typically a gold layer of 47-50 nm is deposited onto a chromium adhesion layer of 1-2 nm. The SPR reflectivity curve is a function of the refraction indexes of the materials near the film surface. A minimum reflectance corresponds to the excitation of electron plasmons, and the minimum angle shifts if the refraction indexes of the layer at the surface are modified by a value of n = nlayer- nbuffer. The geometrical thickness (d) of the adsorbed material can be determined from the angle shift () by knowing the refractive indexes of each layer. A refractive index value of 1.5 is taken for long chain alkyl-thiols on gold as well as for phospholipids and proteins [11-13]. Reflectivity is recorded as a function of the incident angle and the optical thicknesses are determined according to Fresnel equations using the Winspall program (MaxPlanck Institute for Polymer Research, Mainz, Germany). Kinetics are measured at a fixed angle of 1° below the minimum angle. The surface coverage of material bound is determined from the optical thicknesses which are converted by using a coefficient of 1 ng/mm2 for an optical thickness of 10 Å [7,14]. BIAcore apparatus are SPR-based instruments that measure the refractive index changes near a sensor surface up to a distance of about 200-300 nm, in which the sensor chip is coupled with a microfluidic cassette. The sensor surface is the bottom of a small flow cell (h = 50 m, b = 500 m, l = 2.1 mm), 20-60 nL in volume, through which an aqueous solution (the running buffer) passes under continuous flow (1-100 L.min-1). In a typical experiment, a molecule (the ligand) is immobilized onto the sensor surface; the interaction is detected when the binding partner (the analyte) is injected as an aqueous solution (sample buffer) through the flow cell, also under continuous flow. Under flow, as the analyte binds to the ligand, the accumulation of binding molecules on the surface results in an increase in the refractive index. This change in refractive index is measured in real time, and the SPR angle shift response plotted as resonance units (RU) (one RU corresponds to 10-4 degree of angle shift) versus time is called a sensorgram. The difference in the refractive indexes between the

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running and sample buffers give a background response. Thus, the actual binding response is obtained after subtraction of the background response from the sensorgram. A reference flow cell is used to record the background response; in this control experiment, the analyte is injected onto a channel having no ligand immobilized to the sensor surface. One RU represents the binding of approximately 1 pg protein or lipid/mm2.

Figure 1. At the top: scheme of the surface plasmon resonance set up. At the bottom: scheme of the SPR measurement cell. The cell utilizes the Kretschman configuration (excitation of surface plasmona via a prism), the sensor surface can be represented as a membrane model attached to a gold covered surface

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Lipidic Structures

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One of the major advantages of membrane models deposited on or attached to a surface is the wide range of surface sensitive techniques that can be applied to study the characteristics of the biomimetic lipidic assembly as well as to analyze the protein/membrane interactions. These techniques are the SPR as described above, the Fluorescence Recovery After Photobleaching method (FRAP), Atomic force microscopy (AFM), Quartz Cristal Microbalance with Dissipation (QCM-D), Impedance Spectroscopy (IS).... Different constructions and combinations of membrane models on a solid support have been developed since the pioneer work of Mc Connell group [15,16] such as supported lipid bilayers on hydrophilic surfaces [15,17-19], hybrid bilayers [4,20], tethered bilayers [6,21,22] and supported vesicle layers [17,23,24]. For biological applications, these structures can be obtained using different routes which all involve the use of phospholipid vesicles (Figure 2). Phospholipid vesicles, also called liposomes, are used for commercial applications from topical preparations to drug delivery but are also membrane models for research concerning membrane proteins [25,26]. Generally, these assemblies can be prepared from a dried lipid film obtained from lipids solubilized in a chloroform solution by removing the organic solvent under vacuum. The lipid film is hydrated by the appropriate working buffer and liposomes are obtained either from ultrasonic irradiation or/and from extrusion of the lipid suspension through calibrated polycarbonate membranes (50-200 nm size) using a syringetype extruder. The hydrodynamic mean diameters of these aggregates are usually determined by quasielastic light scattering.

Figure 2. The formation of the different membrane models can be obtained from liposomes. For anchored models, as the supported vesicles layers (A) and the tethered lipid layers (B), tether molecules can be directly incorporated in the liposomes. The spontaneous fusion of lipid vesicles on hydrophilic or hydrophobic surfaces led to supported lipid bilayers (C) or to hybrid bilayer models (D), respectively Proposition => ajoute a, b c, d pour les 4 types de structure (later ref to Fig 2a pour hybrid layer etc….)

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Hybrid Bilayer Membranes (HBM) HBM are now commonly used as biomimetic membrane models and numerous examples of theses structures have been described and characterized [4,20,27]. For SPR measurements, HBM are currently built onto gold surfaces. The naked gold surface is cleaned by using a non ionic detergent solution (ex: Hellmanex, octyl glucoside). HBM are obtained in two steps, the distal layer made of alkyl thiols (ex: 1-octadecanethiol) which self assemble as a monolayer on the metal surface, the proximal layer being obtained by hydrophobic adsorption of liposomes (Figure 2D). After injection of a vesicular suspension, vesicles adsorb, rupture and phospholipids self-assemble on the top of the alkylthiol monolayer, forming the lipid leaflet. Thus, a planar monolayer of phospholipids is formed with phospholipids having their hydrocarbon tails oriented toward the chains of the alkylthiols and their polar head oriented toward the buffer solution. The composition of the phospholipid leaflet is directly related to the vesicle lipid composition. Excess lipids such as partially fused vesicles are removed after a sodium hydroxide (50 mM) washing step. The optical thickness of the lipid layer formed can be determined by SPR spectroscopy. After the injection of Egg-PC vesicles, thicknesses of 22  4 Å on a home made surface plasmon resonance apparatus and/or of 1700  100 RU on a BIAcore apparatus, have been routinely measured in good accordance with the formation of a lipid monolayer leaflet on the top of the alkyl-thiol leaftlet [10].

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Supported Lipid Bilayers and Tethered Lipid Bilayers The construction of a tethered bilayer model is envisaged using a simple and reliable procedure. Tethered lipid bilayers (TLB) can be obtained by covalent coupling of lipid vesicles onto a monolayer of cysteamine (1-aminoethane thiol) precoated onto the gold surface. The covalently coupled vesicles are forced to disrupt under osmotic stress, to form a continuous phospholipid bilayer tethered to the surface by a PEG cushion (Figure 2B). The gold-coated glass slides are coated by overnight immersion in a cysteamine solution. After rinsing with ethanol, the coated slides are dried under a nitrogen stream and stored under nitrogen before being used. The coated slide is assembled with the SPR cell. Then 4 ml of vesicle suspension of Phospholipid/DSPE-PEG3400-NHS (Phospholipid 1 mM with DSPEPEG3400-NHS molar ratios from 0 to 10 mol % in buffer) are injected over the surface at a flow rate of 3 ml/min. After stopping the flow, one hour minimum reaction time is needed for vesicles anchoring and disruption. The cell is rinsed with the same buffer (10 min at a flow rate of 1 ml/min), then with 4 ml of 4 mM NaOH (5 min) and finally flushed during one hour with buffer at a flow rate of 1 ml/min. Covalent coupling of the reactive NHS moiety with the amino group on the surface is achieved after about 2 hours of reaction at room temperature. Finally, the surface is flushed with buffer in order to obtain a continuous phospholipid bilayer as a result of osmotic stress. The most challenging part remains to determine the experimental conditions which lead to the spontaneous formation of a fluid and homogenous tethered bilayer. For example, Rossi and colleagues [28] have applied a Doehlert experimental design (second order experimental design) to determine the experimental conditions leading to the formation of a biomembrane. Four important factors involved in the bilayer formation were studied: the lipid concentration

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Surface Plasmon Resonance Spectroscopy for Biomimetic Membrane Assembly… 207 in the vesicles suspension, the mass percentage of anchoring molecules in the vesicles, the time of contact between vesicles and the surface and the time during which the lipidic assembly was left at rest (without flow) after buffer rinse. A restraint experimental domain which led to properties in accordance with a bilayer presence was delimited. The measured properties of the biomimetic membrane were the thickness by SPR on gold substrate, the fluidity by FRAP on gold and/or glass substrates. Analysis and fitting of the SPR reflectivity curves give a thickness between 50 Å to 55 Å for the adsorbed bilayer. A diffusion coefficient of 3.0 ± 0.8 10-8 cm²/s and a mobile fraction of 96 ± 4 % were determined from fluorescence experiments. The vesicle adsorption kinetics was highly reproducible for molar ratios of anchoring molecules between 0.4 and 6 mol%, which was not the case when the molar ratios were higher than 6 mol%.

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Binding of Acylated Proteins towards Bio-Mimetic HBM Kinetics of the interaction of acylated proteins (analyte) with hybrid alkanethiol/phospholipid bilayer (ligand) was determined using a BIAcore instrument. The on and off rates for this binary complex formation are related to the equilibrium dissociation constant by the equation: kd/ka = KD. Equilibrium constants can be determined by measuring the concentration of free interactant and complex at equilibrium. The concentration of complex can be measured directly as the steady state response. Since the concentration of free analyte is equal to the bulk analyte concentration [A]o, the steady state binding level Req, measured in resonance units (RU), is related to the analyte concentration by the equation Req = Rmax KDReq /[A]o where Rmax is the initial concentration of ligand corresponding to the maximum analyte binding capacity. A plot of Req//[A]o against Req at different analyte concentrations thus gives a straight line from which Rmax and KD can be calculated as in a standard Scatchard plot. In the framework of Langmuir model kinetics, a one-to-one reaction between the analyte A and the ligand B (mass transport and rebinding limitations being ignored) is represented by the reaction A + B AB. The SPR signal is noted R, the total SPR response corresponds to R + Ri, Ri being the bulk refractive index contribution. This simple bimolecular model is described by the following relationship: dR/dt = ka[A]o(Rmax - R) - kdR , in which [A]o is the initial concentration of analyte (e. g. acylated protein), R is the concentration of the molecular complex formed (R = [AB]), and Rmax corresponds to the concentration of ligand at time 0 ([B]o = Rmax). Many interactions do not fit the simple bimolecular model (Langmuir model) as shown by the inability of the experimental curve to fit a single binding isotherm. When the binding kinetics are more complex, analytical or numerical integrations are the only methods capable of accurately estimating rate constants. This may be due to physical effects of the biosensor itself such as mass transport limitations or steric hindrance, or it may indicate the presence of a more complex interaction. Usually, the association phase is fitted to the sum of two integrated rate equations. In the framework of the heterogenous ligand model, one analyte A binds independently to two ligand sites (B1 and B2)) : A + B1 AB1 and A + B2 AB2. The total response R = R1 + R2 + Ri and the following relationships are

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208

Joël Chopineau, Laure Beven, Daniel Ladant et al. [B1]o = Rmax1 [B2]o = Rmax2 [AB1] = R1 [AB2] = R2 dR1/dt = ka1[A]o(Rmax1 - R1) – kd1R1 dR2/dt = ka2[A]o(Rmax2 – R2) – kd2R2

another possibility for complex kinetics is a two-step binding model with conformational change, where analyte A binds to ligand B, then complex AB changes to AB*, which cannot dissociate directly to give A+B. This model is described by the following reactions AB AB AB* [AB] = R1 [AB*] = R2 The relationships are:

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dR1/dt = ka1[A]o(Rmax – R1 – R2) – ka2 R1 – kd1 R1 + kd2R2 dR2/dt = ka2 R1 – kd2 R2 Total response: R1 + R2 + Ri The binding of an artificially acylated model protein, bovine pancreatic ribonuclease A (Rnase A), to electrically neutral phospholipids was studied by using BIAcore technology [29]. The association and dissociation kinetics for the binding of Rnase A monoacylated on its N-terminal lysine with fatty acids of 10, 12, 14, 16 or 18 carbon atoms, were measured using sensor chips covered with HBM having a dimyristoyl phosphatidyl choline (DMPC) outer lipid leaflet. Non-acylated Rnase did not bind to the DMPC monolayer, Acylated Rnases C10 and C12 bound only very slightly whereas C14, C16, C18 acylated proteins associated strongly with the DMPC lipid layer. Hence, increasing the acyl chain length above C12 significantly enhanced the affinity of the acylated protein for the phospholipid monolayer. For the same protein concentration (10M), the steady-state responses (Req) were 530 RU for Rnase C16; 450 RU for RNase C18 and 125 RU for Rnase C14. For the C10 and C12 proteins, the kinetics of dissociation were too fast and could not allow a reliable analysis of

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Surface Plasmon Resonance Spectroscopy for Biomimetic Membrane Assembly… 209 the sensorgrams. (Association phase was taken from 20 s to 320 s after injection and dissociation phase was considered in the time interval 80 to 240 s after buffer injection). For C14, C16 and C18 RNases, the observed binding and dissociation kinetics apparently did not follow a simple pseudo-first-order progress curve. In all cases, integrated equations for a twostep binding mechanism had to be used to correctly fit the sensorgrams. Typical sensorgrams for several protein concentrations in the range of 2_300 M: for Rnase C-14 and for Rnase C16 are reported in Figure 3. Using the so-called global method (BIAcore, Biaeval software) and a two-step kinetic model, the series of curves allowed a good evaluation of the kinetic constants for Rnase C14, C16, C18 as reported in the following table. ka1 [1/Ms] 1.93 x 103 3.79 x 103 7.09 x 103

kd1 [1/s] 7.66 x 10-2 1.51 x 10-1 1.29 x 10-1

ka2 [1/Ms] 5.81 x 103 8.88 x 103 4.16 x 103

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Protein Rnase C14 Rnase C16 Rnase C18

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kd2 [1/s] 2.82 x 10-3 9.84 x 10-3 5.88 x 10-3

KD [M] 18.97 4.42 2.57

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Figure 3. Interaction of various concentrations of RNase C14 (top) [6, 10, 15, 30, 50, 100, 295, 500 M] and Rnase C16 (bottom) [2, 4, 6, 8, 15, 30, 50, 100, 135, 200, 270 M]. The filled horizontal bar represents the time course of the protein injection.

The apparent equilibrium constants (KD) for C14 and C16 Rnase binding towards the DMPC hybrid bilayer were also derived from Scatchard plots. A series of experiments was performed using a large range of protein concentrations in the 2_300 M range. For Rnase C14 the data could be fitted with a 1:1 Langmuir model with a KD of 10.4 M. Although, for Rnase C16 a two-phase model was necessary to fit the data. Two straight lines were obtained, the first one (concentrations of 2_30 M) gave a KD of 8.7 M and the second one (concentrations of 50_270 M) gave a KD of 56 M. These KD values (10.4 M and 8.7 M for Rnase C14 and C16, respectively) were close to those calculated from the kinetic rate constants (19 M and 4.4 M for RNases C14 and C16, respectively). We also used HBM to characterize the membrane binding properties of neurocalcin, a neuronal calcium sensor that belongs to the superfamily of EF-hand Ca2+-binding proteins. Neurocalcin is naturally myristoylated on its N-terminus in vivo (by an N-myristoyl transferase) and can associate with biological membranes in a calcium and myristoyldependent manner, a process known as “Ca2+-myristoyl switch”. Alternatively a recombinant non-myristoylated neurocalcin was chemically acylated at its N-terminus with fatty acids of different lengths (from C12 to C16). The influence of the acyl chain length on the membrane binding properties of chemically acylated neurocalcin derivatives were examined by using HBM biomimetic membranes. In this case, a monolayer of brain lipids was formed at the surface of an 1-octadecanethiol layer adsorbed on the bare gold surface. Equilibrium constants (KD) were determined from Scatchard plots. Myristoylated-neurocalcin bound efficiently to the lipid monolayer in the presence of Ca2+ whereas only a residual binding was observed in the absence of Ca2+. In the same conditions, unmyristoylated-neurocalcin did not bind to the lipid coverage neither in the presence nor in the absence of Ca2+. Therefore, the Ca2+ -myristoyl switch of neurocalcin could be monitored in real time by SPR spectroscopy using this hybrid layer configuration. The affinity of myr-neurocalcin was determined by recording sensorgrams at increasing protein concentrations from 0.02 to 2 M in the presence 2 mM Ca2+ [30]. The steady state response (Req) obtained were used for determination of equilibrium constant from Scatchard plots which give a KD value of 0.35  0.05 M. The binding properties of C14 and C16neurocalcins towards the phospholipid supported layer were examined. In the absence of Ca2+, the chemically modified C14 and C16-neurocalcins did not bind significantly to the lipid monolayer: at the highest protein concentration tested, less than 250 RU could be detected. In the presence of Ca2+, C14 and C16-neurocalcins bound efficiently to the phospholipid coated chips. Importantly, C14-neurocalcin (900 RU) gave a steady state response highly similar to myr-neurocalcin (850 RU) while C16-neurocalcin (2000 RU) was about twice higher (protein concentration 0.5 M). Scatchard plots derived from data analysis of those experiments gave a KD value of 0.39  0.05 M for C14-neurocalcins and 0.28  0.05 M for C16-neurocalcins. These results indicate that the binding of C16-neurocalcin was facilitated as compared to that of the C14/myr-neurocalcins although the increase in affinity appeared less than expected for the presence of two additional methylene groups. The HBM biomimetic surfaces can be made of different outer lipid layers depending on the composition of the vesicles. They are well defined and robust structures but they are only

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Surface Plasmon Resonance Spectroscopy for Biomimetic Membrane Assembly… 211 poor equivalent of authentic biological membranes. Indeed, integral membrane proteins can hardly be inserted in such structures in a functional state, mainly because of the rigidity of the covalently attached alkyl under-layer. For integral membrane proteins binding or reconstitution, tethered lipid bilayers appeared to be the preferred biomimetic membrane models as described below.

Tethered Lipid Membrane for the Study of a Bacterial Toxin

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The TLB model system was built as described in the section entitled lipidic structures. Firstly, in order to validate the TLB platform, the “Ca2+-myristoyl switch” functionality of myristoylated neurocalcin was studied using this membrane model and compared with the results obtained on the HBM substrate (Figure 4). Myr-neurocalcin bound efficiently to the TLB membrane model in the presence of Ca2+ while only a residual binding was observed in the absence calcium. As expected, the protein associated with the outer phospholipid layer without insertion as it could be almost completely removed upon washing by carbonate buffer. Similar binding assays carried out at different protein concentrations were performed. At the highest protein concentration tested, the maximal amount of myr-neuro per unit of surface determined from the SPR data, was about 150-160 ng/cm2. In the presence of calcium the apparent affinity of myr-neurocalcin was found to be 0.35-0.39 M. These results were similar to those obtained above with the HBM membrane model, thus comforting the potentialities of the TLB model for further investigations (Figure 4). TLB is composed of two authentic phospholipids layers which delimit two distinct aqueous compartments (“cis” and “trans”) on both sides of the membrane. Thus, it can be used for the study of the translocation mechanism of a bacterial toxin and for the functional reconstitution of integral membrane proteins.

Figure 4. The SPR kinetic represents the binding of Myr-neurocalcin at 2 µM in the absence or in the presence of calcium on HBM and on TLB membrane models. The binding properties of MyrPlasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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neurocalcin are identical on both membrane models in the presence of calcium. The absence, the presence of calcium and the buffer washes are indicated by arrows.

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The bacterial toxin, the adenylate cyclase produced by Bordetella pertussis (CyaA) is able to bind and translocate across the plasma membrane of eukaryotic target cells in a calcium-dependent manner. Inside the target cells, CyaA is activated by interacting with the endogenous calmodulin to produce supra-physiologic levels of cyclic AMP. CyaA is a 1706 amino acid long protein, made of three main regions: (i) the catalytic domain in its N-terminal part; (ii) a central region containing several hydrophobic segments that can insert into the plasma membrane of target cells to allow the translocation of the catalytic domain inside the cell; (iii) and a C-terminal domain that mediates the toxin association with the target cells. The C-terminal domain is also involved in the binding of calcium, which is an essential cofactor for CyaA entry into target cells. Calcium-bound CyaA should behave as an intrinsic membrane protein, with the hydrophobic polypeptide segments putatively inserted within the lipid bilayer. SPR spectroscopy was used to monitor the binding properties of CyaA with artificial TLB membranes in the presence or absence of calcium. Our results indicated that, in the presence of calcium, CyaA associated with the polymer tethered membrane in an irreversible manner, suggesting that the polypeptide chain had readily inserted into the lipid bilayer and therefore behaved as an authentic integral membrane protein (Figure 5) [31].

Figure 5. Binding properties of CyaA on HBM and TLB. The SPR kinetics were obtained by injection of CyaA at 0.05 µM in the absence or in the presence of calcium on both membrane models. The Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

Surface Plasmon Resonance Spectroscopy for Biomimetic Membrane Assembly… 213 quantity of bound proteins differs drastically for the two models. The absence, the presence of calcium and the buffer washes are indicated by arrows.

In contrast, the CyaA protein bound to the hybrid bilayer membrane in a fully reversible manner, irrespectively of the presence or absence of calcium (Figure 5). This suggested that the rigid alkyl monolayer underneath the phospholipid layer prevented the insertion of the polypeptide into the hemimembrane. Alltogether these data indicated that the HBM model could be very useful for the study of peripheral membrane proteins while TLB are particularly appropriate to study the insertion of protein into membrane or to reconstitute or integral membrane proteins in a membrane-like environment.

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CONCLUSION Ideal biomimetic membranes should be fluid structures with two authentic layers of phospholipids. These nanostructures should be stable for long periods of time and their preparation should be easy, reproducible, and robust. They should present two well-separated aqueous compartments (“cis” and “trans”) on both sides of the bilayer. They should be defect free in order to avoid potential channels that could allow free diffusion of electrons, ions, and/or small molecules across the lipid barrier. These membranes like structures should be assembled on different types of surfaces in order to allow various types of detection or analysis. SPR spectroscopy is a well adapted technique to follow the process of biomimetic membrane construction when it is coupled with other complementary techniques such as fluorescence and AFM. SPR can then be used to monitor the interaction properties of proteins with membranes, thus allowing detailed analysis of various biological processes such as protein insertion into membranes, protein translocation across membranes, role of acyl-chains in protein targeting to membrane, etc. When appropriate, the kinetic parameters and/or the equilibrium constant could be directly obtained from the SPR data. TLB is a much better model for biological membranes as compared to the hybrid bilayer. In particular, we have shown that it is much more appropriate to study the association properties of proteins that insert into the membrane bilayer and it should be useful to reconstitute integral membrane protein in a functional state.

REFERENCES [1] [2] [3]

[4]

Homola, J. (2008). Surface plasmon resonance sensors for detection of chemical and biological species. Chem Rev, 108, 462-93. Karlsson, R. (2004). SPR for molecular interaction analysis: a review of emerging application areas. J Mol Recognit, 17, 151-61. Besenicar, M. P. & Anderluh, G. Preparation of lipid membrane surfaces for molecular interaction studies by surface plasmon resonance biosensors. Methods Mol Biol, 627, 191-200. Plant, A. L. (1993). Self-assembled phospholipid/alkanethiol biomimetic bilayers on gold. Langmuir, 9, 2764-2767.

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[15] [16] [17] [18] [19]

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Joël Chopineau, Laure Beven, Daniel Ladant et al. Cooper, M. A. (2004). Advances in membrane receptor screening and analysis. J Mol Recognit, 17, 286-315. Kiessling, V., Domanska, M. K., Murray, D., Wan, C. & L. K. T. (2008). Supported Lipid Bilayers: Development and Application in Chemical Biology. . In Sons, J.W. (ed.), Wiley Encyclopedia of Chemical Biology, Hoboken, vol. 4, 411-422. Knoll, W. (1998). Interfaces and thin films as seen by bound electromagnetics waves. Annu. Rev. Phys. Chem., 49, 569-638. Nuzzo, R. G., Zegarski, B. R. & Dubois, L. H. (1987). Fundamental studies of the chemisorption of organosulfur compounds on gold(111). Implications for molecular self-assembly on gold surfaces. J Am Chem Soc, 109, 733-740. Kretschmann, E. & Raether, H. (1968). Radiative decay of non-radiative surface plasmons excited by light. Z. Naturforsch., 23A, 2135-2136. Lingler, S., Rubinstein, I., Knoll, W. & Offenhausser, A. (1997). Fusion of Small Unilamellar Lipid Vesicles to Alkanethiol and Thiolipid Self-Assembled Monolayers on Gold. Langmuir, 13, 7085-7091. Cuypers, P. A., Corsel, J. W., Janssen, M. P., Kop, J. M., Hermens, W. T. & Hemker, H. C. (1983). The adsorption of prothrombin to phosphatidylserine multilayers quantitated by ellipsometry. J Biol Chem, 258, 2426-31. Lang, H., Duschl, C. & Vogel, H. (1994). A new class of thiolipids for the attachment of lipid bilayers on gold surfaces. Langmuir, 10. Spinke, J., Liley, M., Schmitt, F. J., Guder, H. J., Angermaier, L. & Knoll, W. (1993). Molecular recognition at self-assembled monolayers: optimization of surface functionalization. J. Chem. Phys., 99, 7012-7019. Stenberg, E., Personn, B., Ross, H. & Urbaniczky, C. (1991). Quantitative determination of surface concentration of protein with surface plasmon resonance and radiolabeled proteins. J. Colloid Interface Sci., 143, 513-526. Brian, A. A. & McConnell, H. M. (1984). Allogeneic stimulation of cytotoxic T cells by supported planar membranes. Proc Natl Acad Sci U S A, 81, 6159-63. Tamm, L. & McConnell, H. (1985). Supported phospholipid bilayers. Biophys. J., 47, 105-113. Nollert, P., Kiefer, H. & Jahnig, F. (1995). Lipid vesicle adsorption versus formation of planar bilayers on solid surfaces. Biophys J, 69, 1447-55. Rädler, J., Strey, H. & Sackmann, E. (1995). Phenomenology and kinetics of lipid bilayer spreading on hydrophilic surfaces. Langmuir, 11, 4539-4548. Schmidt, E. K., Liebermann, T., Kreiter, M., Jonczyk, A., Naumann, R. & Offenhausser, A. (1998). Incorporation of the acetylcholine receptor dimer form torpedo california in a peptide supported lipid membrane investigated by surface plasmon and fluorescence spectroscopy. Biosensors and Bioelectronics, 13, 585-591. Plant, A. L. (1999). Supported Hybrid Bilayer Membranes as Rugged Cell Membrane Mimics. Langmuir, 15, 5128-5135. Knoll, W., Frank, C. W., Heibel, C., Naumann, R., Offenhausser, A., Ruhe, J., Schmidt, E. K., Shen, W. W. & Sinner, A. (2000). Functional tethered lipid bilayers. J Biotechnol, 74, 137-58. Tanaka, M. & Sackmann, E. (2005). Polymer-supported membranes as models of the cell surface. Nature, 437, 656-63.

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[23] Jung, L. S., Shumaker-Parry, J. S., Campbell, C. T., Yee, S. S. & Gelb, M. H. (2000). Quantification of Tight Binding to Surface-Immobilized Phospholipid Vesicles Using Surface Plasmon Resonance: Binding Constant of Phospholipase A2. J. Am. Chem. Soc., 122, 4177-4184. [24] Yoshina-Ishii, C. & Boxer, S. G. (2003). Arrays of mobile tethered vesicles on supported lipid bilayers. J Am Chem Soc, 125, 3696-7. [25] Lasic, D. D. (1998). Novel applications of liposomes. Trends Biotechnol, 16, 307-21. [26] Rigaud, J. L. & Levy, D. (2003). Reconstitution of membrane proteins into liposomes. Methods Enzymol, 372, 65-86. [27] Plant, A. L., Gueguetchkeri, M. & Yap, W. (1994). Supported phospholipids/alkanethiol biomimetic membrane: insulating properties. Biophys. J., 67, 1126-1133. [28] Rossi, C., Briand, E., Parot, P., Odorico, M. & Chopineau, J. (2007). Surface response methodology for the study of supported membrane formation. J Phys Chem B, 111, 7567-76. [29] Roy, M. O., Pugniere, M., Jullien, M., Chopineau, J. & Mani, J. C. (2001). Study of hydrophobic interactions between acylated proteins and phospholipid bilayers using BIACORE. J Mol Recognit, 14, 72-8. [30] Beven, L., Adenier, H., Kichenama, R., Homand, J., Redeker, V., Le Caer, J. P., Ladant, D. & Chopineau, J. (2001). Ca2+-myristoyl switch and membrane binding of chemically acylated neurocalcins. Biochemistry, 40, 8152-60. [31] Rossi, C. & Chopineau, J. (2007). Biomimetic tethered lipid membranes designed for membrane-protein interaction studies. Eur Biophys J, 36, 955-65.

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In: Plasmons: Theory and Applications Editor: Kristina N. Helsey, pp.217-235

ISBN: 978-1-61761-306-7 © 2011 Nova Science Publishers, Inc.

Chapter 9

PLASMONIC PHENOMENA IN NANOARRAYS OF METALLIC PARTICLES Victor Coello1, Rodolfo Cortes1, Paulina Segovia2, Cesar Garcia2 and Nora Elizondo2 1

Centro de Investigación Científica y Educación Superior de Ensenada (U. Monterrey). Km 9.5 Carretera Nueva al Aeropuerto. PIIT. C.P 66629. Apodaca NL. México 2 Doctorado en Ingenieria Fisica Industrial. FCFM UANL. Ciudad Universitaria, Pedro de Alba S/N. San Nicolas de los Garza NL. Mexico

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ABSTRACT Current experimental and theoretical investigations of plasmonic phenomena are indented to be the basis for miniaturization of photonics circuits with length scales much smaller than currently achievable, inter-chip and intra-chip applications in computer systems, and bio/sensor-systems. In this chapter experiments and numerical developments conducted to the understanding of this area are outlined. We focus our attention in the interaction of Surface Plasmon Polaritons (SPP) with arrays of nanoparticles. Numerical simulations and experimental results of different SPP elastic (inplane) scattering orders, and the operation of simple plasmonic devices are presented. Furthermore, non-linear microscopy, with a tightly focused laser beam scanning over a sample surface with different densities of nano-particles is presented. Finally, a scanning near-field microwave microscope is presented as an alternative technique that is reliable enough to be used as a check of potential plasmonic components that are based on nanoparticle arrays. In general, the stability with respect to geometrical parameters and dispersion were the main features investigated in all the presented plasmonic phenomena.

1. INTRODUCTION Current progress in optics of surface plasmon polaritons (SPPs) [1] offers new and broad ranges of scientific and technological perspectives. For instance, SPP are applied to

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efficiently channel light using scatterers in subwavelength structures [2]. This could lead, in principle, to a novel generation of nano-optics circuits. This atractive idea is based on the similitude between SPPs and waves propagating in planar waveguides since both are two dimensional waves propagating in the surface. However, one has to born in mind that this similitude stops here. SPP field have its maximum intensity at the surface plane, in contrast with the (two dimensional) guided waves in integrated optics. Moreover, it is easier to scatter SPP out of the plane than along to it. Actually, as a SPP field is strongly confined in the direction perpendicular to the surface, a direct observation of SPP localization is only possible by means of scanning near-field optical microscopy (SNOM) techniques 3. Near-field optical microscopy of SPPs has corroborated the existence of both weak and strong SPP localization [4]. However a local control (at a desirable surface place) of such SPP optical enhancement started to take form only with the birth of the two-dimensional optics of SPPs [5]. The so called Plasmonics has as a purpose the manipulation and controlling of SPP beams along the surface plane by using artificially created nano-components. In particular, metallic nanoparticle arrays played an important role for the development of novel plasmonic structures [5]. Artificially fabricated SPP nano-optical structures such as nano-bumps acting as a nanolens (focusing the SPP field) have been investigated for different films and wavelengths 5. In general, working in optics in the nano-scale regime is not trivial. Structures smaller than the wavelength, may not lead to the expected results and the investigations revealed several features such as wavelength dispersion and stability (with respect to geometric parameters) of the nano-components that still have to be elucidated. The studies are well complemented by using numerical simulations. For example, an scalar multiple-scattering approach was used for simulations of SPP optical nano-components [6] and photonic band gap structures formed by sets of individual scatterers [7]. Later, the approach has been extended into a vector dipolar multiple-scattering theory [8] and used to calculate SPP scattering produced by band-gap structures and for modelling the operation of a SPP interferometer formed by equivalent scatterers lined up and equally spaced [9]. Another interesting approach, for the sub-wavelength studies, is the use of the microwave radiation. The first proposal for a scanning near field microwave microscope (SNMM) came from Ash and Nichols in 1972 10. Since then, the technique has demonstrated its potential in areas such as magneto-resistivity characterization, superconductivity, and dielectric constant of individual samples [11,12]. Recently, it was proposed a simple SNMM designed to adopt dissimilar illumination operations modes and with capabilities for studies of potential twodimensional optical devices [13]. In this work, we present an overview of some of the problems, developments and current progress related with our research in the surface polariton nano-optics. We begin with an introduction to SPPs in section 2. Experimental results on SPP localization exhibited in a metallic surfaces covered with nano-particles will be presented in section 3. Direct evidence of strong and spatially localized SH enhancement in random metal nanostructures is presented in section 4. Section 5 describes a multiple scattering model that was used for simulation of SPP elastic scattering. In section 6, the above mentioned model was extended into a vectorial dipolar model. Thus, the operation of a plasmonic line mirror formed by equivalent nano-particles lined up and equally spaced was analyzed. The effects of SPPs in artificially nanostructured surfaces will be discussed in section 7. A near-field microwave technique used as an experimental approach for checking

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potential two-dimensional nano-SPP components will be analysed in section 8. Finally, in section 9, the conclusions are outlined.

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2. SURFACE PLASMON POLARITONS SPPs are oscillations of surface electron charge density that can exist at a metal/dielectric interface (Figure 1). Associated with them, there exists an electromagnetic field that propagates along the interface exhibiting exponential decays perpendicular to it. Therefore, SPPs show a high sensitivity to surface properties such as roughness and surface adsorbates [14]. As it is characteristic for evanescent fields [15], for the SPP to exist, the wavenumber associated with it must be larger (in absolute value) than the light wavenumber in the neighbour media. Surface polaritons obey Maxwell’s equations and they do represent (quasi) two-dimensional waves. The electromagnetic derivation of the SPP modes results in the fact, that such modes are possible only for p-polarization of light (TM-waves), since s-polarized waves (TE) do not satisfy the boundary conditions. Due to their electromagnetic nature, it is not difficult to infer that SPPs can diffract, reflect, and interfere. Those properties are clearly exhibited in the course of SPP scattering. Scattering of SPPs is usually caused by randomly placed surface imperfections (as even the most carefully prepared surfaces are not completely flat). Hereafter, we should distinguish between two kinds of SPP scattering: inelastic and elastic SPP scattering. For inelastic scattering, we will consider, propagating field components scattered away from the surface decreasing the total energy stored in SPPs. Elastic scattering occurs when SPPs are scattered by surface imperfections along the surface plane, i.e. into other SPPs preserving the total SPP energy. Concerning the mechanisms for SPP excitation, two techniques have been extensively developed: excitation by means of light, and excitation by means of electrons. SPP excitation by electrons is beyond the scope of this work (an overview can be found in Ref. 14). Otto and Kretschmann configurations1 are the mechanism most widely used for SPP excitation by light. They include a dielectric-metalair system, in which a light beam is impinging on the metallic surface under an angle larger than the critical angle. The excitation occurs at the interface between air and metal, and is recognized as a minimum in the angular dependence of the reflected beam power. An angular spectra analysis of SPP excitation allows one to deduce the SPPs characteristics, whose knowledge is indispensable for any kind of SPPs studies. However, those configurations have some drawbacks such as the separation of the source and plasmonic devices on different chips, and the large size of the prism couplers that make them not suitable for photonic integrated circuits. An alternative is the use of a normal incident light to excite the SPPs through a ridge [16] or subwavelength-hole arrays [17,18] located at the top of an air/metal interface. Such SPP launching mechanisms have been used for quantitative experimental analysis of a SPP interferometer [16], nano-parabolic chains [19], and testing of refractive plasmonic structures [20].

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2.1. Surface Polaritons Properties In order to show the SPP characteristics, first let us consider the interface between two semi-infinite media as air-metal. The SPP electric field existing in such a system (Figure 1) can be represented as: E ( x, y )  E 0 e i  xˆ  e  zˆ

,

(1)

which is an electromagnetic mode propagating in the x-direction along the surface and with an exponential decay perpendicular (z-direction) to it (Figure 1). The SPP wave vector, β, and the air decay constant, γ, are derived through the use of Maxwell`s equations and the boundary conditions, yielding the expressions: 

2

0

m 1   m    2  k 02 , ,

(2)

being λ0 the incident wavelength, εm the dielectric constant of metal, and k0 the incident wave number. The SPPs modes have an exponential decay into each of the media, being then the SPP decay constant, γm , in the metallic medium given by:  m    2   m k 02

.

(3)

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Other important SPP characteristics are, in turn, the SPP wavelength,  SPP 

2

 ,

(4)

the propagation length i.e.: the length at which the intensity decreases to 1/e (along the surface), LSPP 

1 2 im

,

(5)

with βim being the imaginary part of β, and the penetration depth i.e.: the length (perpendicular to the surface) at which the field amplitude decrease to 1/e. It is given by: d1 

1

 (air),

d2 

1

 m (metal).

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Figure 1. Schematic representation of a SPP-field (E) that exists at a metal/dielectric interface (XZ plane)

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3. SPP LOCALIZATION SPP localization is difficult to achieve since even for rough films it is not an automatically obtained effect [4,14]. It is quite difficult to find media in which one can get short mean free paths. One cannot make the volume fraction of scatterers larger and larger, since this leads not only to elastic but also to inelastic SPP scattering which may result that the optical signal will be dominated by propagating components. Apparently, to optimize the amount of scattering, it is necessary a large volume of scatterers whose sizes should approximately correspond to the SPP wavelength, SPP. Scatterers smaller (in size) than SPP are necessary for the near-field interactions responsible of the SPP confinement whereas those bigger than SPP would result in strong multiple scattering indispensable for the SPP localization to occur. Then, in order to observe strong localization, ideally one would like to have a medium which scatters light very strongly and preferably with negligible absorption. Under similar sample conditions like the ones just above described (Figure 2a), direct observation of SPP localization, in the form of bright round spots, was obtained (Figure 2b). The metallic films were fabricated by means of thermal evaporation technique. The rough gold films were evaporated on the base of a glass prism which had previously been covered with a sublayer of colloidal gold particles (diameter ~ 40 nm) dried up in atmosphere. Therefore, the introduced surface roughness was randomly distributed along the sample surface. SPPs were resonantly excited by means of the Kretschmann configuration. The position of such spots resulted to be angle dependent and not correlated to surface topography. The size of the bright spots was estimated at ~300-400 nm (Figure 3). The observations were directly recorded by use of scanning near-field optical microscopy (SNOM) [3] and related to the phenomenon of strong SPP localization [4,14]. The images exhibited an enhancement ratio up to 5 times the background signal (Figure 3).

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(a)

(b)

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Figure 2. Gray-scale topographical (a) and near-field optical (b) images of 4.53.6 µm2 obtained with the gold film with nano-particles randomly distributed. The maximum height surface roughness in the topographical image is ~ 42 nm. Contrast of the optical images is ~ 98%

Figure 3. Horizontal cross section of the near-field optical image of the bigger bright spot shown in Figure 4 (b)

4. SECOND-HARMONIC FAR-FIELD MICROSCOPY OF LOCALIZED SPPS Novel optical phenomena arising from the propagation of SPPs at a weakly-corrugated metal surface originated original contributions in the area of SPP-enhancement of second harmonic diffraction [21,22]. The experimental setup (Figure 4) used, for this kind of experiments, was a so called second-harmonic scanning optical microscope (SHSOM). It consisted of a scanning optical microscope in reflection geometry built on the base of a commercial microscope and a computer controlled two-dimensional (XY) translation stage.

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The XY stage is able of moving in steps down to 50 nm with the accuracy of 4 nm within a scanning area of 25x25 mm2. A mode-locked pulsed Ti-Sapphire laser tunable was used in the range of 730-920 nm with pulse duration of ~200 fs, 80 MHz of repetition rate and an average output power of ~300 mW. The linearly polarized light beam from the laser is used as a source of sample illumination at the fundamental harmonic (FH) frequency. The laser beam pass through a wavelength selective splitter and it is focused at normal incidence on the sample surface (spot size ~1μm) with a Mitutoyo infinity-corrected long working distance x 100 objective. In order to avoid thermal damage on the sample, the average incident power

Figure 4. Schematic of the experimental setup. OI: optical isolator, λ/2: halfwave-plate, P: polarizer, BS: beam splitter, F1 and F2: filters, WSBS: wavelength selective beam splitter, L: objective, S: sample, XY: stage, A1 and A2: analysers, PMT photomultiplier tube, and PD: photodiode.

Figure 5. Sample structure. It consists of a high density scattering regions composed of gold bumps (~ 70 nm high) randomly distributed over a thin (~50 nm) gold film.

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Figure 6. Gray scale FH (a-c) and corresponding SH (d-f) images of 8x8μm2 obtained for 770 (a,d), 780 (b,e),and 790 (c,f) nm of the FH wavelength. The polarization of the FH (incident) and the SH (detected) signal were both kept vertical with respect to the presented images respectively. The maximum of the SH signal was 1200 cps (d). Representative dark (a-c) and bright (d-f) spots were enclosed in circles for better visualization

was kept at the level of ~20 mW (intensity at the surface ~2 ×106 W/cm2 ). The sample reflects the FH and the generated SH radiation back to the same objective. The FH and SH signals are separated with the beam splitter and, after passing the appropriated filters and polarizers, detected with a photodiode and a photomultiplier tube respectively. Finally both signals are recorded as a function of the scanning coordinate obtaining, simultaneously, FH and SH images of the sample surface. The resolution of the SHSOM was evaluated in ~0.7μm using domain walls of an electric field poled KTiOPO4 quasi-phase matching crystal [23]. The sample under investigation was fabricated by using thermal evaporation and electron beam lithography techniques (more information about the procedure is found in Ref. [24]). The final sample structure consists of high density scattering regions composed of gold bumps (~ 70 nm high) randomly distributed over a thin (~50 nm) gold film (Figure 5). The scattering regions density is approximately 50 scatterers per 1μm2 and contains 2μm-wide channels free from scattereres that were used for investigations of guiding of SPPs [24]. There were recorded simultaneously FH and SH images of a dense scattering region in the wavelength range of 750-830 nm. The overall behavior of images of the same particular signal was very similar (Figure 6 a-f). In general one can appreciate bright and dark regions which are a collection of small and round bright spots similar to those reported as evidence of localized SPPs [4]. The presence of the observed spots suggested that the total detected FH radiation can be considered as a superposition of the FH beam reflected from the flat gold surface and the FH field scattered by strongly interacting gold bumps. The latter contribution is related to the regime of multiple scattering of the light which exhibits strong polarization and frequency dependence [4] and can be expected to occur, for example, due to localization of resonant dipolar excitations at nanostructured surfaces. In general, one should expect the most efficient excitation for well-localized modes with one strong field maximum. Light scattering via excitation of such a mode (arising from the nano particles) should be similar to the dipole scattering resulting in the excitation of SPP modes. SPPs are scattered, in turn, in the surface plane and into the substrate as well as absorbed due to the internal damping. These

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processes contribute to the decrease of the total flux in the direction of reflection and, thereby, formation of dark spots (see spots enclosed in circles in Figure 6 a-f). The images showed a noticeable re-distribution of the intensity. The bright spot (enclosed in circle) seen clearly in Figure 6d was re-distributed in Figure 6f. Taking this into account, one can claim that the bright spots were not correlated with surface defects since they show illumination wavelength dependence. It is also possible to notice that dark FH spots coincide with bright SH ones (enclosed spots in Figure 6 a-f). About this fact, an explanation can be formulated. Excitation of an FH eigenmode (leading to the local FH enhancement) results in a strong SH signal only if the SH field, which is associated with the generated nonlinear polarization, is further enhanced due to excitation of the corresponding SH eigenmode. That is to say, FH and SH eigenmodes should overlap in the surface plane. However, in general, such a correspondence is very difficult to observe because of the relative low contrast of FH images.

5. SPP SCALAR MULTIPLE SCATTERING MODEL Typically, elastic scattering of SPP and related plasmonic phenomena (e.g weak and strong localization) have been investigated by direct evaluation of the near-field optical image obtained at the place where the SPP is being resonantly excited [3]. This task could be numerically well complemented by using a SPP scalar multiple scattering model. Such model is based on two assumptions:

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(i) The elastic SPP scattering is dominant with respect to the inelastic scattering (ii) The SPP scattered by each scatterer represents an isotropic cylindrical SPP. These assumptions allow one to avoid some of the complicated mathematical treatments involved in the problem of SPP scattering by surface inhomogeneities [25]. The field at a point in the plane pointed at by the vector r is given by:

  

N

E r   E0 r     j E r j G r,r j j 1

,

(7)

where is E0(r) the incident field, αj is the effective polarizability of the jth dipole, E(rj) is the self–consistent field at the site of the jth dipole G(r,rj) is the field propagator, describing the scattered propagation of the scattered field from the jth dipole located at the source point rj to the observation point r. The self-consistent field to each dipole E(rj) can be determined as

 

   E r Gr ,r 

E r j  E0 r j 

N

l 1,l  j

l

l

j

j

l

.

(8)

The total field at the site of the dipole j is the incoming field at the site of the scatterer and the sum of the scattered fields from all dipoles surrounding dipole j. The field in (8) then

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has to be inserted in (7) to find the total field at a point in the plane. The field propagator is given as:

 

G r ,r j 





1 1 H 0  r  rj 4 ,

(9)

where H0(1) is the zero-order Hankel function of first kind and β is the propagation constant for the SPPs. The Hankel function first kind of order n is defined as





     J  r  r  J  r  r cosn  J  r  r  i .

H 01  r  r j  J n  r  r j  iYn  r  r j



n

j

n

j

sin n 

n

j

(10)







Y  r  rj J  r  rj where n is the Bessel function of the first kind and, n is the Bessel function of the second kind, and n is the order. Often it is appropriate and easier to use the farfield approximation for the Hankel Function. The farfield corresponds to large values of the argument, and the farfield approximation reads for large arguments:





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H 01  r  r j 

2



e

i

 4

e

i r  r j

 r  rj

.

(11)

The estimation of the magnitude of , the effective polarizability of the individual scatterers has been done by fitting  such that the calculated (parabolic) interference pattern generated by an individual scatterer has the same contrast of an experimental (near-field) intensity distribution generated in analogue form [4]. Thus, α = 3 was a typical value that was used in the calculations. The SPP elastically scattered has been simulated by using a light wavelength λ = 633nm and a dielectric constant ε = -16+i which corresponds to a silver film at the wavelength of illumination. The calculated total elastic cross section of the scatterer (with a = 3) was found to be: σ = 0.22 µm. The value of σ can be used as a check of the estimation of , i.e., for a symmetric surface defect considered theoretically [25] the same cross section would correspond (in the first Born approximation) to a scatterer of 0.1 µm of height and 0.7mm of radius. As a first case, we have considered the scattering from a single particle. The propagation constant for the SPPs is taken real, so no damping across the surface is presented. Since only a single particle is included, there is no scattering contribution from neighbors, and (7) therefore reduces to the following:

 

 .

E r j  E0 r j

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Now a plane wave of unit amplitude travelling from left to right is incident on the nanoparticle that is:

 

E0 r j  e

ix j

.

(13)

Figure 7. Intensity distribution of a single particle calculated with the scalar model. The illumination wavelength, λ0 = 750nm and the radius of the scatterer is r = 50nm. The white arrow indicates the SPP incident direction

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With the use of (12) and (13), the far-field approximation for the Hankel function in (10) and (11), the total field from a plane wave and spherical wave can be written:

E r   e ix  e

i x j





i 2 i 4 e 4 

e

i r  r j

 r  rj

.

(14)

Leading right to the intensity:

I r   E r 

2



2  1  r  rj 8



1







  x  x j  ...  sin        r  rj  4



2

2

  .  

(15)

For maximum intensity the sine oscillation has to equal 1, then:

  x  x j  

x  x   y  y  2

j

2

j

      n 2 .  4 2

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where n is an integer. For the sake of simplicity, the position of the particle rj(xj,yj) is placed at the origo. Bearing this in mind and using b the SPP propagation constant now reads for n > 0:

n SPP  SPP  SPP 1  y2  x1       . 2 8 128n  8n  2n SPP

(17)

For growing values of n the last term on the left hand side and the last two terms on the right hand side becomes vanishing compared the rest. Then, one can see, that a parabolic fringes in y with a distance of ΛSPP/2 between terms is expected. Figure 2 shows a simulation of (x) for a single nano-particle in a silver/air interface, where one can see that the fringes indeed have parabolic shape.

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6. SPP VECTORIAL MULTIPLE SCATTERING MODEL Despite the apparent success, the scalar model presents limitations, one of them being that the effective polarizability of an individual scatterer is a phenomenological quantity which is difficult to relate to scatterers parameters such as size and dielectric susceptibility. The model was extended into a vector dipolar multiple-scattering theory and used, among other things, to calculate SPP scattering produced by band-gap structures [8]. The approach entails point-like dipolar scatterers interacting via SPPs so that the multiple-scattering problem in question can be explicitly formulated, making it very attractive for modelling of SPP plasmonic phenomena. The validity of the model was established for relatively large inter-particle distances, whereas for smaller distances it was more accurate to use a total Green’s tensor and include multipolar contributions in the scattered field. The self-consistent polarization of each scatterer established in the process of multiple scattering is obtained by solving the following equation:

Pi   i  E 0 ri  

k 02

0

 n i

i

 G ri , rn   Pn .

(18)

where Pi is the polarization of the particle i, α is the polarizability tensor for particle i with the multiple scattering between the particle and the metal surface taken into account E0 is an incoming electric field, k 0 is the free space wave number, λ0 is the vacuum permittivity and G(ri,rn) is the Green´s tensor for the reference structure (total field propagator). The Green`s tensor G is the sum of a direct contribution Gd, in this case the free space Green`s tensor, and an indirect contribution Gs that describes both reflection from the metal/dielectric interface and excitation of SPPs. The incoming E0 describes a Gaussian SPP field impinging on the arrangement of scatterers. For a spherical particle made of the same metal as the substrate, the polarizability tensor is given by

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1

  1  1  1 1 1  0 α  I   xˆxˆ  yˆ yˆ  zˆzˆ   α .   1   2 8 8 4   

(19)

where I is the unit dyadic tensor, ε is the metal dielectric constant, ˆ ˆ ˆ are unit vectors in a Cartesian coordinate system with zˆ being perpendicular to the air–metal interface, and

x, y, z

α 0   0 I 4a 3   1 /   2

is the free space polarizability tensor in the longwave electrostatic approximation with a being the sphere radius. The polarizations and the total field,

E r   E 0 r  

k 02

0

 G r,r  P n

n

n

.

(20)

can be calculated using the appropriate Green`s tensor for the reference structure G(r,rn). Considering both the source and observation points being close to a meal surface but far away from each other, one can approximate the total Green dyadic (which includes the direct and indirect terms) with the part of the indirect Green dyadic concerned with the excitation of SPPs [8]. In this approximation which is actually asymptotically correct as the in-plane separation of source and observation points increases towards infinity, the Green dyadic can be represented by

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GSPP r , rn    zz  eiK z  z h  H 01 K    [ zˆzˆ    zˆˆ  ˆzˆ  K z  ˆˆ  K z K  K

2

  ],  

(21)

  rII  rII ˆ  rII  rII  /  where H01 is the zero-order Hankel function of the first kind, , , xy II with referring to the projection of the radius vector on the plane, which coincides with the metal air interface, and z refers to the height of observation point r above de surface, while h refers to the height of the source point r´ Finally, Kz and Kρ are the components of the three-dimensional SPP wave vector.

K   k0

  1

K z  k02  K 2

.

.

and

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(23)

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K   1  1     zz       1  2  2       .

(24)

The complete analysis of the validity domain of such an approximation is beyond the scope of this work and can be found elsewhere [8]. As previously mentioned, a single circular nanoparticle is assumed to scatter light as cylindrical waves, so no preferred direction of scattering is presented. By placing particles in line of nano-arrays, a common plane wavefront of the scattered light can be achieved. From an application point of view, it seems obvious to exploit line arrays to reflect the wavefront to the applied field in order to make a mirror effect. The idea has been modelled for a line array of 3µm length and whose inclination compared to the applied field is 450 [26]. The line arrays are far from being perfect mirrors since much of the incident light may pass through the structures. This fact can be compensated for by placing an array of lines that satisfy the Bragg condition, 2dsinθ = nλ, where d is the separation distance, θ is the angle the beam makes with the mirror and n is a whole number. Here, we chose an inter-particle distance of 200nm and an inter-line separation of 350nm with an incident beam angle of 600. The simulation has been performed with an incoming Gaussian beam of the form:

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E0  e

 y2    ix   w2   

.

(25)

The incoming wavelength, λ0, the beam waist, w, and the particle radii, r were fixed to 750nm, 2µm, and 60nm respectively. Figure 6 shows the mirroring effect from the nanoparticles. However, much of the signal is still transmitted. This fact is expected when only a few particles lines are used. With many layers of particles in an array, it is possible to obtain almost all of the reflected power when certain wavelengths of the incoming beam. That is the principle of the SPP band gap (SPPBG) structures. SPPBG phenomena are beyond the scope of this work but an overview can be found in Ref [8].

7. NANO OPTICS OF SURFACE PLASMON POLARITONS Two-dimensional optics of surface plasmon polaritons is an exciting novel area. In this context, there exists a revolution in scientific and technological aspects in specific phenomena such as enhancement of an optical signal at a desirable surface place [1,2], waveguiding of SPP [1,2], and in a more general form in integrated optics. Experimental observations on nano-bumps acting as a nano-lens (focusing the SPP field) yielded conclusions about optimal parameters for a good efficiency of SPP nano components. The observations suggested that nano scatterers sufficiently large in size and with smooth borders maximize their strength and preserve an adiabatic perturbation. Thus, some of the first plane (Figure 9) and corner square [26] nano mirrors were successfully reported.

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Another plasmonic device, realizable by a certain array structure of nano-particles, is a focusing micromirror. We show numerically such a possibility. The concept is to place the

particles along a parabolic curve  y  y0   4 F x  x0  where the coordinate x0 , y0  is located at the bottom of the mirror, F is the focal length and x is along the optical axis. In principle, this works for the two-dimensional case of SPP propagation in the same way as the three-dimensional case of a parabolic screen of a solar cooker, which concentrates reflected solar light at a cooking pot. Hence, at the point x0  F , y0  a concentration of light is 2

expected. In Figure 10, we simulated a nanomirror with F = 8µm and where the focusing effect was clearly seen. For applications matters, focusing nano-mirrors give the possibility to enhance SPP signal locally in a controllable way. One can think to exploit that feature in, for example, biosensors, and surface enhanced Raman spectroscopy.

8. TWO-DIMENSIONAL OPTICS WITH SURFACE ELECTROMAGNETIC MICROWAVES

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In the microwave range the mechanical designs as well as the precision requirements are not as demanding as in the optical range. Taking advantage of that fact, experimental approaches for a wavelength, λ, of 2.85 cm. were proposed as a check of potential (twodimensional) nano-components [13]. Several spheres were aligned keeping approximately λ/2 distance of separation between them. Thus, distinct two-dimensional microwave components were fabricated e.g.: a 5-scatterer two-dimensional line mirror (Figure 8a,b), and a curved mirror focusing that reflected signal at 2.5 cm distance from the mirror (Figure 9a,b).

Figure 8. Intensity distribution of a multiline mirror with 5 lines of 30 nano-particles with radius r = 63nm Inter-particle and inter-layer distances are 200 and 350nm respectively. The incoming SPP beam has a wavelength, λ0 = 750nm . The white arrow indicates the SPP incident direction

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(a)

(b)

Figure 9. Gray-scale topographical a) and near-field optical images b) 4.44.3 µm2 of a potential plane nano-mirror. In the figure, the SPP travels from bottom to top

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Figure 10. Intensity distribution map in an area of 30x30µm2 calculated for a curved micromirror with a focal length F = 15µm and composed of 140 nano-particles with inter-particle distance of 220nm. The dotted line represents the curved nanomirror. All else is as in Figure 8

(a)

(b)

Figure 11. Gray-scale near-field image of 4.5 x 10 cm2 a) due to the elastic scattering of an evanescent microwave mode travelling from bottom to top a) on the line of scatterers placed along a wax surface prism and corresponding surface digital picture b). The dot line in a) helps to distinguish the boundary of evanescent wave and line of spheres interaction

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Figure 12. Gray-scale near-field image of 4 x 6 cm2 due to the elastic scattering of an evanescent microwave mode travelling from left to right on the parabolic line of scatterers placed along a wax surface prism a), and corresponding surface digital picture (not in scale) b).

The image for the line mirror showed the interference between a specular reflected evanescent microwave and the incident one. The behaviour, for the line mirrors, was not as good as it had to be. For example in Figure 11a,b the line mirror does not completely backward reflect the incident evanescent field (travelling from bottom to top in the horizontal position) as one would expect from a normal incidence. Concerning the focusing mirror (Figure 12a,b), ideally a focusing mirror should consist of scatterers placed along a parabolic curve y2= 4Fx where (x, y) is the orthogonal system of coordinates in the surface plane, the x axis is oriented along the optical axis and F is the focal length. Based on that, a mirror with F = 2.5 cm was fabricated. The operational principle of the component resulted rather a b satisfactory. For example the expected focusing was exhibited along the axis of the mirror. ) )

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9. CONCLUSION In this chapter, the basic ideas of SPPs and experimental results concerning the phenomenon of SPP localization were reviewed. As far as the SPP localization phenomenon is concerned, it should be stressed that there are many basic and technological aspects that have to be yet explored. The physics behind such a phenomenon is not simple. For instances, the influence of dissipation and wave-wave interaction are not clarified yet [27]. In another hand, results concerning the modelling of plasmonic phenomena were presented. Even though in the last years, there has been a great progress in the understanding of plasmonic phenomena, there exists not at the moment a complete theoretical model to deal with such studies. In that context, a relative simple scalar multiple SPP scattering model was presented. Nano-particles were considered in a two- dimensional geometry as isotropic point-like particles characterized by their effective polarizabilities. The scalar model has limitations on the accuracy of numerical results. For example, the effective polarizability of an individual particler is a phenomenological quantity that is difficult to relate to particle parameters such as size, susceptibility, etc. The scalar approach was extended into a vectorial dipolar model for SPP multiple scattering and used to model the operation of a SPP multiline mirror whose main element represented individual scatterers lined up and equally spaced. One can try to further improve this model by, for example, developing another (but analytical as well) approximation of the Green´s tensor for relatively small inter-particle distances. In the area of

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surface plasmon polariton nano-optics many features have already been clarified. The progress in both theoretical and experimental aspects has significantly increased with the advent of the near-field optical microscopy. Thus, nano-particles were used to create almost any kind of conceivable SPP nanocomponent such as a parabolic nano-mirrors. During the course of the research, several devices composed of a set of nano-scatterers have been numerically simulated and its stability (to geometric parameters) and dispersion dependence were studied in detail. The corresponding experimental results demonstrated that the model is accurate enough and that can be used, in straightforward manner, to design SPP nanocomponents. Finally, we have designed and constructed a stand-alone SNMM, including electronics and software, and demonstrated that it works. The SNMM showed capabilities for imaging of evanescent microwaves. The technique can be used, with certain limitations, as a check for testing potential micro and nano components assembled of individual scatterers, e.g., beam-splitters and interferometers. We conduct further investigations in this area.

ACKNOWLEDGMENT Two of us (R.C. and V.C.) acknowledge financial support from CONACyT Mexico, project No 123553.

REFERENCES [1]

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[2] [3]

[4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

Surface Plasmon Nanophotonics; Eds, Brongersma Mark L.; Kik, Pieter G.; Series in Optical Sciences; Springer-Verlag: Berlin, GR, 2007, Vol. 131, 271. Plasmonic Nanoguides and Circuits, Ed, Sergey I. Bozhevolnyi. Pan Stanford Publishing, World Scientific, Singapure, 2008, Vol 1, 452. Near-Field Optics and Surface Plasmon Polaritons, Eds. Kawata, Satoshi; Ohtsu, Motoichi; Irie, Masahiro; Series: Topics in Applied Physics, Springer-Verlag: Berlin, GR, 2001, Vol 10, 210. V.Coello, Surf Rev and Lett, 2008, Vol 15, 867-879. Maier, Stefan Alexander. Plasmonics: Fundamentals and Applications; SpringerVerlag: Berlin, GR, 2007, XXV, 223. Bozhevolnyi, SI; Coello, V; Phys. Rev, B. 1998, Vol 58, 10899-10910. Bozhevolnyi, SI; Volkov, VS. Opt. Commun, 2001, Vol 198, 241-245. Søndergaard, T; Bozhevolnyi, SI; Phys Rev, B. 2003, Vol 67, 165405-165412. Coello, V; Søndergaard, T; Bozhevolnyi, SI. Opt. Comm, 2004, Vol 240, 345-350. Ash, E; Nicholls, G. Nature, 1972, Vol 237, 510- 512. Wei, T; Xiang, XD; Wallace-Freedman, WG; Schultz, PG. Appl. Phys. Lett, 1996, Vol 68, 3506-3508. Vlahacos, CP; Black, RC; Anlage, SM; Amar, A; Wellstood, FC. Appl. Phys. Lett, 1996, Vol 69, 3272-3274. Coello, V; Cortes, R; Villagomez, R; Lopez, R; Martinez, C. Rev. Mex. de Fís, 2005, Vol 51, 426-430.

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[14] Raether, H. Surface Plasmons, Springer Tracts in Modern Physics, 1998, Springer, Berlin, GR, Vol. 111, 136. [15] de Fornel, F. Evanescent Waves: From Newtonian Optics to Atomic Optics; SpringerVerlag: Berlin, GR, 2001, I, 282. [16] Drezet, A; Hohenau, A; Stepanov, AL; Ditlbacher, H; Steinberger, B; Aussenegg, F; Leitner, A; Krenn, J. Plasmonics, 2006, Vol 1, 141-145. [17] Kim, DS; Hohng, SC; Malyarchuk, V; Yoon, YC; Ahn, YH; Yee, KJ; Park, JW; Kim, J; Park, QH; Lienau, C. Phys. Rev. Lett, 2003, Vol 91, 143901-143904. [18] Devaux, E; Ebbesen, TW; Weeber, JC; Dereux, A. Appl. Phys. Lett, 2003, Vol 83, 4936-4939. [19] Radko, IP; Bozhevolnyi, SI; Evlyukhin, AB; Boltasseva, A. Optics Express, 2008, Vol 15, 6576-6582. [20] Radko, IP; Evlyukhin, AB; Boltasseva, A; Bozhevolnyi, SI. Optics Express, 2008, Vol. 16, 3924-3930. [21] Coello, V; Beermann, J; Bozhevolnyi, S. physica status solidi (c), 2003, vol.0, 30703074. [22] Bozhevolnyi, SI; Beermann, J; Coello, V. Phys. Rev. Lett, 2003, Vol 90, 197403197406. [23] Vohnsen, B; Bozhevolnyi, SI. J. Microscopy, 2001, Vol 202, 244-249. [24] Bozhevolnyi, SI; Volkov, VS; Leosson, K. Phys. Rev. Lett, 2002, Vol 89, 186801186804. [25] Shchegrov, AV; Novikov, IV; Maradudin, AA. Phys. Rev. Lett, 1997, Vol 78, 42694272. [26] Bozhevolnyi, SI; Coello, V. Phys. Rev. B, 1998, Vol. 58, 10899-10910. [27] John, S; in Sheng Scattering, P. and Localization of Clasical Waves in Random Media, World Scientific, Singapore, 1990, 1.

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In: Plasmons: Theory and Applications Editor: Kristina N. Helsey, pp.237-246

ISBN: 978-1-61761-306-7 © 2011 Nova Science Publishers, Inc.

Chapter 10

APPLICATION OF LOCALIZED SURFACE PLASMONS TO STUDY MORPHOLOGICAL CHANGES IN METAL NANOPARTICLES T. A. Vartanyan1*, N. B. Leonov2 and S.G. Przhibelskii3 Center of Information Optical Technologies, St. Petersburg State University of Information Technologies, Mechanics and Optics, St. Petersburg, Russian Federation

ABSTRACT

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All characteristics of surface plasmons localized in small metal particles are known to depend strongly on the particles size and shape. In the size regime 5 to 50 nm these dependencies approximately separate. The form of the nanoparticle is almost solely responsible for the spectral position of the plasmon resonance, while the particle size defines its strength. Although this separation is only valid in the quasistatic limit and disregards scattering and other radiation effects, it leads to many useful applications. In particular, this approximation was used to extract information about the homogeneous widths of localized surface plasmons hidden under huge inhomogeneous broadening of as grown ensembles of supported metal nanoparticles. Quasistatic approximation is also instrumental when one decides to go beyond the plane statement of the existence of correlations between the optical spectra of the particles ensembles and their morphology and to draw some quantitative conclusions. In our recent study, the results of which are presented in this chapter along with a short overview of the results of others, we employ the optical absorption spectroscopy in conjunction with the simple theory based on the quasistatic approximation to characterize several new mechanism of the morphological changes that the nanoparticles undergo in the course of the light and heat treatments. In particular, we report on the reversible changes of the particles shape in the cycles of heating and cooling as well as the possibility to speed up these changes via illumination.

*

Corresponding author: E-mail address: [email protected]

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1. INTRODUCTION Surface plasmons localized in small metal particles have attracted a lot of attention recently. The reasons for that are manifold. First, being collective electronic excitations they couple strongly with optical fields. Hence, on resonance, absorption cross section of a metal nanoparticle exceeds the geometrical cross section by orders of magnitude. Second, the electrical field in the particle interior as well as on its surface is largely enhanced. This enhancement leads to the enhancement of all liner and, to a still larger amount, nonlinear effects in the particles themselves and in their surroundings. Third, the spatial extent of the enhanced optical fields outside the particle is reduced as compared to the wavelength of the light of the same frequency in free space. Thus, diffraction limit on the localization of optical fields may be overridden. It is not our purpose here to mention all applications of localized surface plasmons based on the properties specified above. Instead, we concentrate on the opportunity to study the morphology of island metal films that is provided by the localized plasmon excitations in metal nanoparticles. This opportunity is related to the dependence of the spectroscopic characteristics of plasmon bands on the size and shape of the nanoparticles that constitute the island metal film. In the size regime 5 to 50 nm the resonance frequencies of localized surface plasmons are almost independent of the particle volume being solely determined by the particle shape. The volume of the particle, in its turn, determines the overall strength of the resonance.

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2. QUASISTATIC APPROXIMATION The extinction cross section of a small metal particle depends on its size as well as on its shape. There are two important simplifications of this dependence for particles in the size range regime 5 to 50 nanometers. Firstly, for such particle sizes the quasistatic approximation can be applied, and the optical extinction is mainly due to absorption, the absorption cross section being directly proportional to the particle volume [2]. Secondly, the spectral dependence of the absorption cross section is determined only by the particle shape and is independent of its volume. In some cases the particles can be described as ellipsoids with three main axes a, b, and c. Then each particle possesses three resonance frequencies, corresponding to plasmon excitations parallel to these axes. The values of these resonance frequencies depend solely on the axial ratios a:b:c. Let Ω be one of such resonance frequencies and denote the absorption cross section of a particle with the resonance frequency Ω and volume V at any particular frequency ω as σ(ω,Ω). If we assume that the particle is in vacuum and that the particle material may be characterized by a frequency dependent complex dielectric permittivity ε=ε1+iε2, the quasistatic formula for σ(ω,Ω) reads as

 ( , ) 

1  1   2   , V c 1   1  2   2 2   2

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(1)

Application of Localized Surface Plasmons to Study Morphological Changes…

239

where c is the speed of light. For Drude-like metals with bulk plasma frequency ωp and damping constant γ, the dielectric function has the explicit analytical form

 p2  ( )  1   (  i )

(2)

and (1) simplifies to

 ( , ) 

V p2 c



2 2

 2



2

  2 2

(3)

It is important to note that in this case both the width and the amplitude of a Lorentzianlike spectral band of the surface plasmon resonance localized in the ellipsoidal nanoparticle are independent of the particle shape, which influences solely on the spectral position of the resonance frequency Ω. In the real world metal nanoparticles are usually completely or partly embedded in a dielectric host with a dielectric permittivity εm and their own dielectric permittivity deviates more or less from that of a Drude-metal. Hence, the width and the amplitude of the plasmon band become frequency dependent. Nevertheless, according to the sum rule [9] the integral over frequency of the optical absorption of small particles is independent of the particle shape.

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3. MORPHOLOGY OF METAL ISLAND FILMS AND INHOMOGENEOUS BROADENING OF THE PLASMON BANDS The common method of preparation of supported metal nanoparticles is through the Volmer-Weber growth mode when the atoms of a thermal atomic beam are deposited on a dielectric substrate. Then, nanoparticles appear due to the surface diffusion followed by nucleation and growth of the atomic clusters. Nanoparticles produced in this way have rather broad size and shape distributions. As the resonance frequency of the localized plasmon band depends strongly on the particle shape, the optical absorption spectrum of the particle ensemble is broadened as well. This inhomogeneous broadening usually dominates over the intrinsic widths of the plasmon bands. A well known spectral hole burning technique was adapted to extract information about the homogeneous widths of the localized surface plasmons hidden under huge inhomogeneous broadening of as grown ensembles of supported metal nanoparticles [16, 7]. The theoretical foundation of this successful method is based on the quasistatic approximation [17]. For a recent review see [12].

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4. HEAT TREATMENT INDUCED REVERSIBLE SHAPE CHANGES OF METAL NANOPARTICLES The heat treatment of the metal nanoparticles ensembles is well documented in the literature. In particular, in the case of nanoparticles obtained on dielectric surfaces by means of the self-organization of atoms adsorbed from a vapor phase, the shapes of the particles are different and vary in time because all of them are far from the equilibrium. As a uniform and stable form distribution is desirable in the most of applications, it is tempting to heat the particle ensembles to speed up the transition to the equilibrium shape. Indeed, the shapes vary more rapidly when the substrate is heated. These facts are well known and reported in detail in the papers devoted to electron microscopy investigations, atomic force microscopy data, and the optical extinction spectra of metal island films [21]. It is a common place to treat the shape change during the heating as an irreversible transition to the equilibrium state. The role of heating itself is simply to accelerate the transition process. On the other hand, there is no doubt that the equilibrium shape of nanoparticles depends on the temperature [8]. As far as we know, experimental results concerning this dependence are scare and the process itself has not yet been studied systematically. In what follows we report on our observations of the reversible changes in the optical extinction spectra of silver and sodium films on dielectric substrates under repeated cyclic variations of their temperatures. Due to the relation between the optical spectra of the metal island films and the shapes of the nanoparticles that comprise the film we conclude that the reversible changes of the particle shapes have been observed. The experiments were performed with silver and sodium films. The sodium films were obtained in the sealed-off evacuated cells with quartz, sapphire, or glass windows. The optical spectra of the films were taken through the transparent windows of the cells. The silver films were obtained by thermal sputtering in a vacuum. The substrates were maintained at room temperature during the growth of the films. The optical spectra of the silver films were taken in air after the extraction of the films from the vacuum chamber. The films formed under the indicated conditions are metastable. The kinetics of their morphology, which can be followed up by recording the optical extinction spectra, is complex and long-term. The annealing of the films accelerates the processes resulting in the stabilization of their structure. The sodium and silver films were annealed at different temperatures for different time intervals in a vacuum. Annealing was followed by fast cooling of the films and aging at room temperature. We report here on the regular and reversible changes in the extinction spectra upon the heat treatment of the metal island films and connect them with the shape changes of the nanoparticles that comprise those films. Figure 1 plots the optical extinction spectra of a silver film just after the deposition (curve 1) and in the course of the heat treatments alternated with aging at room temperature (curves 2 to 5). Curve 2 was obtained after annealing in a vacuum at 200°C for 100 minutes and fast cooling to room temperature. The observed narrowing of the localized surface plasmon band and the blue shift of the extinction spectrum maximum is connected with the change of the nanoparticles shape. Being flat just after the deposition they acquire more spherical form during the annealing process. Quasistatic approximation corroborates this direction of the spectral shift. The third spectrum (curve 3) was measured after aging at room temperature for 24 hours. Contrary to the expectation that the spectrum evolution should proceed in the same

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direction, may be, much slowly because of the reduced temperature, the extinction maximum undergoes now a red shift. Moreover, when the annealing process was repeated blue shift of the localized surface plasmon band was observed again (curve 4). Similarity of the positions and widths of the curves 2 and 4 suggests the conclusion that these spectra correspond to the equilibrium shape of the nanoparticles at 200°C that is frost-bound by rapid cooling. The following room temperature kinetics of the extinction spectrum at the aging step is due to the evolution of the nanoparticles to another equilibrium shape that corresponds to the lower temperature. Curve 5 in the figure 1 taken 1 hour after the end of the second annealing process shows that the transition to the equilibrium shape corresponding to room temperature is much faster than it may be expected. Indeed, the localized plasmon band represented by curve 5 is very similar to curve 3 obtained after aging during 24 hour. It shows almost complete restoration of the spectral position of the extinction spectra maximum as well as its absolute value. That means that the particles simply change their shapes rather than loose appreciable amount of their material. The cycles of changes in the shape of the spectrum under the heating and cooling of the film were repeated many times. In all cases, the almost complete reversibility of the changes in the extinction spectrum was observed, indicating a direct relation of the shapes of the islands with the temperature of the substrate. 0,6

3 5

1

0,5

Optical density

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4 2

0,4

0,3

0,2

0,1 300

400

500

600

700

wavelength, nm

Figure 1. Extinction spectra of a silver film measured in the course of the repeated cycles of the heat treatment followed by aging: 1 – just after the deposition, 2 – after annealing in a vacuum at 200°C for 100 minutes and fast cooling to room temperature, 3 – after aging at room temperature for 24 hours, 4 – after second annealing in a vacuum at 200°C for 60 minutes and fast cooling to room temperature, 5 – after aging at room temperature for 1 hour.

Of course, the temperature rise during the annealing step is too low to cause any appreciable evaporation of silver particles. The observed reduction of the spectrally integrated extinction (as a matter of fact, both the width and the amplitude of the localized surface plasmon bands represented by curves 2 and 4 are smaller than that of the bands represented by curves 3 and 5) is due to the peculiarities of the optical properties of silver. Indeed, Drude

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model that ensures the preservation of the spectrally integrated extinction for an ellipsoidal nanoparticle irrespective of the ratios of its axes does not hold for silver in the relevant spectral domain because of the interband transition. We conclude this section with the statement that the measurements of the optical extinction spectra in the course of the heat treatment of a metal island film enable us to observe reversible changes in the shape of the silver nanoparticles deposited on sapphire surface. Similar behavior was found for sodium island films that demonstrate even more intriguing behavior under illumination which is described in the following section.

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5. LIGHT TREATMENT OF THE METAL ISLAND FILMS Shaping nanoparticles and their optical spectra with photons is a long standing problem. The thermal action of light beams derived from the pulsed laser sources was widely employed to this purpose [22, 13]. Although non-thermal light-induced processes on metal surfaces are rather inefficient because of extremely fast decay times of the electronic excitations in the presence of a metal, they too present a fundamental as well as practical interest and were studied as well. In 1984 non-thermal photodetachment of metal atoms from the metal surface was observed for the first time [1]. More details including the temperature dependence and the resonance enhancement of the photodetachment process about 500 nm were given in [3, 4]. This resonance behavior of the photodetachment cross section was further studied and explained by the collective electronic excitations localized in the metal particles comprising the island film – localized surface plasmons - in 1988 [10, 11], while the role of the singleelectron resonances discovered theoretically in the same spectral region [14] is not yet clarified. An unequivocal experimental evidence of the connection between the spectral position of the resonance in the photodetachment cross section and the island film morphology was given in [20]. In this publication the use of the non-thermal photodesorption process for size manipulation of metal nanoparticles was suggested and demonstrated. Extensive research in the nature of the photodetachment process [5, 6] enables us to foresee another photo-induced non-thermal process at the metal surface, namely, the photo-induced diffusion [15]. In what follows we describe how the surface plasmon resonances localized in sodium nanoparticles were employed to prove the existence of the photo-induced diffusion. Figure 2 plots the optical extinction spectrum of the sodium island film deposited on the sapphire substrate and annealed at 50°C for 10 minutes in a vacuum. Annealing was followed by fast cooling of the films. Then, the film was aged for an hour at room temperature. To study the light action on the film morphology one half of the film was illuminated while the other half was kept in the dark in the course of the aging process. The light beam used for illumination was derived from a cw diode laser operating at the wavelength of 810 nm. The power density at the laser spot on the island metal film was 4 mW cm-2. The extinction spectra of both halves of the film were measured and the results are displayed in Figure 3 in the form of the difference between the optical density of the aged film and that of the film just after annealing. Curve 1 represents the excess of the optical density in the dark region as compared to the state before the aging process while curve 2 represents the excess of the optical density in the illuminated region as compared to its initial state (the minor difference between the initial states in the dark and illuminated regions were accounted for).

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0,6

Optical density

0,5

0,4

1

0,3

0,2

400

600

800

1000

wavelength, nm Figure 2. Extinction spectrum of a sodium film measured just after the end of annealing at 50°C for 10 minutes and fast cooling to room temperature

Optical density difference

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0,04

0,02

1

0,00

-0,02

-0,04 400

2

500

600

700

800

900

1000

1100

wavelength, nm Figure 3. Extinction differences due to the aging process for 1 hour in a vacuum: 1 - in the dark and 2 – under illumination at the wavelength of 810 nm

The overall behavior of the sodium film is similar to that of the silver film described above. The curve 1 in the Figure 3 obtained in the dark region represents the red shift of the extinction maximum. That is, the direction of the shift is in accord with the curves 2 and 3 in the Figure 1 corresponding to silver films. A new and very interesting phenomenon manifests

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itself by the curve 2 in the Figure 3. The shape changes of the nanoparticles that were observed in the dark are accelerated by illumination. It is seen that the integral of the difference extinction spectrum is much smaller than a change in extinction. For this reason, it can be assumed that the light-induced change in the volume of nanoparticles is insignificant and the observed effect is attributed to a change in their shape.

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6. DISCUSSION OF THE RESULTS The optical extinction spectra of metal nanoparticles are determined primarily by their shapes. The influence of the nanoparticles size on the extinction spectrum is much weaker than the influence of the shape. Hence, the extinction spectra of the ensembles of particles are determined by the shape distribution of the particles. For this reason, the revealed reversible changes in the extinction spectra of silver and sodium films with the variation of their temperature can be attributed to changes in the shapes of individual islands. Other possible mechanisms, which are associated with the material transport between the particles or the motion of particles themselves on the substrate, can be rejected in view of the transmission electron microscopy data. It is seen on the microphotographs [19] of our annealed silver films that the nanoparticles with a mean diameter of 14 nm are well separated. Moreover, all of the changes associated with the mass transport on the substrate should be irreversible while we restrict our discussion to the reversible changes in the nanoparticles ensembles. The irreversible cumulative effects were observed at higher temperatures or intensities of illumination than those used in the experiments presented above. Theoretical description of the shape changes of nanoparticles in not well developed yet. Numerical simulations [8] indicates that the rate of the relaxation of the shapes of the crystal nanoparticles owing to surface mass transport is limited not by the displacement of atoms through terraces, which is a very fast process, but by the escape of atoms from relatively stable positions near the steps and the attachment of diffusing atoms at the positions corresponding to the final equilibrium shapes. The activation energy of the escape of the atoms from the positions attached to the steps to a terrace is obviously lower than the evaporation heat, but can be comparable with the latter. The activation energy of the incorporation of atoms into stable positions can be low, but the incorporation process can be very long owing to the complexity of the path of the assembly of the stable final shapes of the nanoparticles. In the light of the above discussion, the observed light-induced acceleration of the relaxation of the shapes of sodium nanoparticles is explained by the fact that photons trigger the mechanism of surface diffusion by the detachment of atoms from the steps and the transfer of them to terraces. It is assumed that light excites electrons localized near the atoms situated on the surface at irregular positions. The energy of the electronic excitation is converted then into the potential energy of the atom displaced from its original position. A similar, but still less probable process results in the separation of atoms from the surface, which is manifested as the photodetachment process [6]. In the absence of illumination, the attachment of atoms diffusing on the terraces to the steps decelerates the relaxation of the shape, and nanoparticles are “frozen” in metastable shapes, because the mean thermal energy is much lower than the energy of the separation of an atom from a step. Under illumination

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Application of Localized Surface Plasmons to Study Morphological Changes…

245

the energy of the absorbed photon may be used to facilitate the separation of an atom from the step. In the case of sodium the energy supplied by a photon is sufficient to initiate separation. The efficiency of the light effect can be estimated as follows. We take as a rough estimate of the absorption cross section of the atoms attached to the steps a molecular value of 10–16 cm–2. The quantum efficiency of the separation of the atom from the step is taken to be 0.01. Then the frequency of the photo-stimulated events of the separation of the atom from the step for the radiation intensity corresponding to a photon flux of 1016 cm–2s–1 is 0.01 s–1. This value is equal to the frequency of the same process activated thermally at room temperature provided the activation energy is estimated by 0.8 eV (this last estimate corresponds to the known evaporation heat of sodium equal to 1.14 eV). According to these estimates, light can noticeably affect the rate of the relaxation of the shape of the sodium nanoparticles. The evaporation heat of silver is larger than that for sodium. For this reason, high energies of photons are required to transfer attached silver atoms from steps to terraces. This circumstance can explain why illumination did not accelerate the relaxation of the shapes of the silver nanoparticles in our experiments.

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CONCLUSION We described general principals of employing the localized surface plasmons to study morphological changes of island metal films and applied the dependence of the surface plasmon resonance on the shape of the nanoparticle in which it is localized to reveal and explore two new effects. The first effect is the reversible change in the shapes of the supported metal nanoparticles with a variation of the temperature of the substrate. The second effect is the light-induced acceleration of the relaxation of the shapes of nanoparticles. The latter effect is due to the non-thermal light-induced diffusion of the own atoms along the metal surface. It is near of kin of the non-thermal photodetachment process [5, 6] and other non-thermal photo-excited surface processes [18]. The authors are grateful to V.V. Khromov for useful advices and stimulating discussions of the results. This work was done in the framework of two State contracts for scientific researches #02.740.11.0211 and #02.740.11.0536 and supported in part by the Russian Foundation for Basic Research (project no. 08-02-00695).

REFERENCES [1] [2] [3] [4]

Abramova, IN; Aleksandrov, EB; Bonch-Bruevich, AM; Khromov, VV. JETP Letters, 1984, 39, 203-205. Bohren, CF; Huffman, DR. Absorption and Scattering of Light by Small Particles, Wiley, New York, 1983. Bonch-Bruevich, AM; Maksimov, Yu, N; Khromov, VV. Sov. Phys. JETP, 1987, 65, 161-164. Bonch-Bruevich, AM; Vartanyan, TA; Maksimov, Yu,N; Przhibelskii, SG; Khromov, VV. Sov. Phys. JETP, 1990, 70, 993-996.

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T. A. Vartanyan, N. B. Leonov and S.G. Przhibelskii

Bonch-Bruevich, AM; Vartanyan, TA; Maksimov, Yu,N; Przhibel’skii, SG; Khromov, VV. Surf. Sci., 1994, 307-309, part A, 350-354. [6] Bonch-Bruevich, AM; Vartanyan, TA; Przhibel’skii, SG; Khromov, VV. Phys. Usp., 1998, 41, 831-837. [7] Bosbach, J; Hendrich, C; Stietz, F; Vartanyan, T; Träger, F. Phys. Rev. Lett., 2002, 89, 257404. [8] Combe, N; Jensen, P; Pimpinelli, A. Phys. Rev. Lett., 2000, 85, 110-113. [9] Fuchs, R. Phys. Rev. B., 1975, 11, 1732-1740. [10] Hoheisel, W; Jungmann, K; Vollmer, M; Weidenauer, R; Träger, F. Phys. Rev. Lett., 1988, 60, 1649 -1652. [11] Hoheisel, W; Vollmer, M; Träger, F; Phys. Rev. B., 1993, 48, 17463-17476. [12] Hubenthal, F. Prog. Surf. Sci., 2007, 82, 378-387. [13] Hubenthal, F. Eur. J. Phys., 2009, 30, S49-S61. [14] Jacob, T; Martin, D; Stietz, F; Träger, F; Fricke, B. Phys. Rev. B., 2002, 66, 233409. [15] Leonov, NB; Przhibel'skii, SG; Vartanyan, TA. JETP Letters, 2010, 91, 125-128. [16] Stietz, F; Bosbach, J; Wenzel, T; Vartanyan, TA; Goldmann, A; Träger, F. Phys. Rev. Lett., 2000, 84, 5644-5647. [17] Vartanyan, T; Bosbach, J; Stietz, F; Träger, F. Appl. Phys. B., 2001, 73, 391-399. [18] Vartanyan, TA; Przhibelskii, SG; Khromov, VV. In New trends in quantum coherence and nonlinear optics (Horizons in World Physics, Volume 263), Nova Science Publishers, N.Y., Edited by Drampyan, R., 2009, 245-263. [19] Vartanyan, TA; Przhibel’skii, SG; Leonov, NB; Khromov, VV. Opt. Spectrosc., 2009, 106, 697-700. [20] Vollmer, M; Weidenauer, R; Hoheisel, W; Schulte, U; Träger, F. Phys. Rev. B., 1989, 40, 12509-12512. [21] Warmack, RJ; Humphrey, SL. Phys. Rev. B., 1986, 34, 2246-2252. [22] Wenzel, T; Bosbach, J; Goldmann, A; Stietz, F; Träger, F; Appl. Phys. B., 1999, 69, 513-517.

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[5]

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In: Plasmons: Theory and Applications Editor: Kristina N. Helsey, pp.247-265

ISBN: 978-1-61761-306-7 c 2011 Nova Science Publishers, Inc.

Chapter 11

S URFACE P LASMON R ESONANCE – A PPLICATIONS IN N ANOPARTICLE D ETECTION FROM A L IQUID M ATRIX U SING THE M AXIMUM E NTROPY M ETHOD

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Jarkko J. Saarinen1∗, Erik M. Vartiainen2, and Kai-Erik Peiponen3 1 Abo Akademi University, Center for Functional Materials, Laboratory of Paper Coating and Converting, Porthansgatan 3, FI-20500 Abo/Turku, Finland 2 Lappeenranta University of Technology, Department of Physics, P.O. Box 20, FI-53850 Lappeenranta, Finland 3 University of Eastern Finland, Department of Physics and Mathematics, P.O Box 111, FI-80101 Joensuu, Finland

PACS 78.20.Ci, 73.20.Mf, 42.30.Rx, 78.67.Bf, 02.70.-c Keywords: Optical constants, Surface plasmons, Phase retrieval (optics), Optical properties of nanoparticles, Computational techniques mathematics

1.

Introduction

Light interaction with a medium depends on the wavelength of the light and the structure of the medium. In the UV-visible spectral range the interaction of the electromagnetic field with a homogenous medium, which is either insulator, semiconductor or conductor, typically occurs with the electron system of the medium, which may be in solid, liquid or gaseous state. The interaction can be fully described with the aid of light absorption and dispersion in the medium. The situation becomes much more complicated when we wish to study light interaction with a binary- or multiphase system. The binary system can be, for instance, a liquid matrix that contains either macroscopic solid particles (suspension) or nanoparticles (colloid). In such a case, in addition to the light absorption and dispersion one has to deal with light scattering, which can be characterized using the models of Mie and ∗

E-mail address: [email protected]

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Rayleigh [1] for spherical and small inclusions, respectively. In this chapter we deal with the optical properties of a binary system, which is a colloid. The study of optical properties of colloids has a rather long history. Indeed, for instance, Maxwell Garnett (MG) studied the colors of metallic solutions already more than a hundred years ago, and presented an elegant theory to predict the optical properties of effective media [2, 3], a theory that is even nowadays widely used in the description of optical properties of effective media. The investigation of metal nanoparticles in liquid matrix has been a popular field for a long time, and different physical laws governing colloidal particles have been described in the literature [4]. The optical properties of a colloid depend on the size, shape and structure of the nanoparticles [5, 6, 7, 8, 9], their complex refractive index and thermodynamic condition of the ambient liquid [10, 11]. These all have a role in the color of the colloid, which is subject to change. This color is due to surface plasmon resonance (SPR) of the incident light with the metal nanoparticle. Visible light, which is incident on a metal nanoparticle, couples with the plasma oscillation of the electrons that are confined in the nanoparticle. The color and also scattering of light is usually measured by recording light transmittance of the colloid. Much of the research work has been on fundamental optical properties of nanoparticles in liquid environments, but also on applications as nanoprobes and imaging in medicine and biology [10, 12, 13, 14, 15, 16]. Nanotoxicology of liquid food is also an issue that has been recently raised [17]. The SPR can be utilized by another way as described above where metal nanoparticles are embedded in a liquid matrix. Indeed, one can construct an active system namely a sensor for monitoring of minute change of refractive index of gases or liquids [18, 19, 20, 21, 22]. In the past such sensors have typically utilized a prism with one face coated with a thin metal film such as gold or silver. The prism face with metal coated film is in contact with the liquid to be measured. A traditional method is to use a laser or a LED as a light source, and a polarizer to produce TM-polarized probe light that is needed to excite surface plasmons. Either by changing the angle of incidence or by focusing the incident light beam it is possible to obtain a dip in the reflectance due to the SPR. As a detector one can use a photodiode, array of photodiodes or a CCD-camera. Different kinds of SPR sensors have been constructed and are commercially available. Imaging SPR has been also realized [23, 24, 25]. In addition to angular interrogation it is possible to record a reflection spectrum with the aid of a SPR sensor [26]. In such a case one has to scan the wavelength of the incident light. This provides an interesting option because in addition to the SPR dip one is able to get spectral fingerprints of the liquid. In this chapter we deal with a case of insulating nanoparticles in liquid phase. Such a binary- or multi-phase system is important, because such inclusions appear both in nature (natural water bodies) and in medicine (drugs). The goal is to find out the optical properties of the matrix and the inclusions, and also find ways to estimate the volume fraction of the nanoinclusions. This is possible for spherical insulator inclusions embedded in the host liquid provided that the scattering of light is negligible. In other words, we are dealing with nanoparticles having dimensions much smaller than the wavelength of the incident light and the media can be described as homogeneous effective medium with effective optical constants. What we show here is that by using a wavelength scanning mode of an SPR sensor it is possible to extract the complex dielectric function ε or complex refractive index N from the measured SPR reflectance. This is possible with the aid of a chemometric method that is based on the so-called maximum entropy method

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Surface Plasmon Resonance – Applications in Nanoparticle Detection from a Liquid...249 (MEM) [27, 28] that will be described in detail. In Sec. 2. of this chapter we describe how surface plasmons can be excited at the dielectric/metal interface using prism coupling for the momentum matching, and the formulas for the SPR reflectance are presented. We will also review the basics of the effective medium theories. In Sec. 3. we present the basic principles of the MEM used for the phase retrieval of the SPR reflectance. The retrieved phase is then applied in Sec. 4. for the identification of nanoparticles having different spectral features and volume fractions in a water matrix. Finally, in Sec. 4. we present the conclusions of our work.

2.

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2.1.

Surface Plasmon Resonance from Nanoparticle Solutions Surface Plasmon Resonance

In 1902 Wood [29] observed extraordinary behavior in diffraction efficiency of metal gratings compared to the classical dielectric gratings. The physical interpretation of such anomalous behavior was initiated by Lord Rayleigh [30] and followed by Fano [31]. Later, the term plasmon was introduced by Pines [32] who studied collective energy losses in metals. The phenomenon was identified as the surface plasmon resonance at the metal film both by Otto [19] and Kretschmann & R¨ather [18] in 1968. The first method devised by Otto [19] used a thin air gap between the prism and the metal film to tunnel the evanescent electromagnetic field across the gap, and coupling light into the plasmons at the air gap/metal interface. However, a much simpler set-up was realized when the thin metal film was directly deposited on the prism, where metal itself transports the evanescent plasmon field to the dielectric/metal interface. The Otto method is highly sensitive to the air gap thickness, whereas Kretschmann’s configuration is much more robust as it is much easier to control the thickness of the deposited metal film than the thickness of the air gap. Therefore, nowadays most SPR studies are performed in the Kretschmann’s configuration shown in Fig. 1 (a). In the Kretschmann’s configuration one side of a glass prism is covered by a thin (thickness d approximately 50 nm) metal film. A prism coupling is needed for momentum matching of incident photons with the surface plasmon modes. A trapped surface mode can exist at the dielectric/metal interface, which decays exponentially into both media. These surface plasmon modes are collective oscillations of electrons on the surface of a metal film [33]. The SPR dispersion relation can be obtained by solving the electromagnetic boundary conditions at the dielectric/metal interface. By requiring continuity of the tangential components of the electric and magnetic fields we arrive in dispersion relation for SPR s ε1 (ω)ε2(ω) ˜ , (1) κsp (ω) = ω ε1 (ω) + ε2 (ω) where κ denotes the wave vector component parallel to the surface of the metal film, and ε1 (ω) and ε2 (ω) are the complex dielectric functions of the investigated sample and the metal film, respectively. Here ω ˜ = ω/c = 2π/λ0 denotes the wave number of the incident light with ω, c, and λ0 being the angular frequency, the speed, and the wavelength of light in √ vacuum. Here the square root of a complex number z is defined such that Im( z) ≥ 0, and

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250

Jarkko J. Saarinen, Erik M. Vartiainen and Kai-Erik Peiponen (b)

(a) investigated sample κsp (ω) silver film

ε1 (ω) ε2 (ω) ε3 (ω)

BK7 prism

ω

ω = cκ ω = cκ/n3

d

ωsp Bound SP mode

θ

κ

Figure 1. (a) A schematic picture of the SPR set-up in Kretschmann’s configuration with dielectric functions of the prism ε3 (ω), metal film ε2 (ω), and the investigated sample ε1 (ω). (b) Dispersion relation curve ω vs. κ for surface plasmons.

√ √ Re( z) ≥ 0 if Im( z) = 0. For propagating wave the wave number κ is a real number and for a metal film we have Re{ε2} < 0. Therefore, we can deduce from the dispersion relation Eq. (1) that |ε2 | > |ε1 |. The dispersion relation curve (ω vs. κ) is plotted in Fig. 1 (b): for all values of κ the surface plasmon dispersion relation do not intersect with the light line in air. Hence, it is impossible to excite surface plasmons directly from air coupling, and the prism is needed for frequency and wave vector (or momentum) matching of the surface plasmon to the frequency of the incident light.

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2.2.

SPR Reflectance in Sensing and Nanoparticle Detection

SPR reflectance was found out to be very sensitive tool of the optical properties of metal surfaces in the 1970’s [34, 35]. Gordon and Swalen [36] pioneered using SPR for monitoring changes in the refractive index at the vicinity of the metal film as they studied shifts in the SPR angle by monolayers of cadmium arachidate deposited on the metal film by Langmuir-Blodgett technique. Adding only two monolayers with a thickness of 5.36 nm was clearly detectable from the SPR reflectance that induced approximately a change of one degree in the angular resonance position. Later, SPR was shown to be a suitable tool for biomolecular sensing purposes with detection of anaesthetic gases being one of the earliest applications of the SPR in biosensing [20, 21]. The accuracy of the SPR measurement was found to be in good agreement when compared to a commercial quartz crystal microbalance (QCM). In general, there are two possibilities for exciting surface plasmon resonance. First, one can do angular scanning with a constant wavelength of the incident light (the angular scanning mode) that allows one to extract the effective refractive index of the investigated sample at a single wavelength. On the other hand, one can keep the angle of incidence constant and perform wavelength scanning (the wavelength scanning mode). The latter approach is more convenient for detection of nanoparticles in a liquid matrix and for retrieval of their optical properties as one can obtain the full spectral features of the nanoparticles from a single measurement. Recently, we have used the wavelength scanning mode for nanoparticle identification from a liquid matrix with the aid of the MEM [37, 38, 39, 40, 41].

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Surface Plasmon Resonance – Applications in Nanoparticle Detection from a Liquid...251 To analyze SPR reflectance it is convenient to split the wave vector k = κˆ κ + wzˆ of the incident light into parallel (given by κ at the xy-plane with a unit vector κ ˆ ) and perpendicular (given by w along the perpendicular unit vector zˆ) pieces compared to the surface of the metal film, which is assumed to be parallel to the xy-plane. We adopt here formalism developed by Sipe [42] to analyze reflectance from the Kretschmann’s configuration illustrated in Fig. 1 (a). In reflection the parallel wave vector component remains the same whereas the perpendicular components in different regions are given by p wi (ω) = ω ˜ 2 εi (ω) − κ2 (2) with i = 1, 2, 3 denoting the sample liquid, the metal film, and the prism. The parallel component κ = ω ˜ n3 sin θ depends on the refractive index of the prism n3 and the angle of incidence θ. From the Maxwell’s equations and the electromagnetic boundary conditions one obtains the following Fresnel coefficients for the TM-polarized light wi(ω)εj (ω) − wj (ω)εi(ω) , wi(ω)εj (ω) + wj (ω)εi(ω) 2ni (ω)nj (ω)wi(ω) tij (ω) = , wi(ω)εj (ω) + wj (ω)εi(ω)

rij (ω) =

(3)

which satisfy rij = −rji and tij tji − rij rji = 1 (note that no summation over i or j). We use the transfer matrix (T-matrix) method to propagate the electromagnetic field through the structure. First, the interface matrix for light passing from medium i to j is [42]   1 1 rij , (4) Mij = tij rij 1

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and the phase accumulation in the layer i with a propagation distance of z is   exp(iwiz) 0 . Mi (z) = 0 exp(−iwiz)

(5)

Therefore, the full T-matrix for the Kretschmann’s configuration for propagating field from region 3 (prism) to 1 (sample liquid) is given by a matrix product   1 T31T13 − R31R13 R31 , (6) T31 = M32M2(d)M21 = −R13 1 T31 where M32 and M21 propagate the fields through prims/metal and metal/sample interface, respectively, and M2 (d) propagates the field through metal film with thickness d. The complex SPR reflectivity is given by r31 =

T31(1, 2) r32 + r21 exp(2iw2d) = . T31(2, 2) 1 + r32r21 exp(2iw2d)

(7)

The observed SPR reflectance is the square of the absolute value of the complex reflectivity i.e. r32 + r21 exp(2iw2d) 2 ∗ , (8) R = r31r31 = 1 + r32r21 exp(2iw2d) where ∗ denotes complex conjugation. In the SPR measurement the amplitude of the reflectance is obtained whereas the phase information is lost. However, for nanoparticle identification it is crucial to have both the amplitude and the phase of the complex SPR reflectivity. This is obtained by the maximum entropy method as we will show in Sec. 3.

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2.3.

Jarkko J. Saarinen, Erik M. Vartiainen and Kai-Erik Peiponen

Effective Medium Theories

Effective medium theory (EMT) has been used for evaluation of the optical properties of composite for long time since the seminal work by Maxwell Garnett [2, 3] in the start of the 20th century. The basis of different EMT formulations lies in the work of Clausius and Mossotti that relates the molecular polarizability γmol to the macroscopic dielectric function ε [43]: 3 ε−1 γmol = , (9) N ε+2 where N is the average number per unit volume of molecules. This is also known as the Lorentz-Lorenz formula [44]. Maxwell Garnett (MG) extended the above formula for binary mixtures, which is followed by the MG effective medium theory

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εMG,eff (ω) − εh(ω) εi (ω) − εh (ω) = fi εeff (ω) + 2εh (ω) εi (ω) + 2εh (ω)

(10)

with εi (ω) and εh (ω) being the frequency-dependent dielectric functions of the inclusions and the host, respectively. There are some limitations with the MG formalisms. First, the MG EMT is only valid at the dilute limit i.e. the number of nanoparticles per unit volume needs to be sufficiently small. In literature volume fractions below 10 % are typically used. Secondly, single scattering is expected in the formalism that is realized in the dilute limit. Finally, the MG formalism is asymmetric: the effective dielectric constant would depend on the choice of the host medium if equal amounts of host and inclusions are intermixed together. Therefore, the MG formalism is only used at the dilute limit. A generalization of the MG formalism was presented by Bruggeman in 1935 [45] who considered the particles to be embedded into the effective medium itself. This is followed by a symmetric equation for the effective dielectric function fh

εh (ω) − εBR,eff (ω) εi (ω) − εBR,eff (ω) + fi = 0. εh (ω) + 2εBR,eff (ω) εi (ω) + 2εBR,eff (ω)

(11)

Here it is assumed that the inclusions are spherical objects – the formalisms has also been extended to take into account non-spherical shaped particles but for the sake of simplicity, we restrict ourselves to spherical particles in this chapter. With simple algebra we can extract the effective dielectric function εBR,eff (ω) for spherical nanoparticles from Eq. (11) as follows: p εBR,eff (ω) = 1/4[c(ω) + c(ω)2 + 8εh(ω)εi (ω) ], (12) c(ω) = (3fh − 1)εh(ω) + (3fi − 1)εi(ω). The effective complex refractive index Neff (ω) can be calculated from the effective complex dielectric function εeff (ω) = [Neff (ω)]2 as follows 1 neff (ω) = √ [Re{εeff } + (Re{εeff }2 + Im{εeff }2)1/2]1/2, 2 Im{εeff } . κeff (ω) = 2neff

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(13)

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3.

The Maximum Entropy Method

Phase retrieval problems arise in many branches of applied physics such as crystallography, astronomy, and optics [46]. A classical example of such a problem was given by Pauli [47] in 1933: is it possible, for a single particle moving in one dimension, to determine the wave function ψ(q) from amplitudes |ψ(q)| and |F {ψ(q)}|, where F denotes the Fourier transform? Here ψ(q) = |ψ(q)| exp[ia(q)] i.e. we only need to find the phase function a(q). However, this is not a trivial task as recently shown by Jaming [48] as there may be an infinity of Pauli partners a(q) satisfying the initial conditions for ψ(q). The general problem setting in the phase retrieval problems is as follows: find a welllocalized smooth complex-valued function f for a given modulus |f |. This is also the case for nanoparticle identification from liquid matrix by using SPR reflectance data. From reflectance measurement we get the amplitude of the reflectance but the phase information is lost. Traditionally such problems in reflection spectroscopy have been analyzed using the Kramers-Kronig type dispersion relations [49, 50, 51, 52, 53, 54], which relate the phase and the amplitude of reflectance to each other. Unfortunately, there are some serious drawbacks in the classical dispersion theory analysis. First, dispersion relations are integral relations with an integration range from zero to infinity, which require, in principle, knowledge of data on the whole spectral range. In reality the limitation to finite spectral range can be overcome with large enough frequency range covering absorption bands of the investigated system since the reflectance spectrum is most strongly affected by the resonances [50]. Secondly, the dispersion relations utilize the logarithm of the reflectance, which may lead to problems with TM-polarized light used for the SPR excitation. There may exist zeros of the reflectance, which are the diverging points of the logarithm. The complex poles in the reflectance data can be taken into account by a complex-valued Blaschke product [55] but knowledge on a whole complex function is needed at these points. Therefore we take a different approach here for the phase retrieval problem and utilize the maximum entropy method (MEM). The roots of the MEM lie in information theory, which was initiated by Shannon at Bell Labs [56], and the theory was quickly applied to statistical mechanics [57, 58]. Here entropy is a measure of the uncertainty represented by a probability distribution and the maximum entropy (ME) estimate is the least biased on the given information. This concept is well explained in an another Shannon’s paper [59] by a simple example of secrecy systems: let us consider a simple substitution cipher, which changes the order of letters in the message. Then you have 26! possibilities for different ciphers and the probability for sorting out a cipher a priori is 1/(26!). Let us assume that you receive a single letter from an English message - then the probabilities for letters are the typical values present in English words. But the situation changes drastically when more letters from the message are received. If the number of letters is large enough (say N = 50), then there is usually a single message posteriori with nearly unity probability. With a smaller value of N e.g. N = 15 there are typically many messages with comparable probability. Hence with large enough sampling one can find the most probable solution by applying the MEM, and the method has been successfully used, for example, to enhance astronomical images [60], to study reaction kinetics and protein folding [61], and to predict atmospheric CO2 content [62]. The phase retrieval of the complex SPR reflectivity is obtained by fitting accurately the

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measured SPR reflectance into the maximum entropy model: |β|2 R(ν) = |AM (ν)|2

with

AM (ν) = 1 +

M X

am exp(−i2πmν),

(14)

m=1

where ν is the normalized frequency ν = (ω − ω1 )/(ω2 − ω1 ) for the measured frequency range ω ∈ [ω1, ω2]. In Burg’s algorithm [63] the unknown ME coefficients am and β are the solutions of a Toeplitz system      2 1 C(0) C(−1) ··· C(−M ) |β|  C(1)   a1   0  C(0) · · · C(1 − M )      (15)  ..   ..  =  ..  . .. .. ..   .    . . . . .  C(M ) C(M − 1) · · ·

C(0)

aM

0

Here the values of the autocorrelation function C(m), m ≤ M are obtained from a Fourier transform of the measured reflectance R(ν) as follows: Z 1 R(ν) exp(i2πtν)dν, (16) C(t) = 0

and the complex reflectivity is given by

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r(ν) =

|β| exp[iφ(ν)] β = . AM (ν) |AM (ν)| exp[iψ(ν)]

(17)

The maximum entropy phase ψ(ν) is obtained from the autoregressive MEM and the last unknown quantity is the error phase function φ(ν). The key benefit of the MEM is that the error phase function is typically a much smoother function than the actual phase function ϕ(ν) = ψ(ν) + φ(ν) i.e. the main features of the actual phase are captured by the MEM. Therefore, the error phase function can easily be approximated by having a priori knowledge about the whole complex function at a few separate points. A polynomial fitting can be utilized to estimate the error phase φ(ν) =

L−1 X

Bk ν k ,

(18)

k=0

where the coefficients Bk are solutions of the Vandermonde system [64]      · · · ν0L−1 1 ν0 B0 φ(ν0 )    1 ν1  · · · ν1L−1    B1   φ(ν1 )   = .    ..   . .. . . .. .. ..   ..    . . . 1 νL−1 · · ·

L−1 νL−1

BL−1

(19)

φ(νL−1 )

The Vandermonde interpolation typically yields a good estimate for the error phase function, which is typically a slowly varying function. In favorable cases only two anchor points are needed and the degree of polynomial can be restricted to one. In less favorable cases we can artificially modify the spectrum to yield a better convergence between the exact and

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Surface Plasmon Resonance – Applications in Nanoparticle Detection from a Liquid...255 retrieved values. We have shown [39, 40] that a simple squeezing procedure yields a better result in the phase retrieval. The squeezing procedure is given by [64]  K R(ω1); 0 ≤ ν < 2K+1      K K+1 R(ω); 2K+1 ≤ ν ≤ 2K+1 , (20) R(ν) → R(ν; K) =      R(ω ); K+1 ≤ ν < 1 2 2K+1 where the squeezing parameter K defines the extend of the original spectrum. For example, with a value of K = 0 we have the original spectrum whereas for the value K = 1 the original spectrum is squeezed into one third of the original width at the range ν ∈ [1/3, 2/3]. The tails of the squeezed spectrum are obtained by adding constant wings at both ends of the spectrum. The squeezing procedure further linearizes the error phase spectrum.

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4.

Retrieval of Optical Properties of Nanoparticles in a Liquid Matrix

In this section we utilize the maximum entropy method for the identification and retrieval of optical properties of nanoparticles in a liquid matrix. The section is divided into three subsections. First, we utilize the MEM to retrieve the error phase function of the known pure host liquid that is assumed to be water. We have recently demonstrated an extraordinary feature of the MEM i.e. the error phase functions of the host liquid and the solution with nanoparticles are essentially the same [38, 39, 40]. Therefore, this allows us to identify nanoparticles without a priori knowledge about their optical properties if the optical properties of the host liquid are known at the investigated spectral range. We first apply the MEM to identify Maxwell Garnett nanoparticles with single or multiple resonances. Finally, the dilute limit restriction of the MG formalism is released by considering Bruggeman nanoparticles in water matrix with higher volume fractions.

4.1.

Retrieval of Refractive Index of the Host Liquid

In our simulations we will consider the Kretschmann’s configuration presented in Fig. 1 (a). We take the glass prism to be BK7 glass, whose refractive index (no absorption at the investigated frequency range) is given by n23 (ω) = 1 +

266.387 17.804 0.0150 + + , 256.237 − ω 2 76.811 − ω 2 0.0148 − ω 2

(21)

with the angular frequency ω of the incident light measured in electron volts. The complex dielectric function of the silver film is given by the classical Drude’s model available from literature [65]. The refractive index of water (host liquid) is given by Partington [66], and the absorption of water can be neglected at the investigated frequency range. The simulations are performed at an energy range of E ∈ [1.0eV − 3.0eV], which corresponds to a wavelength range of λ ∈ [413nm − 1240nm]. Figure 2 (a) shows the SPR reflectance from water with an incident angle of 70 ◦ and metal film thickness of 50 nm. A typical reflectance dip is observed around 2 eV, where

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256

Jarkko J. Saarinen, Erik M. Vartiainen and Kai-Erik Peiponen −0.6

0.9

−0.8

Error phase φh(ω) (radb

SPR reflectance R(ω)

(a) 1

0.8 0.7 0.6 0.5 0.4 0.3 0.2

(b)

−1 −1.2 −1.4 −1.6 −1.8 −2 −2.2

0.1 0 1

1.5

2

2.5

−2.4 1

3

K=1 K=20 1.2

1.4

1.6

2

2.2

2.4

2.6

2.8

3

Energy (eV)

Energy (eV) 1.81

1.8

(c)

1.345

(d)

Refractive index nh (ω)

Dielectric function εh(ω)

1.8 1.79 1.78 1.77 1.76

1.34

1.335

1.33

1.325

1.75 Exact MEM 1.74 1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

Exact MEM 3

1.32 1

Energy (eV)

1.5

2

2.5

3

Energy (eV)

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Figure 2. SPR reflectance from the pure host liquid (a) and the corresponding error phase function with two different K-values (b). The retrieved dielectric function (c) and refractive index (d) of the host liquid show a good agreement between theory and MEM values.

reflectance is drastically decreased. In SPR sensoring applications the shift of the resonance dip yields information about the change of the refractive index of the investigated sample. The sensitivity of such sensor set-up depends on the ratio of the reflectance dip shift to the full-width at half maximum (FWHM) value of the dip. This means that the broader the reflectance dip, the more difficult it is to observe small shifts in the resonance frequency. The SPR reflectance dip width can be minimized by an optimal choice of the metal film thickness or by adding an intermediate dielectric film between the glass prism and the metal film as shown by Matsubara et al. [67]. Our approach by the MEM is not based on the detection of the reflectance dip, which could be problematic in the case of multiple spectral peaks as shown later in this section. The error phase function of pure water is shown in Fig. 2 (b) with two different squeezing parameter values. The error phase shows almost linear behavior and it is used with the ME phase ψ(ω) obtained from the ME model for retrieval of the complex SPR reflectivity r31(ω) = R1/2(ω) exp[iϕ(ω)]. From Eq. (7) we can solve for the unknown Fresnel coefficient between the metal film and the sample r21(ω) =

r32(ω) − R1/2(ω) exp[iϕ(ω)] , {r32(ω)R1/2(ω) exp[iϕ(ω)] − 1} exp[2iw2(ω)d]

(22)

where the actual phase ϕ(ω) = ψ(ω) + φ(ω) is the sum of the ME phase and the error phase. Extracting w2 (ω) from both Fresnel coefficients r32(ω) and r21(ω) and setting them Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

Surface Plasmon Resonance – Applications in Nanoparticle Detection from a Liquid...257 equal, we can solve for the unknown sample dielectric function ε1 (ω) a formula   ε3 (ω) 1 + r32(ω) 1 + r21(ω) w1(ω). ε1 (ω) = w3 (ω) 1 − r32(ω) 1 − r21(ω)

(23)

Utilizing Eq. (2) for w1(ω) we obtain a quadratic equation for ε1 (ω) of the form ε1 (ω) =

o 1n B(ω) + [B 2 (ω) − 4B(ω)n23 (ω) sin2 θ]1/2 2

with B(ω) =



n3 (ω)[1 + r32(ω)][1 + r21(ω)] cos θ[1 − r32(ω)][1 − r21(ω)]

2

,

(24)

(25)

where we have used the expression w3(ω) = ω ˜ n3 (ω) cos θ. The formulas (22)–(25) can be used to retrieve the optical properties of the investigated sample. In Fig. 2 (c) and (d) we show a comparison of the retrieved MEM estimates (dots) to the exact values (lines) for the dielectric function and for the refractive index of water, respectively. Here both values are real as no absorption is assumed for water at the investigated spectral range. We obtain an excellent fit between the retrieved and exact values.

4.2.

Identification of MG Nanoparticles

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The optical properties of nanostructures have attracted much interest during the past two decades. Here we utilize the MEM to identify single and triple resonance Lorentzian nanoparticles embedded in water matrix described in a dilute limit by Maxwell Garnett formalism (10). The Lorentzian model for the multi-resonance dielectric function of embedded nanoparticles is εi (ω) = ε∞ +

X m

2 ωpm , 2 − ω 2 − iΓ ω ω0m m

(26)

where m describes the number of resonances, ε∞ is the high-frequency value, and ωpm , ω0m and Γm are the plasma frequency, the resonance frequency and the half-width of the mth peak, respectively. The used parameters for the Lorentzian lineshape nanoparticles are given in Table 1. The SPR reflectance from a single (a) and triple (b) resonance MG nanoparticle solutions are shown in Fig. 3. As noted earlier it would be difficult to identify dips caused by the SPR in the case of three resonance particles with a volume fraction of 10 %. The retrieved effective complex dielectric function of single resonance nanoparticles with three different values of the squeezing parameter K are presented in Fig. 4 (a) and (b). The solid line is the exact value and the MEM estimates are shown by dots. With a low squeezing parameter value K = 1 we observe a good fit around resonances whereas divergence from exact values occurs at both ends of the spectrum. The situation is significantly improved as the squeezing parameter is increased to a value K = 10. A slightly better agreement is found with a value K = 20, which is used throughout the rest of the calculations. From the MG effective medium theory it is straight-forward to retrieve the

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258

Jarkko J. Saarinen, Erik M. Vartiainen and Kai-Erik Peiponen Table 1. The used parameters for the simulations Angle of incidence θ 70◦ Silver film thickness d 50 nm Single resonance MG particles High frequency value ε∞ 2.0 Resonance frequency ω0 1.95 eV Plasma frequency value ωp2 0.1 eV2 Half-width Γ 0.1 eV Three resonance MG and BR particles ε∞ 2.0 High frequency value Resonance frequency 1 ω01 1.80 eV Resonance frequency 2 ω02 1.95 eV Resonance frequency 3 ω03 2.10 eV 2 2 2 Plasma frequency values ωp1 =ωp2 =ωp3 0.1 eV2 Half-widths Γ1 =Γ2 =Γ3 0.1 eV (b) 1

0.9

0.9

0.8

0.8

SPR reflectance R(ω)

SPR reflectance R(ω)

(a) 1

0.7 0.6 0.5 0.4 0.3 0.2

0.7 0.6 0.5 0.4 0.3 0.2

f=0.05 f=0.10

f=0.10 0.1 1

1.5

2

2.5

3

0.1 1

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Energy (eV)

1.5

2

2.5

3

Energy (eV)

Figure 3. The SPR reflectance from a single (a) and triple (b) resonance MG nanoparticle solutions.

optical constants of the embedded nanoparticles. These results are shown in Fig. 4 (c) and (d). A similar trend in these retrieved values is observed as with the effective dielectric functions i.e. increased squeezing parameter value reduces the error. Obviously the spectral features are well produced by the MEM procedure (dots). The deviation from the exact curves (solid lines) occurs only at the wings of the spectra. However, in practical spectroscopical techniques we are mainly interested about the resonances of the nanoparticles as the identification of nanoparticles as well as the assessment of their concentration is mainly based on the height, half-width and location of the spectral peaks. It is noteworthy here that no a priori knowledge about the nanoparticles is needed for obtaining the results in Figs. 4 (a)–(d). The single resonance case can be generalized by allowing adjacent absorption peaks for nanoparticles, which is a more realistic case e.g. when dealing with organic nanoparticles and dyes. The SPR reflectance will have four adjacent peaks of which three originate from

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Surface Plasmon Resonance – Applications in Nanoparticle Detection from a Liquid...259 (a)

Real part Re{εMG,eff (ω)}

1.83 1.82 1.81 1.8 1.79 Exact f=0.10 MEM K=1 MEM K=10 MEM K=20

1.78 1.77 1

1.2

(b)

0.05

Imaginary part Im{εMG,eff (ω)}

1.84

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

Exact f=0.10 MEM K=1 MEM K=10 MEM K=20

0.04

0.03

0.02

0.01

0

−0.01 1

3

1.5

(c)

0.6 Exact MEM K=1 MEM K=10 MEM K=20

Real part Re{εi(ω)}

2.3

Imaginary part Im{εi(ω)}

2.4

2.2 2.1 2 1.9 1.8 1.7 1

1.5

2

2

2.5

3

Energy (eV)

Energy (eV)

2.5

3

(d) Exact MEM K=1 MEM K=10 MEM K=20

0.5 0.4 0.3 0.2 0.1 0 −0.1 1

Energy (eV)

1.5

2

2.5

3

Energy (eV)

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Figure 4. The retrieved effective complex dielectric function for a single resonance MG nanoparticle solution with a volume fraction of 10 %. The retrieved dielectric function for the embedded nanoparticles are shown with the real part in (c) and the imaginary part in (d). The exact values are shown by solid lines and the MEM values with different squeezing parameter values by dots.

nanoparticle absorption and one from SPR as shown in Fig. 3 (b). With a volume fraction f = 0.10 the four spectrally separated reflectance dips have practically the same levels of minimum and identification of SPR dip may be problematic. The results of the MEM procedure are presented in Fig. 5 where the retrieved effective complex dielectric function of three resonance MG nanoparticles are shown with two different volume fractions in a dilute limit in (a) and (b). We obtain almost perfect fit between the exact values (solid lines) and the MEM estimates (dots with squeezing parameter value K = 20). Finally, the spectral properties of individual nanoparticles are calculated in Figs. 5 (c) and (d) with an excellent agreement between the exact values (solid lines) and MEM estimates (dots).

4.3.

Identification of BR Nanoparticles

The last example of the nanoparticle identification from a liquid matrix is SPR reflectance from Bruggeman nanoparticle solution where the restriction to dilute limit is removed. We use two different volume fractions of 15 % and 30 %, and consider three resonance nanoparticles. In BR formalism the system is treated as completely random media with indistinguishable constituent particles. The SPR reflectance is presented in Fig. 6 (a) showing four adjacent reflectance dips. The corresponding retrieved effective complex dielectric func-

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Jarkko J. Saarinen, Erik M. Vartiainen and Kai-Erik Peiponen (a)

(b)

0.06

Imaginary part Im{εMG,eff (ω)}

Real part Re{εMG,eff (ω)}

1.84

1.82

1.8

1.78 Exact f=0.05 MEM K=20 Exact f=0.10 MEM K=20

1.76 1

1.5

2

2.5

Exact f=0.05 MEM K=20 Exact f=0.10 MEM K=20

0.05 0.04 0.03 0.02 0.01 0 −0.01 1

3

1.5

Energy (eV)

(c)

0.7 Exact f=0.05 f=0.10

2.4

Real part Re{εi(ω)}

2.3 2.2 2.1 2 1.9 1.8 1.7

3

(d) Exact f=0.05 f=0.10

0.5 0.4 0.3 0.2 0.1 0

1.6 1.5 1

2.5

0.6

Imaginary part Im{εi(ω)}

2.5

2

Energy (eV)

1.5

2

2.5

3

−0.1 1

Energy (eV)

1.5

2

2.5

3

Energy (eV)

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Figure 5. The retrieved complex effective dielectric function of three resonance MG nanoparticle solution with a volume fraction of 5 % and 10 %. The real part (c) and imaginary part (d) of the complex dielectric function of the nanoparticles. The exact values are shown by solid lines and the MEM values by dots.

tions are shown in Figs. 6 (b) and (c) with squeezing parameter value K = 20. As in the case of MG solutions, a small deviation between the exact values (solid lines) and MEM estimates (dots) are found at the tails of the spectra. However, in the vicinity of the spectral peaks we obtain excellent agreement between exact values and MEM estimates. As a summary we can conclude that the MEM is a suitable tool for identification of both MG and BR nanoparticles in a liquid matrix. Here we have used water as the host liquid that is probably the most important and the most typically used liquid in SPR devices, especially in biological applications. Nevertheless, the approach described here is not restricted to water only but can be used to arbitrary host liquids as long as the optical properties of the liquids are known. There is also an alternative route for solutions with unknown host liquid as one can utilize the classical MEM approach with anchor points for the evaluation of the error phase.

5.

Conclusion

In this chapter we have shown how to extract optical properties and how to identify nanoparticles from water solution. The basic principles of the SPR reflectance measurements were described that has found applications in, for example, biosensing. For nanoparticle identi-

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Surface Plasmon Resonance – Applications in Nanoparticle Detection from a Liquid...261 (a) 1

SPR reflectance R(ω)

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

f=0.15 f=0.30

0.1 1

1.5

2

2.5

3

Energy (eV) (c)

(b)

0.18

Exact f=0.15 MEM K=20 Exact f=0.30 MEM K=20

1.95

Imaginary part Im{εBR,eff (ω)}

Real part Re{εBR,eff (ω)}

2

1.9

1.85

1.8

1.75

1.7 1

1.5

2

2.5

3

Exact f=0.15 MEM K=1 Exact f=0.30 MEM K=20

0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 1

Energy (eV)

1.5

2

2.5

3

Energy (eV)

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Figure 6. The SPR reflectance from a three resonance BR nanoparticle solution with a volume fraction of 15 % and 30 % (a) showing multiple adjacent peaks. The retrieved complex dielectric function with real part (b) and imaginary part (c). The exact values are shown by solid lines and the MEM values by dots.

fication from SPR reflectance a phase retrieval procedure is needed as the reflectance measurement only yields information about the amplitude of the reflectance but the phase information is lost. Instead of using the conventional Kramers-Kronig type dispersion relation we utilize the maximum entropy method that has been successfully applied in optical phase retrieval problems for the past few decades. The MEM has an extraordinary feature with SPR reflectance from the nanoparticle solution as the error phase function can be solved from the pure host liquid. This allows us to retrieve the optical properties and to identify nanoparticles from the solution without a need for a priori knowledge about the properties of the investigated nanoparticles. We have verified the potential of the SPR-MEM procedure by investigating SPR reflectance from single and multiple resonance Lorentzian nanoparticles with different volume fractions in solution. We have shown that it is possible to get an excellent agreement between the exact values and the MEM estimates obtained from the MEM procedure. We have also examined the effects of squeezing, where the original spectrum is squeezed into a narrower range. This will result in a better agreement between exact values and the MEM estimates. However, a small deviation is observed at the end of the spectra even with sufficient squeezing but this does not play any role in practical spectroscopy, where the spectral peaks are of interest and those are well captured by the MEM. We wish to emphasize that the procedure described here is not limited to water host

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Jarkko J. Saarinen, Erik M. Vartiainen and Kai-Erik Peiponen

liquid or to Lorentzian nanoparticles. The effective complex dielectric function of the investigated sample can always be retrieved from the SPR reflectance if the optical properties of the arbitrary host liquid are known. For individual nanoparticle identification a model for the effective medium is needed. Furthermore, the MEM is also suitable for arbitrary shaped nanoparticles. It is believed that nanoparticle identification will become a more important topic in the future. Currently a large number of products containing nanoparticles are entering into the market but very little is known about their biological or environmental impacts, especially in the long term.

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[7] Daniel, M.-C.; Astruc, D. Chem. Rev. 2004, 104, 293–346. [8] Scaffardi, L. B. Tocho, J. O. Nanotechnology 2006, 17, 1309–1315. [9] Kubo, S. ; Diaz, A. ; Tang, Y. ; Mayer, T. S.; Khoo, I. C.; Mallouk, T. E. Nano Lett. 2007, 7, 3418–3423. [10] Haes, A. J.; Van Duyne, R. P. Expert Rev. Mol. Diagn. 2004, 4, 527–537. [11] Palpant, B. ; Guillet, Y. ; Rashidi-Huyeh, M. ; Prot, D. Gold Bull. 2008, 41, 105–115. [12] Han, M. ; Gao, X. ; Su, J. Z.; Nie, S. Nat. Biotechnol. 2001, 19, 631–635. [13] Alivisatos, P. Nat. Biotechnol. 2004, 22, 47–52. [14] Jain, P. K.; Lee, K. S.; El-Sayed, I. H.; El-Sayed, M. A. J. Phys. Chem. B 2006, 110, 7238–7248. [15] Urban, G. A. Meas. Sci. Technol. 2009, 20, 012001. [16] Scampicchio, M. ; Arecchi, A. ; Mannino, S. Nanotechnology 2009, 20, 135501. [17] Weiss, J. ; Takhistov, P. ; McClements, D. J. J. Food Sci. 2006, 71, R107–R116. [18] Kretschmann, E. ; R¨ather, H. Z. Naturforsch. A 1971, 23, 2135–2136. Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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[31] Fano, U. J. Opt. Soc. Am. 1941, 31, 213–222. [32] Pines, D. Rev. Mod. Phys. 1956, 28, 184–198. [33] R¨ather, H. , Surface plasmons on smooth and rough surfaces and on gratings ; Springer: Berlin, 1988. [34] Kretschmann, E. Z. Phys. 1971, 241, 313–324. [35] Barker, Jr., A. S. Phys. Rev. B 1973, 8, 5418–5426. [36] Gordon II, J. Swalen, J. Opt. Commun. 1977, 22, 374–376. [37] Saarinen, J. J.; Peiponen, K.-E.; Vartiainen, E. M. Appl. Spectrosc. 2003, 57, 288– 292. [38] Saarinen, J. J.; Vartiainen, E. M.; Peiponen, K.-E. Appl. Phys. Lett. 2003, 83, 893– 895. [39] Vartiainen, E. M.; Saarinen, J. J.; Peiponen, K.-E. J. Opt. Soc. Am. B 2005, 22, 1173– 1178. [40] Saarinen, J. J.; Vartiainen, E. M.; Peiponen, K.-E. J. Mod. Opt. 2006, 53, 1047–1059. Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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[41] Saarinen, J. J.; Vartiainen, E. M.; Peiponen, K.-E. Sens. Actuators B: Chem. 2006, 53, 1047–1059. [42] Sipe, J. E. J. Opt. Soc. Am. B 1987, 4, 481–489. [43] Jackson, J. D. , Classical Electrodynamics, 3rd ed.; Wiley: New York, 1998. [44] Born, M. Wolf, E. , Principles of Optics, 7th ed.; Cambridge University Press: Cambridge, 1999. [45] Bruggeman, D. A. G. Ann. Phys.; Leipzig 1935, 24, 636–679. [46] Hurt, N. E. , Phase Retrieval and Zero Crossings: Mathematical Methods in Image Reconstruction; Kluwer: Dordrecht, 1989. [47] Pauli, W. , “Die allgemeinen Prinzipien der Wellenmechanik,” in Handbuch der Physik, Vol. XXIV, Geiger, K. Scheel, H. , eds.; Springer, 1933. [48] Jaming, P. J. Fourier Anal. Appl. 1999, 5, 309–329. [49] Velicky, B. Chezh. J. Phys. B 1961, 11, 541–543. [50] Roessler, D. M. Brit. J. Appl. Phys. 1965, 16, 1119–1123. [51] Ahrenkiel, R. K. J. Opt. Soc. Am. 1971, 61, 1651–1655. [52] Goedecke, G. H. J. Opt. Soc. Am. 1975, 65, 146–149. [53] Smith, D. Y.; Manogue, C. A. J. Opt. Soc. Am. 1981, 71, 935–947. Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

[54] Nash, P. L.; Bell, R. J.; Alexander, R. J. Mod. Opt. 1995, 42, 1837–1842. [55] Toll, J. S. Phys. Rev. 1956, 104, 1760–1770. [56] Shannon, C. E. Weaver, W. , The Mathematical Theory of Communication; University of Illinois Press: Urbana, 1949. [57] Jaynes, E. T. Phys. Rev. 1957, 106, 620–630. [58] Jaynes, E. T. Phys. Rev. 1957, 108, 171–190. [59] Shannon, C. E. Bell System Technical J. 1949, 28, 656–715. [60] Cornwell, T. J.; Evans, K. F. Astron. Astrophys. 1985, 143, 77–83. [61] Steinbach, P. ; Ionescu, R. ; Matthews, C. R. Biophys. J. 2002, 82, 2244–2255. [62] Mannermaa, J. ; Karras, M. Geophysica 1989, 25, 37–46. [63] Burg, J. P. Geophysics 1972, 37, 375–376. [64] Vartiainen, E. M.; Peiponen, K.-E.; Asakura, T. Appl. Spectrosc. 1996, 50, 1283– 1289. Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

Surface Plasmon Resonance – Applications in Nanoparticle Detection from a Liquid...265 [65] Lynch, D. W.; Hunter, W. R. , in Handbook of Optical Constants of Solids, Palik, E. D.; Academic Press: Boston, 1985, pp. 275–367. [66] Partington, J. R. , An advanced treatise on physical chemistry; Longmans, Green: New York, 1960; Vol. 4.

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[67] Matsubara, K. ; Kawata, S. ; Minami, S. Opt. Lett. 1990, 15, 75–77.

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Chapter 12

E NHANCED T RANSMISSION OF L IGHT AND M ATTER T HROUGH S UBWAVELENGTH N ANOAPERTURES BY FAR -F IELD M ULTIPLE -B EAM I NTERFERENCE S. V. Kukhlevsky ∗ Institute of Physics, University of Pecs, Ifjusag u. 6, Pecs, H-7624, Hungary

Abstract

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Subwavelength aperture arrays in thin metal films can enable enhanced transmission of light and matter (atom) waves. The phenomenon relies on resonant excitation and interference of the plasmon-polariton or matter waves on the metal surface. We show a mechanism that could provide a great resonant and nonresonant transmission enhancement of the light or de Broglie’s particle waves passed through the apertures not by the surface waves, but by the constructive interference of diffracted waves (beams generated by the apertures) at the detector placed in the far-field zone. According to the model, the light beams generated by multiple, subwavelength apertures can have similar phases and can add coherently. If the spacing of the apertures is smaller than the optical wavelength, then the phases of the multiple beams at the detector are nearly the same and beams add coherently (the light power and energy scales as the number of light-sources squared, regardless of periodicity). If the spacing is larger, then the addition is not so efficient, but still leads to enhancements and resonances (versus wavelength) in the total energy transmitted (radiated). We stress that the plasmon-polaritons do not affect the principle of the enhancement based on the constructive interference of diffracted waves (beams) generated by the subwavelength apertures at the detector placed in the far-field zone. Naturally, the plasmonpolaritons could provide additional enhancement by increasing the power and energy of each beam. The Wood anomalies in transmission spectra of gratings, a long standing problem in optics, follow naturally from the interference properties of our model. The point is the prediction of the Wood anomaly in a classical Young-type two-source system. Our analysis is based on calculation of the energy flux (intensity) of a beam ∗

E-mail address: [email protected]

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S. V. Kukhlevsky array by using Maxwell’s equations for classical, non-quantum electromagnetic fields. Therefore the mechanism could be interpreted as a non-quantum analog of the superradiance emission of a subwavelength ensemble of atoms (the light power and energy scales as the number of light-sources squared, regardless of periodicity) predicted by the well-known Dicke quantum model. In contrast to other models, the enhancement mechanism depends on neither the nature (non-quantum electromagnetic waves, quantum light or matter) of beams (continuous waves or pulses) nor material and shape of the multiple-beam source (arrays of one- and two-dimensional subwavelength apertures, fibers, dipoles, and atoms). The quantum reformulation of our model is also presented. The Hamiltonian describing the phenomenon of interference-induced enhancement and suppression of both the intensity and energy of a quantum optical field is derived. The basic properties of the field energy determining by the Hamiltonian are analyzed. Normally, the interference of two or more waves causes enhancement or suppression of the light intensity, but not the light energy. The model shows that the phenomenon could be observed experimentally, for instance, by using a subwavelength array of the coherent quantum light-sources (one- and two-dimensional subwavelength apertures, fibers, dipoles, and atoms).

PACS 42.25.Bs, 42.50.Nn, 42.79.Ag, 0.3.70.+k, 03.75.-b. Keywords: Enhanced transmission, subwavelength nanoapertures, multiple-beam interference, superradiance, quantum electromagnetic fields.

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1.

Introduction

The scattering of waves by apertures is one of the basic phenomena in the wave physics. The most remarkable feature of the light scattering by subwavelength apertures in a metal screen is enhancement of the light by excitation of plasmons in the metal. Since the observation of enhanced transmission of light through a 2D array of subwavelength metal nanoholes [1], the phenomenon attracts increasing interest of researchers because of its potential for applications in nanooptics and nanophotonics [1, 2]. The enhancement of light is a process that can include resonant excitation and interference of surface plasmons [3, 4, 5], Fabry-Perot-like intraslit modes [6, 7, 8, 9, 10], and evanescent electromagnetic waves at the metal surface [11]. Most of the related published work concerns the transmission though thick (many skin depths) metals. It is clear that there would be almost no transmission through a thick metal in the absence of waveguide and plasmon resonances. In the case of a thin screen whose thickness is too small to support the intraslit resonance, the extraordinary transmission is caused by the excitation and interference of plasmons on the metal surface [3, 4, 5]. For some experimental conditions, many studies [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] indicated a nonessential role of the surface plasmons in the enhancement of light waves. For an example, the study [18] showed that a perfect conductor whose surface is patterned by an array of holes can support surface polaritons, which just mimic a surface plasmon in channeling of additional energy into the aperture. Nowadays, it is generally accepted [24] that the excitation and interference of surface plasmon-polaritons play a key role in the process of enhancement of light waves in the most of experiments (also, see the recent comprehensive reviews [25, 26]). In the present study [27], we show a new mechanism that could provide great resonant and nonresonant

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transmission enhancements of the light waves passed through the apertures not by the surface waves, but by the constructive interference of diffracted waves (beams generated by the apertures) at the detector placed in the far-field zone.

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2.

Non-quantum Light Waves

In this section we consider a mechanism that could provide great resonant and nonresonant transmission enhancement of the classical (nonquantum) light waves passed through subwavelength aperture arrays. The section is reprinted from Physical Review A, Vol. 78, S.V. Kukhlevsky, Enhanced transmission of light through subwavelength nanoapertures by far-field multiple-beam interference, pp. 023826. Copyright (2008), with permision from the American Physical Society. The transmission enhancement by the constructive interference of diffracted waves at the detector can be explained in terms of the following theoretical formulation. We first consider the transmission of light through a structure that is similar, but simpler than an array of holes, namely an array of parallel subwavelength-width slits in the metal screen. In some respects, the resonant excitation and interference of surface plasmon-polaritons in these two systems are different from each other [28]. The difference, however, is irrelevant from a point of view of our model. Indeed, the excitation of plasmon-polaritons and coupling between the apertures do not affect the principle of the enhancement based on the constructive interference of diffracted waves (beams generated by the independent apertures) at the detector placed in the far-field zone. The resonant excitation of the plasmons or trapped electromagnetic modes, as well as the coupling between apertures could provide just additional, in comparison to our model, enhancement by increasing the power (energy) of each beam. Therefore, our model considers an array of slits, which are completely independent from each other. We also assume, for the sake of simplicity, that the metal is a perfect conductor. Such a metal is described by the classic Drude model for which the plasmon frequency tends towards infinity. The beam produced by each independent slit is found by using the Neerhoff and Mur model, which uses a Green’s function formalism for a rigorous numerical solution of Maxwell’s equations for a single, isolated slit [29, 30, 31, 32, 33, 34]. In the model, the screen placed in vacuum is illuminated by a normally incident TM-polarized wave with the wavelength λ = 2πc/ω = 2π/k. The mag~ netic field of the incident wave H(x, y, z, t) = U (x)exp(−i(kz + ωt))~ey is supposed to be time harmonic and constant in the y direction. The transmission of the slit array is determined by calculating all the light power of the ensemble of beams in the observation plane. To clarify the numerical results, we then present an analytical model, which quantitatively explains the resonant and nonresonant enhancement in the intuitively transparent terms of the constructive interference of diffracted waves (beams generated by the apertures) at the detector placed in the far-field zone. Finally, we show that the mechanism depends on neither nature of the beams (continuous waves and pulses) nor material and shape of the multiple-beam source (arrays of 1-D and 2-D subwavelength apertures, fibers, dipoles, and atoms). Let us first investigate the light transmission versus the wavelength by using the rigorous numerical model. The model considers an ensemble of M waves (beams) produced by M independent slits of width 2a and period Λ in a screen of thickness b. The transmission of

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the slit array is determined by calculating all the light power P (λ) radiated by the slits into the far-field diffraction zone, x∈[−∞, ∞] at the distance z  λ from the screen. The total per-slit transmission coefficient, which represents the per-slit enhancement in transmission achieved by taking a single, isolated slit (beam) and placing it in an M -slit (M -beam) array, is then found by using an equation TM (λ) = P (λ)/M P1 , where P1 is the power radiated by a single slit. Figure 1 shows the transmission coefficient TM (λ), in the spectral region 500-2000 nm, calculated for the array parameters: a = 100 nm, Λ = 1800 nm, and b = 5 × 10−3λmax . The transmitted power was computed by integrating the total energy flux

Figure 1. The per-slit transmission TM (λ) of an array of independent slits of period Λ in a thin (b  λ) screen versus the wavelength for the different number M of slits. There are three Fabry-Perot like resonances at the wavelengths λn ≈Λ/n, where n=1, 2 and 3. at the distance z = 1 mm over the detector region of width ∆x = 20 mm. The transmission spectra TM (λ) is shown for different values of M . We notice that the spectra TM (λ) is periodically modulated, as a function of wavelength, below and above a level defined by the transmission T1(λ) = 1 of one isolated slit. As M is increased from 2 to 10, the visibility of the modulation fringes increases approximately from 0.2 to 0.7. The transmission TM exhibits the Fabry-Perot like maxima around wavelengths λn = Λ/n. The spectral peaks

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max increase with increasing the number of slits and reach a saturation ( TM ≈ 5) in amplitude by M = 300, at λ ≈ 1800 nm. The peak widths and the spectral shifts of the resonances from the Fabry-Perot wavelengths decrease with increasing the number M of beams (slits). An analysis of Fig. 1, indicates that the power (energy) enhancement and dispersion are the general interference properties of the ensemble of beams. Therefore, the enhancement and suppression in the transmission spectra could be considered as the natural properties also of the periodic array of independent subwavelength slits. The spectral peaks are characterized by asymmetric Fano-like profiles. Such modulations in the transmission spectra are known as Wood’s anomalies. The minima and maxima correspond to Rayleigh anomalies and Fano resonances, respectively [35]. The Wood anomalies in transmission spectra of gratings, a long standing problem in optics, follow naturally from the interference properties of our model. The new point, in comparison to other models [36, 37]), is the prediction of a weak Wood anomaly in a classical Young-type two-source system (see, Fig. 1). The above-presented analysis is based on calculation of the energy flux of a beam array, in which the electromagnetic field of a single beam is evaluated numerically. The transmission enhancement and dispersion were achieved by taking a single, isolated slit (beam) and placing it in a slit (beam) array. The interference of diffracted waves (beams generated by the slits) at the detector placed in the far-field zone could be considered as a physical mechanism responsible for the enhancement and dispersion. To clarify the results of the computer code and gain physical insight into the enhancement mechanism, we have developed an analytical model, which yields simple formulas for the electromagnetic field of the beam produced by a single slit. For the field diffracted by a narrow ( 2a  λ, b ≥ 0) slit into the region |z| > 2a, the Neerhoff and Mur model simplifies to an analytical one [38]. ~ = (Ex, 0, Ez ) components of the single ~ = (0, Hy , 0) and electric E For the magnetic H beam we found the following analytical expressions:

Hy (x, z) = iaDF01 (k[x2 + z 2 ]1/2),

(1)

Ex (x, z) = −az[x2 + z 2 ]−1/2DF11 (k[x2 + z 2 ]1/2),

(2)

Ez (x, z) = ax[x2 + z 2 ]−1/2DF11 (k[x2 + z 2 ]1/2),

(3)

D = 4k−1 [[exp(ikb)(aA − k)]2 − (aA + k)2 ]−1

(4)

and

where

and A = F01 (ka) +

π ¯ [F0 (ka)F11 (ka) + F¯1 (ka)F01(ka)]. 2

(5)

Here, F11 , F01 , F¯0 and F¯1 are the Hankel and Struve functions, respectively. The beam is spatially inhomogeneous, in contrast to a common opinion that a subwavelength aperture diffracts light in all directions uniformly [39]. The electrical and magnetic components of the field produced by a periodic array of M independent slits (beams) is given by

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P PM ~ ~ ~ ~ ~ E(x, z) = M m=1 Em (x + mΛ, z) and H(x, z) = m=1 Hm (x + mΛ, z), where Em and ~ Hm are the electrical and magnetic components of the m-th beam generated by the respec~ and H ~ calculated tive slit. As an example, Fig. 2(a) compares the far-field distributions E by using the analytical formulae (1-5) to that obtained by the rigorous computer model. We ~ H) ~ is found by notice that the distributions are undistinguishable. The field power P (E,

Figure 2. The electrical and magnetic components of the field produced by an array of M independent slits (beams). (a) The distributions Re( Ex (x)) (A and D), Re(Hy (x)) (B and E), and Re(10Ez (x)) (C and F ) calculated for M = 10 and λ = 1600 nm. The curves A, B, and C: rigorous computer code; curves D, E, and F : analytical model. (b) Re(Ex (x)) for M =1: analytical model. (c) Re(Ex (x)) for M =5: analytical model. ~ ∗ × H. ~ Therefore, the analytical model ac~ =E ~ ×H ~∗ +E integrating the energy flux S curately describes also the coefficient TM of the system of M independent subwavelength slits (beams). The analytical model not only supports results of our rigorous computer code (Fig. 1), but presents an intuitively transparent explanation (physical mechanism) of the enhancement and suppression in transmission spectra in terms of the constructive or destructive interference of the waves (beams produced by the subwavelength-width sources) at the detector placed in the far-field zone. The array-induced decrease of the central beam divergence by the far-field multiple beam interference (Figs. 2(b) and 2(c)) is relevant to the beaming light [40], as well as the ”diffraction-free” light and matter beams [41, 42]. The amplitude of a beam (evanescent spherical-like wave) produced by a single slit rapidly decreases with increasing the distance from the slit (1-3). However, due to the multiple

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beam interference mechanism of the enhancement and beaming, the array produces in the far-field zone a propagating wave with low divergence. Such a behavior is in agreement with the Huygens-Fresnel principle, which considers a propagating wave as a superposition of secondary spherical waves. We now consider the predictions of our analytical model in light of the key observations published in the literature for the two fundamental systems of wave optics, the one-slit and two-slit systems. The major features of the transmission through a single subwavelength slit are the intraslit resonances and the spectral shifts of the resonances from the FabryPerot wavelengths [7]. In agreement with the predictions [7], the formula (4) shows that the transmission T = P/P0 = (a/k)([Re(D)]2 + [Im(D)]2) exhibits Fabry-Perot like maxima around wavelengths λn = 2b/n, where P0 is the power impinging on the slit opening. The enhancement and spectral shifts are explained by the wavelength dependent terms in the denominator of Eq. (4). The enhancement (T (λ1)≈b/πa [38]) is in contrast to the attenuation predicted by the model [7]. Although, our model considers a screen of perfect conductivity, polarization charges develop on the metal surface. The surface polaritons do not adhere strictly to traditional surface plasmons. Nevertheless, at the resonant conditions, the system redistributes the electromagnetic energy by the surface polaritons in the intra-slit region and around the screen. Thus, additional energy could be channeled thought the slit in comparison to the energy impinging on the slit opening. The mechanism is somewhat similar to that described in the study [18]. This study showed that a perfect conductor whose surface is patterned by an array of holes can support surface polaritons that mimic a surface plasmon in the process of channeling of additional energy into the slit. We considered TM-polarized modes because TE modes are cut off by a thick slit. In the case of a thin screen, TE modes propagate into slit so that magneto-polaritons develop. Because of the symmetry of Maxwell’s equations the scattering intensity is for~ and H ~ swapping roles. Again, the magneto-polaritons could provide mally identical with E channeling of additional energy into the slit. This enhancement mechanism is different from those baced on the constructive interference of the waves (beams produced by the subwavelength-width sources) at the detector placed in the far-field zone. The Young type two-slit (two-beam) configuration is characterized by a sinusoidal modulation of the transmission spectra (for an example, see T2(λ) in Refs. [36, 37]). The modulation period is inversely proportional to the slit separation Λ. The visibility V of the fringes is of order 0.2, independently of the slit separation. In our model, the transmission T2 depends on R the interference-like cross term [F11 (x1)[iF01(x2 )]∗ + F11 (x1 )∗iF01 (x2)]dx, where x1 = x and x2 = x + Λ. The high-frequency interference-like modulations with the sidebandfrequency fs (Λ) ≈f1 (λ) + f2(Λ, λ)∼1/Λ (Figs. 1 and 3) are produced like that in a classic heterodyne system by mixing two waves having different spatial frequencies, f1 and f2 . Although our model ignores the enhancement by the plasmon-polaritons, its prediction for the transmission (T2max ≈1.1), the visibility (V ≈0.1) of the fringes and the resonant wavelengths λn ≈ Λ/n compare well with the plasmon-assisted Young’s type experiment [36] (Fig. 3). It should be noted that in the case of b ≥ λ/2, the far-field interference resonances at λn ≈ Λ/n could be accompanied by the intraslit polariton resonances at λn ≈ 2b/n. One can easily demonstrate such behavior by using the analytical formulas (1-5). The interference of two beams at the detector is not the only contribution to enhanced transmission. There could be enhancement also due to the energy redistribution by the resonant

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Figure 3. The per-slit transmission coefficient T2(λ) versus wavelength for the Young type two-slit experiment [36]. Solid curve: experiment; dashed curve: analytical model. Parameters: a = 100 nm, Λ = 4900 nm, and b = 210 nm. intraslit plasmon-polaritons and/or by the surface waves with resonant coupling through the slits. We stress, however, that the plasmons or trapped electromagnetic modes do not affect the principle of the enhancement based on the constructive interference of diffracted waves (beams generated by the independent subwavelength-width apertures) at the detector placed in the far-field zone. The plasmon-polaritons could provide just additional enhancement by increasing the power and energy of each beam. This kind of enhancement is of different nature compared to our model, because the model requires neither resonant excitation of the intraslit plasmon-polaritons nor coupling between the slits (see, also Refs. [61, 44, 22]). In order to gain physical insight into the mechanism of plasmonless and polaritonless enhancement in a multiple-slit or multiple-beam (M ≥2) system, we now consider the dependence of the transmission TM (λ) on the slit (beam) separation Λ. According to the Van Citter-Zernike coherence theorem, a light source (even incoherent) of radius r = M (a + Λ) produces a transversely coherent wave at the distance z≤πRr/λ in the region of radius R. In the case of Λ  λ, the collective coherent emission of an ensemble of slits (beams) ~ ~ ~ = PM E generates the coherent electromagnetic field ( E m=1 m exp(iϕm )≈M E1 exp(iϕ) and ~ ~ 1exp(iϕ)) in the far-field zone of the region of radius R = ∞. This means that H≈M H the beams arrive to the detector with the nearly same phases ϕm (x) ≈ ϕ(x) (see, also Ref. [45]). Consequently, the beams add coherently and the power (energy) of the emitted light scales as the number of beams squared, regardless of periodicity, P ≈ M 2 P1 . Thus, the transmission enhancement (TM = P/M P1 ) grows linearly with the number of slits, TM ∼M . For a given value of M , in the case of Λ  λ, the transmission TM (λ) monotonically (nonresonantly) varies with λ (see, Fig. 4). At the appropriate conditions, the transmission can reach the 1000-times nonresonant enhancement ( M = λz/πR(a + Λ)). In the case of R > λz/πr or Λ>λ, the beams arrive to the detector with different phases ϕm (x). Consequently, the power and transmission enhancement grow slowly with the number of beams (Figs. 1-4). The constructive or destructive interference of the beams leads respectively to the enhancement or suppression of the transmission amplitudes. Although, the addition of beams is not so efficient, the multiple beam interference leads to enhancements and resonances (versus wavelength) in the total power transmitted. In such a case, the transmission coefficient TM exhibits the Fabry-Perot like maxima around the wavelengths

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Figure 4. The per-slit transmission TM (λ) versus wavelength for the different values of Λ and M : (A) Λ = 100 nm, M = 2; (B) Λ = 500 nm, M = 2; (C) Λ = 3000 nm, M = 2; (D) Λ = 100 nm, M = 5; (E) Λ = 500 nm, M = 5; (F) Λ = 3000 nm, M = 5. Parameters: a = 100 nm and b = 10 nm. There are two enhancement regimes at Λ  λ and Λ>λ. λn = Λ/n. We stress again that the constructive or destructive interference of beams at the detector requires neither the resonant excitation of plasmon-polaritons nor the coupling between radiation phases of the slits. The plasmon-polariton effects could provide just additional enhancement by increasing the power and energy of each beam. Our consideration of the subwavelength gratings is similar in spirit to the dynamical diffraction models [12], Airy-like model [13], and especially to a surface evanescent wave model [11]. In the case of Λ >> λ, our model is in agreement with the theories of conventional (non-subwavelength) gratings [46]. In the above-presented multiple-beam interference model, we have considered a particular light-source, namely an array of subwavelength metal slits. One can easily demonstrate the interference mediated enhancement and suppression in the transmission and reflection spectra of an arbitrary array of subwavelength-dimension sources of light by taking into account the interference properties of Young’s double-source system. At the risk of belaboring the obvious, we now describe the phenomenon. In the far-field diffraction zone, the radiation from two pinholes of Young’s setup is described by two spherical waves. The light intensity at the detector is given by I(~r) = |(E/r1)exp(ikr1 + ϕ1) + (E/r2)exp(ikr2 + ϕ2)|2 = RI1R + I2 + 2(I1 I2)1/2cos([kr1 + ϕ1] − [kr2 + ϕ2]). The corresponding energy [I1 + I2 + 2(I1I2 )1/2cos([kr1 + ϕ1 ] − [kr2 + ϕ2])]dxdy. Here, we use the is W = units c∆t = 1. In conventional Young’s setup, which contains the pinholes separated by the distance Λ >> λ, the interference cross term (energy) vanishes. Therefore, the energy is

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RR

given by W = (I1 + I2 )dxdy = W1 + W2 = 2W0 , where W1 = W2 = W0. In the case of Young’s subwavelength system ( Λ λ could lead to a new effect, namely the enhancements and resonances (versus period of the array) in the total power emitted by the periodic array of quantum oscillators (atoms). It should be stressed again that the cross correlation energy (19-22) vanishes if the light waves have the spatial

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dependence appropriate for an optical resonator. In the present study, we have considered photon fields. A quantum reformulation of our model for other boson and fermion fields will be presented in our next paper. In conclusion, the Hamiltonian describing the phenomenon of interference-induced enhancement and suppression of both the intensity and energy of a quantum optical field has been derived. The basic properties of the field energy determining by the Hamiltonian were analyzed. The model shows that the interference-induced enhancement and suppression of both the intensity and energy of a quantum EM field could be observed experimentally, for instance, by using a subwavelength array of the coherent quantum light-sources (1D and 2D subwavelength apertures, fibers, dipoles and atoms). It was also showed that the phenomenon could associate with several basic optical phenomena, such as the extraordinary transmission of light through subwavelength apertures, the scattering of entangled photons in Young’s two-slit experiment and the Dicke’s quantum super-radiance.

Acknowledgments This study was supported in part by the Framework for European Cooperation in the field of Scientific and Technical Research (COST, Contract No MP0601) and the Hungarian Research and Development Program (KPI, Contract GVOP 0066-3.2.1.-2004-04-0166/3.0).

References [1] T.W. Ebbesen et al., Nature (London) 391, 667 (1998).

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[2] W.L. Barnes et al., Nature (London) 424, 824 (2003). [3] U. Schr¨oter and D. Heitmann, Phys. Rev. B 58, 15419 (1998). [4] M.B. Sobnack et al., Phys. Rev. Lett. 80, 5667 (1998). [5] J.A. Porto, F.J. Garcia-Vidal, and J.B. Pendry, Phys. Rev. Lett. 83, 2845 (1999). [6] S. Astilean, P. Lalanne, and M. Palamaru, Opt. Comm. 175, 265 (2000). [7] Y. Takakura, Phys. Rev. Lett. 86, 5601 (2001). [8] P. Lalanne et al., Phys. Rev. B 68, 125404 (2003). [9] A. Barbara et al., Eur. Phys. J. D 23, 143 (2003). [10] S.V. Kukhlevsky et al., Phys. Rev. B 70, 195428 (2004). [11] H.J. Lezec and T. Thio, Opt. Exp. 12, 3629 (2004). [12] M.M. Treacy, Phys. Rev. B 66, 195105 (2002). [13] Q. Cao and P. Lalanne, Phys. Rev. Lett. 88, 057403 (2002). [14] M. Sarrazin, J.P. Vigneron, and J.M. Vigoureux, Phys. Rev. B 67, 085415 (2003). Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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[15] W.L. Barnes, W.A. Murray, J. Dintinger, E. Devaux, and T.W. Ebbesen, Phys. Rev. Lett. 92, 107401 (2004). [16] K.J.K. Koerkamp, S. Enoch, F.B. Segerink, N.F. van Hulst, and L. Kuipers, Phys. Rev. Lett. 92, 183901 (2004). [17] A.E. Miroshnichenko and Y.S. Kivshar, Phys. Rev. E 72, 056611 (2004). [18] J. B. Pendry, Science 305, 847 (2004). [19] R. Gomez-Medina, M. Laroche, and J.J. Saenz, Opt. Express 14, 3730 (2006). [20] E. Moreno, L. Martin-Moreno, and F.G. Garcia-Vidal, J. Opt. A: Pure Appl. Opt. 8, S94 (2006). [21] B. Ung and Y. Sheng, Opt. Express 15, 1182 (2006). [22] X.R. Huang, R.W. Peng, Z. Wang, F. Gao, and S.S. Jiang, Phys. Rev. A 76, 035802 (2007). [23] Y. Ben-Aryeh, Appl. Phys. B - Lasers and Optics 91, 157 (2008). [24] H. Liu and P. Lalanne, Nature (London) 452, 728 (2008). [25] F. J. Garcia de Abajo, Rev. Mod. Phys. 79, 1267 (2007).

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[26] A.K. Sarychev and V.M. Shalaev, Electrodynamics of Metamaterials , World Scientific, New Jersey, p. 185 (2007). [27] The study was presented at the Nanoelectronic Devices for Defense and Security (NANO-DDS) Conference, 18-21 June, 2007, Washington, USA. [28] E. Popov, M. Niviere, S. Enoch, and R. Reinisch, Phys. Rev. B 62, 16100 (2000). [29] F. L. Neerhoff and G. Mur, Appl. Sci. Res. 28, 73 (1973). [30] R.F. Harrington and D.T. Auckland, IEEE Trans. Antennas Propag. AP28, 616 (1980). [31] E. Betzig, A. Harootunian, A. Lewis, and M. Isaacson, Appl. Opt. 25, 1890 (1986). [32] S.V. Kukhlevsky, M. Mechler, L. Csapo, K. Janssens, and O. Samek, Phys. Rev. B 72, 165421 (2005). [33] M. Mechler, O. Samek, and S. V. Kukhlevsky, Phys. Rev. Lett. 98, 163901 (2007). [34] S.V. Kukhlevsky, Resonant enhancement and near-field localization of fs pulses by subwavelength nm-size metal slits, in Lasers and electro-optics research at the cutting edge, Steven B. Larkin, Ed., Ch.1, 1-42, Nova Science Publishers, New York (2007). [35] A. Hessel and A.A. Oliner, Appl. Opt. 4, 1275 (1965). Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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[36] H.F. Schouten et al., Phys. Rev. Lett. 94, 053901 (2005). [37] P. Lalanne et al., Phys. Rev. Lett. 95, 263902 (2005). [38] S.V. Kukhlevsky, M. Mechler, O. Samek, and K. Janssens, Appl. Phys. B: Lasers and Optics 84, 19 (2006). [39] H.J. Lezec et al., Science 297, 820 (2002). [40] L. Martin-Moreno, F.J. Garcia-Vidal, H.J. Lezec, A. Degiron, and T.W. Ebbesen, Phys. Rev. Lett. 90, 167401 (2003). [41] S.V. Kukhlevsky, Diffraction-free subwavelength-beam optics on nanometer scale, in Localized waves, H.E. Hernandez-Figueroa, M. Zamboni-Rached, E. Recami, Eds, Ch.10, 273-297, John Wiley, Hoboken - New Jersey (2008). [42] S.V. Kukhlevsky et al., Phys. Rev. E 64, 026603 (2001). [43] R. Gordon, J. Opt. A: Pure Appl. Opt. 8, L1 (2006). [44] S.V. Kukhlevsky, Enhanced Transmission of Light and Matter through Nanoapertures without Assistance of Surface Waves, physics/0602190 (February 2006). [45] C. Genet et al., J. Opt. Soc. Am. A 22, 998 (2005). [46] R. Petit, Electromagnetic theory of gratings (Springer-Verlag, London, 1980).

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[47] R.W. Schoonover and T.D. Visser, Opt. Commun. 271, 323 (2007). [48] M.O. Scully and M.S. Zubairy, Quantum Optics (Cambridge University Press, New York, 1997). [49] R.H. Dicke, Phys. Rev. Lett. 93, 439 (1954). [50] E. Altewischer et al., Nature (London) 418, 304 (2002). [51] L.D. Landau and E.M. Lifshitz, Classical Theory of Fields (Nauka, Moscow, 1972). [52] V.B. Beresteckii, E.M. Lifshits, and L.P. Pitaevskii, Quantum Electrodynamics (Nauka, Moscow, 1980). [53] R. Loudon, The Quantum Theory of Light (Oxford University Press, New York, 1983). [54] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interaction (Wiley, New York, 1992). [55] S. Weinberg, Theory of Quantum Fields (Cambridge University Press, London, 1995). [56] E.R. Pike and S. Sarkar, Quantum Theory of Radiation (Cambridge University Press, London, 1995). Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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[57] E. Notte-Cuello and W. A. Rodrigues, e-print arXive:math-ph/0612036. [58] U. Leonhardt, Nature (London) 444, 823 (2006). [59] M.I. Stockman1, S.V. Faleev, and D.J. Bergman, Phys. Rev. Lett. 88, 067402 (2002). [60] N. Gauthler, Am. J. Phys. 71, 787 (2003). [61] R. Gordon, J. Opt. A: Pure Appl. Opt. 8, L1 (2006). [62] P.H. Souto Ribeiro, Braz. J. Phys. 31, 478 (2001).

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[63] S.V. Kukhlevsky, Phys. Rev. A 78, 023826 (2008).

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In: Plasmons: Theory and Applications Editor: Kristina N. Helsey, pp.289-326

ISBN: 978-1-61761-306-7 c 2011 Nova Science Publishers, Inc.

Chapter 13

G RADED P LASMONIC S TRUCTURES AND T HEIR P ROPERTIES Jun Jun Xiao,1,∗ Kousuke Yakubo2 and Kin Wah Yu3 1 Department of Electronic and Information Engineering, Key Laboratory of Network Oriented Intelligent Computation Shenzhen Graduate School, Harbin Institute of Technology Shenzhen 518055, China 2 Division of Applied Physics, Graduate School of Engineering Hokkaido University, Sapporo 060-8628, Japan 3 Department of Physics, The Chinese University of Hong Kong Shatin, N. T., Hong Kong, China

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Abstract Plasmonic structures have extraordinary optical properties in terms of engineering surface plasmon for optical waveguiding, switching, sensing, and enhanced optical spectroscopies such as surface enhanced Raman spectroscopy (SERS) and strong nonlinear optical effects. In particular, there is a type of plasmonic structures which have spatially varying characteristics—graded plasmonic structures—that enable extremely flexible and strong confinement of surface plasmon in real space, making it easier to control hot spots, surface signal propagation distance and direction. We show that a peculiar kind of confined plasmon modes, called plasmonic gradons, can occur in such graded plasmonic structures. The plasmonic gradons are different from either disorder-induced Anderson-type localization in random media or Bragg-gap type confinement that is supported by periodic structures. We identify several localizationdelocalization transitions between various plasmonic gradons which are of distinct flavors and are uniquely sustainable in these graded plasmonic structures. To this end, we also discuss the interplay between gradon confinement and a variety of oscillations, such as Bloch oscillation and breathing-like oscillation. To have a deep understanding of the plasmonic gradon physics, an analogous problem in graded elastic lattices with tunable on-site potentials are proposed.

∗ E-mail

address: [email protected]

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PACS 42.82.Et, 71.45.Gm, 63.20.Pw Keywords: Graded structure, Graded plasmonic structure; Localization-delocalization transition

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1.

Plasmonic gradons,

Introduction

Manipulating electromagnetic (EM) waves inside nanostructures has attracted ever increasing interest over the past decades due to the rapid advancement of nanofabrication techniques. Particularly, interactions of light with structures modulated at the wavelength or subwavelength scale offer an opportunity to achieve novel properties and designated functionalities in nanophotonics. Notable examples include photonic crystals, metamaterials, and plasmonic structures [1–3]. Plasmonic structure refers to metallic systems which possess collective motion of the conduction electrons that is called plasmon mode [3]. When an external light excites such a mode, the EM field of the light can be enhanced greatly in the vicinity of the nano-object [4]. Plasmonics structures offer the potential to confine and guide light below the diffraction limit and promises a new generation of highly miniaturized photonic devices. Plasmonics– the coupling between light and collective oscillation of free electrons in metal–becomes one of the important branches in nanooptics due to its applications in nanodevices breaking the diffraction limit. There are two major ingredients of plasmonics, surface plasmon polaritons (SPP) at metallic interfaces and localized surface plasmons (LSP) in nanostructures [4]. Such unprecedented ability of plasmonic structures has propelled their use in a vast array of nanophotonic technologies and research endeavors, for example, in deep subwavelength waveguiding [5], super-imaging, sensing and surface-enhanced Raman spectroscopy (SERS) [6], one-way EM wave transmission [7], negative refraction [8], dramatic absorption and emission modulation [9], photovoltaic devices [10], optical cloaking and illusion [11], extraordinary transmission, optical force enhancement [12], and negativeindexed metamaterials. Flexibility in the structure configuration and easy controlling mechanism for plasmon mode confinement have created a unique context of graded plasmonic structures. Fine control over the material structure along certain direction or within a volume enables novel physical phenomena and previously unthinkable design freedom for spatial, spectral, and temporal functions. Describing that context and functionalities is the aim of this Chapter. We shall first present the results in graded elastic lattices which include 1D monotonic chains, 2D lattice with orthogonal gradient, 2D lattice with diagonal diagonal, and 1D diatomic chains. These results are useful in understanding the gradon physics. Then we turn to describe various graded plasmonic crystals such as 1D nanoparticle chains and 1D nanoshell chains. Despite of the normal and quasi-normal mode analysis, we further present the wave dynamics analysis in some of these graded plasmonic structures.

1.1.

Overview of Graded Wave Functional Structures

Generating complex functionalities from simple and well-understood building blocks of various crystals and artificial structures has been of great interest, from which new physics Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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291

and phenomena may emerge. Manipulation of electronic wave is highly accurate in a wide variety of systems, however, the same has been a long-standing challenge for classical waves, hindered by the difficulty of applying field-responsive control over their fundamental excitations (e.g., phonon and photon) that possess neither charge nor spin. In such situation, one usually relies on structural or physical parameter variation. Graded wave functional materials and structure is such an example [13–16], which could be useful in classical wave generation, manipulation, and detection. In the meantime, thermal management via graded nanostructures has been proposed [17]. In contrast to the traditional inhomogeneous media, i.e., periodically modulated system and randomly disordered system, graded system demonstrates a unique way to control classical wave, resulting in a new type of localization-delocalization transition, which can confine the primary excitations (e.g., photons, phonons, and surface plasmons) and redistribute them spatially. This is not only of fundamental significance, but will also pave new avenue for various applications, for example, in surface elastic waves, nanooptics, and plasmonics. It also has implications with practical problems in industry such as oil probing and earthquake study. With the numerous new nanofabrication techniques, the past several decades have witnessed a large shift of researches on various inhomogeneous systems to a special kind of materials–functionally graded materials (FGMs). This is a new generation of engineered materials wherein the microstructural details are spatially varied through nonuniform distribution of the reinforcement phase(s). Engineers accomplish this by using reinforcements with different properties, sizes, and shapes, as well as by interchanging the roles of the reinforcement and matrix phases in a continuous manner. The result is a microstructure that produces continuously or discretely changing physical properties at the macroscopic or continuum scale. FGMs are ideal candidates for applications involving severe thermal gradients, ranging from thermal structures in advanced aircraft and aerospace engines to computer circuit boards. Along with these ideas, there are now many newly emerging dielectric, optical, ferroelectric, ferromagnetic, ferroelastic, piezoelectric, and plasmonic FGMs. The realm of the classical wave behavior in graded wave functional systems–gradon physics–has yielded many interesting results [13, 14, 18–22]. In particular, there are many useful graded plasmonic structures whose certain characteristics vary along one spatial dimension [23–26]. We attempt to give an overview of studies on this topic. However, no review of a large field can be exhaustive and we have chosen not to discuss important efforts on fully retardation effect, which constitutes a larger body of research of retardation-based structures.

1.2.

Graded Plasmonic Nanostructures

The most popular system of graded plasmonic structure is a chain of plasmonic nanoparticles. Typical graded spherical metallic nanoparticle chains with varying characteristics such as particle spacing and radii have been studied theoretically [24–27]. A cascaded plasmonic “snowman" chain of several metal nanospheres with progressively decreasing sizes and separations is shown to be able to focus excitation light into nanogaps [28]. An optical analog of the xylophone is realized by a metallic chains of varying width on a plasmonic

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substrate [29]. A nano-optical Yagi-Uda antenna can also be made by a one-dimensional array of nanoparticle of different size [30]. Nevertheless, despite of such discrete arrays, continuous plasmonic graded plasmonic structures have also been proposed. Adiabatic concentration of light into a tapered waveguide tip was studied by Stockman [31]. “Trapped rainbow” has been proved and realized in gradually narrowing channel [32] and demonstrated in graded plasmonic crystal [33]. Plasmonic waveguide arrays with incremental spacing have been shown to enable sub-wavelength focusing and steering of light [34]. A common feature in such “spatially varying” structures is that the localization of the electric field can be controlled temporally by illuminating it with either a continue or a chirped pulse. The frequency, polarization, and the sign of the excitation controls the excitation sequence of the particles with great flexibility.

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2.

Classical Wave Excitations in Graded Structures

Over the past decades, a lot of exotic phenomena were observed in graded photonic crystal. For example, enhanced coupling to slow photon modes at the band-edge of graded threedimensional, higher structural and optical quality colloidal photonic crystals was demonstrated. For the gradation, polystyrene latex is etched by exposure to air plasma, yielding a top-to-bottom filling-fraction gradient. By monitoring transmission and corresponding phase spectra upon etching, spectral blue-shifts and enhanced mode coupling were observed [35]. A superbending effect was reported in two-dimensional graded photonic crystal [36, 37]. Focusing (lens) and guiding (waveguide) effects are also realizable in graded index photonic crystals [38, 39]. Emmanuel and Cassagne proposed using graded photonic crystals to enhance the control of light propagation. Gradual modifications of the lattice periodicity make it possible to bend the light at the micrometer scale. This effect is tailored by parametric studies of the isofrequency curves [40]. In addition, there is now a probability of using transformation optics to design the optical ray (signal) road, which is then realized by graded photonic crystal or anisotropic and variable metamaterials. It is known that classical waves have common properties in many aspects. For example, electromagnetic wave can be similar to acoustic wave, liquid surface wave, and sound wave in terms of scattering, waveguiding, and cloaking. Therefore understanding in one type of classical wave could give useful insights for the others. To this end, we have described the excitation spectrum in both elastic vibrational lattices and plasmonic arrays and waveguides. We show that a simple spatial variation of certain characteristic along a specific dimension leads to very interesting physics– that is the gradon physics.

2.1.

Harmonic Vibrations in Graded Elastic Networks

A quite easy toy model to understand excitations in a graded system is the spring-mass system [20,21]. The vibrational excitation in spring-mass lattices can be studied easily, which, however, reveals the fundamental underlying mechanics that governs various fundamental excitations in graded system, either discrete or continuous.

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Graded Plasmonic Structures and Their Properties M1

Mn

M2

MN-1

293

MN

(a) (b)

K1

K2

KN-1

Kn

Figure 1. Schematics of the graded elastic networks: (a) The graded mass model with Mn = M0 −CM n/N, and (b) the graded force constant model with Kn = K0 +CK n/N. 2.1.1. One-dimensional Chains The one-dimensional system of spring-mass chain is a textbook example which can give substantial insights to graded systems. Two different graded vibrational lattice models, namely, the graded spring constant model and the graded mass model (see Figure 1) have been studied and basically show the same results [13, 22]. The motion of each unit in the graded chain with N units is described by (1)

where un is the scalar displacement of the nth unit with mass Mn , and Kn denotes the force constant of the linear spring connecting the nth unit to the (n + 1)th unit. The most prominent difference of introducing a gradient with respect to a homogenous elastic chain is that the vibrational modes over certain transition frequency ωc becomes localized, and the density of states (DoS) maximizes at this frequency, opposite to the divergence in this point in homogeneous chain. The details can be found in Figure 2. Equation (1) can be formulated to an eigenproblem by assuming harmonic solution. The eigenspectrum is then obtained numerically with a specific N [13]. 3 Ck = 0.2

gradon

2

CK = 0.5

phonon

1.0

K = 1.0 K = 1.2 K = 1.5 K = 1.7

1

0.5

0.0 0.0

CK = 0.7

ω

1.5

DoSK (ω)

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Mn u¨n = Kn−1 (un−1 − un ) + Kn (un+1 − un ) ,

0 0

0.5

π/2 k

1.0

π

1.5

2.0

2.5

Excitation frequency ω

3.0

Figure 2. DoS’s for one-dimensional graded spring-mass chain. Inset shows the connection of phonon band with gradon band from a picture of band overlapping. Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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en

0.1

ω=1.568

(a)

0.0

-0.1 1

N/2 Site index n

0.1

(b)

ω=2.000

0.1

0.0

0.0

-0.1

-0.1

N

1

N/2 Site index n

N

ω=2.191

(c)

1

2N/5

N/2 Site index n

N

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Figure 3. Three typical excitation states of the graded force constant model with K0 = 1.0 and CK = 0.5 at (a) ω = 1.568(< ωc ), (b) ω = 2.0(= ωc ), and (c) ω = 2.191(> ωc ). It has been further shownpthat modes with ω > ωc are qualitatively different from modes with ω < ωc , where ωc = 2 K0 /M0 for both types of models. Modes of higher frequency than ωc are partially extended within lighter (harder) regions, whereas lower frequency modes are fully extended. We call these partially extended modes with ω > ωc gradons. A mode with a frequency ω larger than ωc cannot have amplitudes in a heavier (softer) region because masses (springs) in this region are too heavy (soft) to vibrate with the frequency ω. As a consequence, the mode pattern has finite amplitudes only in the lighter (harder) region, while no amplitude in the heavier (softer) region, as p shown in Figure 3(c). The boundary between these two regions is determined by ω = 2 Kn /Mn . Therefore, the position of the gradon front (the site with the largest amplitude, also is the classical turning point mentioned later) nc is given by nc = Ξ(ω)N with Ξ(ω) = (M0 /Cm )(1 − ω2c /ω2 ) for the graded mass model and Ξ(ω) = (K0 /Ck )(ω2 /ω2c − 1) for the graded force constant model [13]. The fact that the gradon front predicted by these relation coincides with numerical results indicates the validity of our interpretation of gradon modes. One of the remarkable features of gradon wave functions is that vibrational amplitudes are significantly enhanced near the gradon front. This hump structure of gradons is relevant to our later discussion. Now we try to obtain the vibrational DoS of a graded elastic chain analytically. First, we concentrate on the graded mass case [13,22], by the argument of dividing an infinite 1D mass graded chain into many sub-chains which are also infinite but enough small compared to the original chain. Thus, each sub-chain can be regarded as an infinite chain with a constant mass of M. The DoS of the sub-chain is given by Dsub (M, ω) =

2 θ[ωD (M) − ω] q , π ω2 (M) − ω2

(2)

D

where θ(x) is the step function ( 1, for x ≥ 0 θ(x) = , 0, for x < 0 and the maximum frequency ωD (M) of the subsystem with mass M is given by p ωD (M) = 2 K/M.

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(3)

(4)

Graded Plasmonic Structures and Their Properties

295

Since contributions of inter-subchain coupling to the total DoS are negligible compared to intra-subchain contributions, the total density of states DM (ω) can be expressed as the sum of Dsub (M, ω), namely, DM (ω) =

1 CM

Z M0 M0 −CM

Dsub (M, ω)dM ,

(5)

where M0 and CM are used to denote the linear graded masses profile as Mn = M0 −CM (n − 1)/(N − 1). Therefore, we can write DM (ω) =

2 πCM

Z M0 M0 −CM

θ(4K/ω2 − M) p dM . 4K/M − ω2

(6)

The step function θ(4K/ω2 − M) which restricts the integration to the range of M < 4K/ω2 and the integral range [M0 −CM , M p0 ] separate the integral into three regions: 2 (1) If 4K/ω < M0 −CM , i.e., ω > ωm ≡ 4K/(M0 −CM ), we have no integral region, and D(ω) is always zero in this case. p (2) If M0 −CM 6 4K/ω2 < M0 , namely, ωc (≡ 4K/M0) < ω 6 ωm , Eq. (6) becomes DM (ω) =

2 πCM

Z 4K/ω2 M0 −CM

p

dM 4K/M − ω2

,

(ωc < ω 6 ωm ) .

(7)

(3) If 4K/ω2 > M0 , i.e., ω 6 ωc , the DoS is given by

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DM (ω) =

2 πCM

Z M0 M0 −CM

dM p , 4K/M − ω2

(0 < ω 6 ωc ) .

(8)

After some calculations, we have    q q 2   2 2 2 2  (M0 −CM ) ωm − ω − M0 ωc − ω   πCM ω2  " ! !#   2 2 2 2  ω − 2ω ω − 2ω 4K  m c  p p tan−1 − tan−1 , +   2 − ω2  πCM ω3 2ω ω2c − ω2 2ω ω  m     (0 < ω 6 ωc ) q 2 (9) DM (ω) = (M0 −CM ) ω2m − ω2   2  πC ω M # " !     4K π ω2m − 2ω2  −1  p , (ωc < ω 6 ωm ) + tan +   πCM ω3 2  2ω ω2m − ω2        0, (ωm < ω) . In similar way, we can obtain the analytical DoS form for a chain with graded force constants. For this case, we write the DoS of the subnetworks as Dsub (K, ω) =

2 θ(K − Mω2 /4) q . π ω2 (K) − ω2 D

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(10)

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1.0

DK(ω)

0.8 0.6

(a) CK = 0.5

0.4 0.2

DM(ω)

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 1.0 (b) Numerical data 0.8 Analytical data 0.6

CM = 0.5

0.4 0.2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

ω

Figure 4. Analytical (curves) and numerical (circles) DoS for vibration modes in graded chains. (a) Graded force constant case and (b) graded mass case. The numerical results are obtained with size N = 1000.

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where θ and ωD are of the same forms as in Eqs. (3) and (4), respectively. Then the density of states DK(ω) of a chain with graded force constants can be expressed as DK (ω) =

1 CK

Z K0 +CK K0

Dsub (K, ω) dK .

(11)

After some manipulations, we have  √  √ M ω2m −ω2 − ω2c −ω2    , (0 < ω 6 ωc )  √ πCK 2 2 (12) DK (ω) = M ωm −ω , (ωc < ω 6 ωm )  πCK   0 , (ωm < ω) . p p Here, we redefine ωc = 4K0 /M and ωm = 4(K0 +CK )/M in Eq. (12). The analytical results from Eqs. (9) and (12) compare favorably with numerical data as shown in Figure 4 for both the graded force constant case [Figure 4(a)] and the graded mass case [Figure 4(b)]. These analytic expressions are very rare examples of a closed form of DoS in inhomogeneous media and would be very useful in examining macroscopic properties of graded system. 2.1.2. Two-dimensional Square Lattices To have further insight of higher dimension, we have studied two-dimensional [14, 21] graded vibrational lattices, with both orthogonal gradient [14] and diagonal gradient [21].

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Graded Plasmonic Structures and Their Properties

DoS

1.5 1.0

K =1.0 K =1.2 K =1.5 Analytical (CK=0.5) Numerical DGSL Numerical OGSL

297

K0=1.0 CK=0.5

0.5 0.0 0.0

2.0 3.0 1.0 Excitation frequency ω

Figure 5. DoS’s of infinite DGSL and OGSL are the same (thick solid line), which can be obtained from DoS (thin lines) of the homogeneous sub-lattices. Symbols are numerical results with finite size N = 1000 for DGSL (2) and OGSL (◦).

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Although the DoS for the two cases are identical (see Figure 5), there is differences in the mode pattern and phase diagram. Let us consider a 2D triangular lattice of size N × L. Following the partial decomposition scheme [14, 21], we get the following effective 1D dynamic equation k  k  s s L un−1 + 2KnL cos un+1 cos mn u¨n = 2Kn−1 2 2 h i L −2 Kn−1 + KnL + KnT (1 − cos ks ) un , (13) where mn is the mass in the n-th site along the gradient direction, un represents the displacement, and ks = 2πs/L (s = 1, 2, · · · , L − 1) denotes the transverse wave number perpendicular to the gradient direction, along which the lattice is translation invariant—periodic boundary conditions are therefore employed in this transverse direction. Both K L and K T are nearest neighbor couplings between the masses in the longitudinal and transverse directions. With the definitions of an effective coupling constant Kn = 2 cos(ks /2)KnL and an on-site potential h i L + KnL + KnT (1 − cos ks ) − Kn − Kn−1 , (14) Un = 2 Kn−1 the dynamical equation Eq. (13) therefore becomes mn u¨n = Kn−1 (un−1 − un ) + Kn (un+1 − un ) −Un un .

(15)

Note that if we eliminate the transverse couplings of KnT , we arrive at a case of applying a uniaxial gradient in a diagonal direction of a square lattice (i.e., DGSL), instead of along the orthogonal directions (i.e., OGSL) as in [14]. In the DGSL case we simply set KiT = 0 in the on-site potential of Eq. (14). Then, to describe the case of a square lattice with gradient along the diagonal direction, we have an effective 1D dynamical equation mn u¨n = Kn−1 (un−1 − un ) + Kn (un+1 − un ) −Vn un ,

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Jun Jun Xiao, Kousuke Yakubo and Kin Wah Yu 0.30

ω(49π/50) = 1.99172 ω(49π/50) = 2.23484 ω(49π/50) = 2.46206

0.20

en

0.10 0.00 -0.10 -0.20 -0.30

1

N/2 N Site index n along the gradient direction

Figure 6. Typical mode patterns for transverse wave number ks = 49π/50 (kc < ks < π) in a DGSL. The modes are soft gradon, soft-hard gradon, and hard gradon, from the left to the right, respectively.

where

 ks KnL , 2
L = 2 Kn−1 + KnL − Kn − Kn−1 .

Kn = 2 cos Vn



(17a) (17b)

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Hereafter, in numerical calculations, we consider the ks -dependent variations of Kn and Vn for a DGSL with KnL = K0 +

CK (n − 1) , N −1

KnT = 0 ,

n = 1, 2, · · · , N ,

(18a)

n = 1, 2, · · · , N ,

(18b)

where K0 = 1, mn ≡ m0 = 1, and it describes a linear gradient profile with CK = 0.5. We observed four kinds of vibrational harmonic normal modes in DGSL, namely, soft gradons (S), hard gradons (H), soft-hard gradons (SH) (for example, see Figure 6), and unbounded modes (U). It is possible to examine these various modes in DGSL from a perspective of dividing the infinite graded lattice into small segments along the gradient direction. Each of them is still infinite in size but can be considered as homogeneous while the couplings between these segments are weak. From this perspective it is expected that the DoS does not depend on the direction of the gradient provided that the same graded profile has been used. This is because the DoS of the whole system is given by the sum of DoS of d−1 such homogeneous systems [14]. In fact the numerical DoS D(ω) = N1d ∑Ns=1 ∑λ∈S(ks) δ(ω− ωλ ) of the DGSL is the same with that of the OGSL as shown in Figure 5, and both match the analytical one as [14] 1 gG(ω) = CK

Z K0 +CK

g(ω, 1, K)dK ,

K0

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where the DoS of 2D square lattice of homogeneous media reads  √4  F(ν) , ω2 (ω2L − ω2 ) > 16β2 ,   π2 ω2L −ω2 ω g(ω, M, K) = (20) F( 1ν ) , 0 < ω2 (ω2L − ω2 ) < 16β2 , π2 β    0, otherwise, p where β1/2 = K/M is the intrinsic frequency, ω2L = 8β denotes the maximum frequency, F(x) is the complete elliptic integral of the first kind, and ν=

2.2.

4β q . ω ω2L − ω2

(21)

Gradon Confinement

The mechanism of gradon modes is basically the same with that of localized impurity modes. The most striking difference between gradon modes and truly localized modes, such as the Anderson localization and localized impurity modes, is that gradons in infinite graded systems vibrate in infinite spatial region, namely, gradon modes are partially extended in the sense that a finite ratio of the whole system has vibrational amplitudes, while usual localized modes are localized within a finite region even in infinite systems. This character of gradon modes appears in the inverse participation ration (IPR) defined as

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P−1 =

∑Nn φ4n , (∑Nn φ2n )2

(22)

where |φi is an eigenfunction for the equations of motion Eqs. (1) or (16). We will give a detailed discussion in the proceeded section by a quantum analogy to classical graded systems. 2.2.1. Quantum Analogy It is noticed that the hump structure of a gradon mode at the gradon front [see Figures 3(c) and 6] is essentially the same with the phenomenon that the existence probability of a quantum particle in a confinement potential becomes large due to the slowing down of the corresponding classical motion of the particle near the (classical) turning point in the potential. According to this idea, we regard a gradon mode as a quantum state in a semiclassical regime in a linearly graded potential. Profiles of wave functions are the same with the corresponding vibrational modes. The system has the graded potential given by V (x) = V0 −

V1 x, N

(23)

where N is the length of the system in a unit of the lattice constant, V0 and V1 the base and modulation of the potential. We assume that the gradon mode at the energy E has its finite amplitudes in the spatial region of [xc, N]. Namely, the position x = xc represents the gradon front. The position xc is determined by the energy E, and can be written as xc = β(E)N ,

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where β(E) corresponds to Ξ(ω) in Section (2.1.1.) and 0 < β(E) < 1. Since the energy E is classically given by V1 1 E = mv2 +V0 − x , 2 N the classical particle velocity reads r 2V1 (x − xc ) . v(x) = mN

(25)

(26)

Here we use the relation V1 V0 − E = xc , N

(27)

which stands for the classical turning point coming from the condition V (xc) = E. The probability to find the moving particle is inversely proportional to the velocity. Thus, the existence probability in the corresponding quantum system is also inversely proportional to the velocity. Therefore, we can write |ψ(x)|2 =

c γ =√ , v(x) x − xc

(28)

where c is a proportionality constant and r mN . γ=c 2V1

(29)

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The simple estimation of Eq. (28) well reproduces the envelope function of a gradon R R mode obtained numerically. The IPR is defined by P−1 = |ψ(x)|4 dx/[ |ψ(x)|2 dx]2 . Using Eq. (28), the denominator of this expression is simply given by Z N xc

√ |ψ(x)|2dx = 2γ N − xc .

(30) R

We should be careful in the integration in the numerator. The quantity xNc |ψ(x)|4 dx diverges, because the wave function given by Eq. (28) diverges at x = xc . Actually, the wave function does not diverge due to the exponential decay in the region of [0, xc]. As we know in quantum mechanics, the wave function in a linear graded potential has a form of the Airy function. Therefore, the actual wave function obeys Eq. (28) in a region of [xc + ε, N] in which we can adopt the semiclassical quantum analogue, and is strongly suppressed in [xc , xc + ε], where the small quantity ε has the order of a typical length (~2 /2mV1)1/2 corresponding to the size-independent exponential tail near x = xc . If we neglect the small R R contribution of xxcc +ε |ψ(x)|4 dx to the numerator, the numerator |ψ(x)|4 dx can be approximated by Z N

4

|ψ(x)| dx '

xc

Z N xc +ε

|ψ(x)|4 dx = γ2 [log(N − xc ) − log(ε)] .

(31)

Using Eqs. (24), (30), and (31), the denominator and the numerator of the IPR are given by 2 Z N |ψ(x)|2dx = 4Nγ2 (1 − β) , (32) xc Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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5

ω = 2.2 4

C 1logN+C 2

3

N*IPR

ω = 2.0 2

ω = 1.0 1

0

0

1

2

3

4

log(N)

Figure 7. System-size dependence of the IPR (solid lines) for a phonon ( ω = 1.0), the transition mode (ω = 2.0), and a gradon (ω = 2.2). Dashed lines are fitted by the formula C1 log N +C2 . Sets of coefficients (C1 ,C2 ) are (0, 1.508), (0.423, 1.040), and (0.983, 0.935) for ω = 1.0, 2.0, and 2.2, respectively. Z N

|ψ(x)|4dx ' γ2 [log N + log(1 − β) − log ε] .

(33)

xc

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Therefore, the IPR for large N is evaluated as P−1 =

log N log[(1 − β)/ε] 1 1 + . 4(1 − β) N 4(1 − β) N

(34)

Thus, we have P−1 (N) = C1

log N 1 +C2 , N N

(35)

where C1 = 1/4(1 −β) and C2 = log[(1−β)/ε]/4(1 −β) are the size-independent constants. It should be noted that the above arguments are valid if the invalid region of the semiclassical approximation [xc , xc + ε], in which the squared amplitude of the quantum wave function cannot be approximated by Eq. (28) due to the violation of the local-wavelength condition |∇λ|  1, is negligibly small compared to the gradon size, namely, ε  N − xc . For enough large systems, therefore, Eq. (35) provides an accurate approximation for any frequency except for very close to the maximum frequency at which xc approaches N. The result of the size dependence of the IPR for one-dimensional graded force constant chains is shown in Figure 7. The system size runs over 100 ≤ N ≤ 10000 in our numerical calculation. We show the results for three typical values of frequencies, namely, ω = 1.0 (lower line) for phonons, ω = 2.0 (middle line) for the transition mode from phonons to gradons, and ω = 2.2 (upper line) for gradons. Three straight dashed lines represent the analytical results described by Eq. (35). The values of C1 and C2 in Eq. (35) are chosen

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as C1 = 0 and C2 = 1.508 for ω = 1.0 (lower dashed line), C1 = 0.423 and C2 = 1.040 for ω = 2.0 (middle dashed line), and C1 = 0.983 and C2 = 0.935 for ω = 2.2 (upper dashed line). As shown in Figure 7, the analytical and numerical results match very well with each other. Fluctuations in numerical data around a straight line are caused by finite gaps between adjacent eigenfrequencies in a finite system. Values of the numerical IPR are calculated for the closest eigenfrequencies to a specific frequency, but these eigenfrequencies have some fluctuations around the specified frequency. The coefficients C1 and C2 have meaningful physical interpretation. For phonons, C1 vanishes identically while C2 takes a large finite value. However for gradons, C1 and C2 become large. Thus we can say that the first term in Eq. (35) mainly comes from a contribution of the gradon hump and the second term from the gradon body. Phonons do not have gradon humps and hence C1 = 0. C1 becomes large for a gradon with narrow spatial extent because of β close to unity in such cases. Although the behavior of Eq. (35) for large N is quite similar to a power law N −δ with δ slightly less than unity, these two are quantitatively different. We should note that the slope of a straight line in this plot, i.e., C1 in Eq. (35), depends only on β describing the spatial extent of the gradon.

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2.2.2. Confinement Mechanism and Transition to Extended Modes From all the results presented above, we can conclude that an infinite graded elastic chain exhibits the localization-delocalization transition at the frequency ωc being the maximum eigenfrequency of the homogeneous elastic chain with Mn = M0 (or Kn = K0 ) for any n. In the frequency regime ω < ωc , we have phonon-type modes extended over the whole system, while modes with ω > ωc (namely, gradons) are localized in the lighter mass side (or harder force constant side). It should be noted that this delocalization-localization transition (the phonon-gradon transition) occurs in a single band as shown in the inset of Figure 2. The mechanism of gradon localization is considered as follows. A mode with a frequency ω larger than ωc cannot have amplitudes in the heavier (or softer) region because masses (or force constants) in this region are too heavy (or too soft) to vibrate with the frequency ω. As a consequence, the mode pattern has finite amplitudes only in the lighter (or harder) region, while no amplitude in the heavier p (or softer) region. The boundary between these two regions is determined by ω = 2 Kn /Mn . Therefore, the localization center site nc which is the position with the maximum amplitude [see Figure 3(c)] is given by    NM0 ω2c   1− 2 graded mass model ,   ω  CM (36) nc =   2     NK0 ω − 1  graded force constant model . CK ω2c The fact that value of nc = 2N/5 of the mode shown by Figure 3(c) is predicted by Eq. (36) shows the validity of our interpretation of gradon localization. This mechanism of localization is essentially the same with that of impurity localized modes. Therefore, gradons belong to a kind of confined modes by impurities, and the phonon-gradon transition in a single band is induced by impurity localized modes.

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Properties of this new type of the delocalization transition are largely different from usual transitions occurring in a single band, such as the Anderson transition. First of all, the gradon transition can occur even in a one-dimensional graded system, while the Anderson transition requires three or more dimensions. This is not surprising, because the scaling theory cannot be applied to the gradon transition. Furthermore, a localized mode in a usual sense has a finite localization length ξ even if the system size is infinite. On the contrary, the localization length of a gradon, which is defined as the size of vibrating region and approximately given by N − nc , becomes infinite if the system size N tends to infinity as seen from Eq. (36). Gradon modes are localized only in the sense that a part of the whole system has vibrational amplitudes. The ratio of the vibrating region to the whole system is given by 1 − M0 (1 − γ2 )/CM for the graded mass model and 1 − K0 (γ−2 − 1)/CK for the graded force constant model, where γ = ω/ωc . Although it is known that there exist one-dimensional delocalization transitions in correlated disordered systems, the gradon transition is distinguished from this type of transitions at this point. The fact that the DoS shows the singularity (see Figures 2 and 5) at the transition frequency is also a feature of the phonon-gradon transition, which contrasts strikingly with the Anderson transition without any spectral singularity at the critical point. For the graded mass model, each state in the vibrational DoS above ωc corresponds to a light-mass impurity mode mentioned in the preceding section. The linear distribution of light-mass impurities in the graded mass model forms a continuous spectrum of impurity modes above ωc . The whole spectral structure shown in Figure 2 is constructed by connecting two different spectra, the phonon-type excitation spectrum and the impurity mode one above ωc . As seen from Figure 3, mode pattern profiles of gradon excitations are highly asymmetric due to the asymmetricity of the graded elastic network. A gradon mode in the heavier (or softer) side of the localization center nc has an exponential tail with a large damping factor d0−1 as ∼ exp[−(nc − n)a/d0 ]. The width of the localization front (i.e., the decay length) d0 seems to be independent of ω. On the contrary, in the lighter (or harder) side of nc , vibrational amplitudes decrease very slowly (almost constant) as increasing n − nc . We should remark that the gradon transition remains sharp even for a finite system. This is because the width d0 is much smaller than the system size L. This also contrasts to a conventional localization-delocalization transition described by the scaling theory, in which the finite system size reduces the transition to a broad crossover. 2.2.3. Phase Diagram: A Band-overlapping Picture By the argument that the graded lattice can be regard as combining a sequence of homogeneous lattices, we are able to construct the band structure of the DGSL and OGSL. For example, one for DGSL is shown in Figure 8(a). The four curves correspond to ωc1 (1, ks), ωc2 (1, ks), ωc1 (N, ks), and ωc2 (N, ks), respectively, where ωc1 (i, ks) and ωc2 (i, ks) are defined as r Vn , (37a) ωc1 (n, ks) = mn r Vn + 4Ki , (37b) ωc2 (n, ks) = mn

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Excitation frequency ω

4.0 ω ( c2 N, k) s

3.0

H

(a)

4.0

(b)

SH 3.0

ω (1 c2 , k s)

2.0 1.0

U

) ,k s

S

ω c1

) ks

2.0

2 3

(N ω c1

, (1

0.0 0

1

K0=1.0 CK=0.5

1.0 K=1.5 K=1.0 K=0.5

π0.00 π π/2 π/2 wave vector ks wave vector ks

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Figure 8. (a) Phase diagram for a DGSL of K0 = 1 and CK = 0.5. (b) Demonstration of band overlapping picture from three HSLs with K = 0.5, 1.0, and 1.5. The two bands represented by solid lines and dashed lines determine the phases in (a). Phases in any DGSL can be constructed in a similar way by the corresponding bands of HSLs.

representing the upper and the lower band edges of a homogenous square lattice (HSL) with characteristic constant Kn , Vn , and mn . Variations of the two characteristic frequencies defined by Eqs. (37) are functions for typical ks . It is in view of these that we construct the phase diagram Figure 8(a) with the help of a band overlapping picture, the essence of which is to examine the overlapping situation of bands for HSLs with various K (in our case m ≡ m0 ). For example, for emergence of unbounded modes (U), we need that the band of a HSL with K = K0 [solid curves in Figure 8(a)] and that of a HSL with K = K0 + CK [dashed curves in Figure 8(a)] can share a normal mode frequency. This is exactly indicated by the grey shaded region. In this distinction, we have implicitly utilized the fact that any of the HSLs with K0 < K < K0 + CK has a band covering this grey shaded region. The S and H phases are determined consistently and similarly in the same way—the S spectrum region is shared only by part of HSLs (K0 < K < Ki ) corresponding to sites n close to the left edge of the DGSL, while the H spectrum region is commonly covered by bands of HSLs (Ki < K < K0 + CK ) corresponding to sites at the other edge of the DGSL. Very interestingly, the SH spectrum region is shared by HSLs corresponding to sites at the the central part in the DGSL, resulting in the peculiar localization modes of soft-hard gradon as exemplified by the dashed lines in Figure 6. It is seen in Figure 8(a) that there exists a critical transverse wave number kc , below which we have unbounded modes at intermediate frequencies, whereas above kc we have soft-hard gradons at intermediate frequencies. It is easy to show that kc is determined by ωc2 (1, kc) = ωc1 (N, kc) ,

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which yields

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kc = 2 arccos [CK /(CK + 2K0 )] ,

(39)

for the case of the graded force constant model. This equation results in kc ≈ 0.87π for the case K0 = 1 and CK = 0.5, in good agreement of numerical results. The various normal modes and the phase boundaries determined by the band overlapping picture in Figure 8(a) are verified by numerical results for a system of size N = 1000. As frequency goes up, when ks < kc , the normal modes change from soft gradons to unbounded modes, and reenter another localized modes (hard gradons), whereas for kc < ks < π, the normal modes change from soft gradons to soft-hard gradons, and to hard gradons, experiencing no delocalized mode at all. From the perspective of excitation frequency, Figure 8(a) suggests that for ω > ωc2 (1, 0)(≈ 2.828) there is a pure hard gradon band, however, for ω < ωc2 (1, 0), we would find mixing of localized and delocalized modes as demonstrated by the IPR. As a matter of fact, using the band overlapping picture to determine the phases can be very effective and successful, in both DGSL and OGSL, as it is in the one-dimensional graded chain (see the inset of Figure 2). We further demonstrate this by Figure 8(b), where we plot one more band (dotted lines) of a HSL with K = 0.5, in addition to the ones of HSLs of K = K0 (= 1.0) (solid curves) and K = K0 +CK (= 1.5) (dashed curves). The latter two can be regarded as direct copies of the curves shown in Figure 8(a). Now from the intersections of the band edges, we can easily identify the phase regions of various DGSLs. For instance, if we focus on the dashed and dotted curves, four phase regions in the spectrum correspond to distinct modes supported by a DGSL with K0 = 0.5 and CK = 1.0. The point labelled by “2" in Figure 8(b) gives a new critical wave number kc ≈ 2.09 in this case. Similarly, the combination of the solid and dotted curves in Figure 8(b) provides the phase diagram for a DGSL with K0 = 0.5 and CK = 0.5. From this picture and the fact that the critical wave number given by Eq. (39) is bounded by 0 < kc < π, we conclude that in DGSL we always have soft-hard gradons for certain ks close to the zone boundary (ks = π) [21]. However, in OGSL, soft-hard gradons can only show up when CK > K0 [14].

2.3. Graded Plasmonic Nanoparticle Chains Inspired by the fascinating behaviors of graded elastic lattice, we consider to combine the novel properties of gradons and surface plasmons (SPs), in an attempt to explore new mechanisms to manipulate SP. The ideal system is graded plasmonic crystals such as plasmonic chains and plasmonic nanoshell chains [24–26, 41], in which variation of typical characteristic leads to interesting plasmonic gradon localization and plasmonic gradons which are of distinct flavors for the various cases. Quite different to the elastic spring-mass lattice model, where soft-hard gradons appear just in 2D DGSL or OGSL, 1D graded plasmonic crystal can already sustain the four typical kind of gradons, e.g., light gradon, heavy gradon, light-heavy gradon, and usual coupled plasmons. It is even shown that light can be slowed down in graded nanoshell chains [42]. We note that metallic nanoparticles can even be of fractal structures [43], in which plasmonic fracton may be possible.

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2.3.1. System and Formalism We consider a chain of N spherical nanoparticles immersed in a dielectric host, and take the lossy Drude form complex dielectric function ε(ω) = 1 − ω2p /ω(ω + iΓ) for the nanoparticles while assuming that the host has its dielectric constant varying from the chain’s lefthand side to the right-hand side along x axis as ε2 (di ) = εleft + cdi /l ,

(40)

where di denotes the position of the n-th nanoparticles, l the total length of the chain, and c the coefficient of dielectric gradient in the host. Obviously c = 0 recovers the case of homogeneous host, while di+1 − di = d0 [1 + (i − 1)∆], (i = 1, 2, . . ., N) characterizes an incrementally spaced case, where d0 ≡ d1,2 is the nearest spacing at the left extremity and ∆ (∆ = 0 recovers periodic chain) is the spacing increment. In the graded plasmonic chains (GPCs), each nanoparticle has multipole moments qilm , where i is the particle index and l, m denote the multipole order. These moments are coupled through multipole interaction constant Qilm; jl 0 m0 , which depends on the moment order, interparticle separation, the ambient dielectric environment, and measures the EM influence of the j-th particle on the i-th particle. Thus, the equation of plasmon motion reads q˙ilm ΓR ... p ilm + q¨ilm = −ω2ilm qilm − τ ω2ilm + ω2Q

0

∑ Qilm; jl m q jl m + ω2i0 a3Eilm (t)δl1 , 0

0

0

0

(41)

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jl 0 m0

where the prime over the sum indicates restricted summation j 6= i, τ and ΓR are the typical relaxation time and radiation damping respectively, and the differential is taken with p respect to time t. Note that ωilm = l/(2l + 1)ωip are the resonance frequencies of the localized Mie plasmons of an individual particle, where ωip is the surface plasmon resonant frequency of the i-th particle (for identical particles, one may omit the subscript i in ωilm ). The last term in Eq. (41) represents the effect of a uniform but time-dependent electric field Ei1m (t) applied to the i-th nanoparticle, whose dipolar resonant frequency is denoted by ωi0 . We consider the case that the diameter a of the nanoparticles is much less than the wavelength of the incident light and the spacing d between any nanoparticles are larger than their diameter. Due to the small size of the particles, retardation effect is neglected and we seek a solution for the local dipole moment in the quasistatic approximation. Hence, in dipole approximation and without radiation damping or relaxation attenuation, Eq. (41) is reduced to the following coupled equation for the graded spacing model   N (L,T ) (L,T ) (L,T ) d1,2 3 + ω20 a3 Ei1 (t) , (42) p¨i10 = −ω20 pi10 − λ(L,T ) ω21 ∑ p j10 d i, j j6=i where λL = −2 for longitudinal (L) polarization (i.e., x-axis) and λT = 1 for transverse (T ) polarization (z-axis). The symbol di, j represents the distance between the i-th and j-th particles. The point dipole model is believed sufficient in capturing the fundamental dispersion characteristics of coupled plasmons in usual plasmonic waveguiding chains [24–26]. In general cases, the coupled equation for point dipoles can be written as G(ω)p = 0 ,

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where p is the N-rowed column vector of the component of a dipole moment oscillating with the frequency ω, Gii (ω) = 1/αi (ω), (i = 1, . . ., N) and Gi j (ω) = −Ti j , ( j 6= i). Here αi (ω) = ε2 (di )a3 [ε(ω) − ε2 (di )]/[ε(ω) + 2ε2 (di )] represents the polarizability of the i-th particle, and Ti j is the near field dipole electromagnetic interaction between the i-th particle ˜ i j = Ti, j /ε2 (xi ) where Ti j denotes the near field coupling in and the jth particle [26], T vacuum. In Cartesian coordinates (x, y, z), the (β,γ) components are given by Tβ,γ (i, j) =

3dnm,βdi j,γ − |di j |2 δβ,γ . |di j |5

(44)

Equation (43) can also be regarded as an eigenequation and the normal mode frequency ωα is determined by det{G(ω)} = 0 with ωα , generally, being complex valued. However, it is hard to determine ωα by directly solving this equation with Γ 6= 0 which is a nonlinear complex transcendental equation with respect to ω. Alternatively, we can employ an inhomogeneous equation G(ω)p = v to study the plasmon excitations. It is possible to linearize Eq. (43) with respect to ω2 [24, 26] (F − ω2 I)p = 0 ,

(45)

2 2 0 2 where I is identity matrix and Fi j = ωp i − Gi j (ωi )/Gii(ωn ). Here the prime indicates deriva2 tive with respect to ω and ωi = ω p / 1 + 2ε2 (di) is the dipole Mie resonant frequency of the i-th particle. Equation (45) represents a good approximation of Eq. (43) as ω ≈ ωi .

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2.3.2. Different Graded Models In the linearized lossless (Γ = 0) case, the coupled plasmon dispersion can be directly obtained by diagonalizing F, resulting in real valued eigenpair {ωα , φα}. Also, Eq. (45) can be mapped into an equivalent chain of graded coupled harmonic oscillators [14] with additional on-site potentials, just like the case of a 2D graded lattice we discussed in Section 2.1.2. The on-site potential is related to dipole resonance ω0 and varies along the chain due to inhomogeneous couplings resulted from either incremental spacing or graded host. In detail, for an incrementally spaced nanoparticle chain in a homogeneous host (c = 0) and for an periodic chain (∆ = 0) in a graded host, we have the following results: (A) Incrementally spaced nanoparticle chain in a homogeneous host In this case, Eq. (45) is equivalent to coupled oscillators with constant M = 1, graded coupling constant Ki and on-site potential Ui −ω2 pi = Ki (pi+1 − pi ) + Ki−1 (pi−1 − pi ) −Ui pi , (i = 1, 2, . . ., N) ,

(46)

where 1 , (i = 1, 2, . . .N − 1) , [1 + (i − 1)∆]3   1 1 2 2 = ω0 + λω1 + , [1 + (i − 1)∆]3 [1 + (i − 2)∆]3 (i = 2, 3, . . ., N − 1) ,

Ki = −λω21 Ui

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(47)

(48)

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while K0 and KN are zero. Here λ = −2 and 1 correspond to longitudinal and transverse polarizations, respectively. Also for the two particles at the chain extremities, we have U1 = ω20 + λω21 /∆3 and UN = ω20 + λω21 /[1 + (N − 2)∆]3 , respectively. (B) Periodic nanoparticle chain in a host with graded refractive index In this case, the particle mass mi , and the strength of the additional harmonic spring Ui are respectively mi = Ui =

1 + 2ε2 (di ) , (i = 1, 2, · · · , N) , 3λω2i ε2 (di ) 1 + 2ε2 (di ) − 2K0 , (i = 1, 2, · · · , N) , 3λε2 (di )

(49) (50)

where K0 = (a/d0)3 is a force constant between adjacent particles that depends on the ratio of the interparticle spacing and the particles radius. We then define two characteristic frequencies that are useful in determining the band boundaries and mode transitions: r r Vi Vi + 4K0 ; ωc2 (i) = . (51) ωc1 (i) = Mi Mi It is easy to notice that ωc1 (i) → ωi and ωc2 (i) → ωi when d0 → ∞, which are as expected because the coupling between the particles vanishes for infinite separation.

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2.3.3. Results and Discussion We mainly focus on the graded host case in this section and seek plasmonic eigenmode solutions Φα of Eq. (45) which is obtained in the absence of the external driving field. It is noteworthy that in the calculations we used the approximation that the dielectric constant of the host around the i-th nanoparticle is homogeneous, e.g., denoted by ε2 (di ). The results for a periodic chain of d0 = 4a in a graded host are shown in Figure 9. The DoS D(ω) = ∑Nα=1 δ(ω − ωα )/N are shown in Figure 9(b). Notice that the DoS of plasmonic mode of the graded case are dramatically modified, as compared to that of homogeneous chain. It is expected that the local DoS of EM modes around the graded plasmonic structures are quite unusual. This suggests a possibility of spontaneous emission enhancement or suppression. The IPRs of the modes are shown in Figure 9(c), which reflects that modes of relatively higher or smaller frequencies are more localized. It is seen that relatively low frequency modes are localized at the right hand side [Figure 9(d)] of the GPW while high frequency modes are localized at the left hand side [Figure 9(f)]. We named them as ‘heavy gradons’ and ‘light gradons’, respectively. More interestingly, coupled plasmon modes at intermediate frequency between two transition points ωL (d0 , c) < ω < ωH (d0 , c) are extended when d0 = 3a (not shown), but are still somehow localized, rather at the central part of the graded plasmonic chain for d0 = 4a [corresponding to Figure 9(e)]. These center-localized modes resemble light gradons in the left while look like heavy gradons in the right, therefore are called ‘light-heavy gradons’. These terminologies are meaningful with regard to the optical effective electron mass, as we shall show later. It was shown that when ωc1 (1) < ωc2 (N), one have extended modes between ωL (d0 , c) < ω < ωH (d0 , c), where ωL (d0 , c) = ωc1 (1) and ωH (d0 , c) = ωc2 (N) [25]. However, if ωc1 (1) > ωc2 (N), there is no extended modes,

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Graded Plasmonic Structures and Their Properties

ω = 0.331ωp

0.36 0.34

ε2 = 4.0

0.32 1 2

DoS

10

N/2 Mode index α

(b)

0.4

(d)

= 3.0

Full NN

"heavy gradon" εlow=3

εhigh = 4

0.0 -0.2

-0.4 N 1 20 40 60 80 100 Site index n

(e) ω = 0.352ωp

0.2 0.0

1

10

ωL

ωH

0.32 0.34 0.36 0.38 ω/ωp ωL

-1

ωH

"light-heavy gradon -0.2

1 20 40 60 80 100 Site index n 0.4

(f)

ω = 0.377ωp 0.2

IPR

10

0.0

(c)

"light gradon"

-2

10

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0.2 pn

ω/ωp

0.38

2

pn

(a) ε

0.32 0.34 0.36 0.38 ω/ωp

pn

0.40

309

-0.2

1 20 40 60 80 100 Site index n

Figure 9. (a) Pseudo-dispersion relations for near field full coupling. Those for cases of homogeneous host with ε2 = 3.0 (dash-dotted line) and ε2 = 4.0 (dashed line) can be regarded as the usual dispersion relations. (b) Density of states (DOS) versus coupled plasmon frequency for the full coupling. The line with circles represent results of graded case for nearest-neighboring coupling. (c) Inverse participation ratio (IPR) versus frequency. (d) Heavy gradon in low frequency regime (ω = 0.331ω p ). (e) Light-heavy gradon near the resonant frequency (ω = 0.352ω p). (f) Light gradon at high frequency (ω = 0.377ω p ).

but light-heavy gradons exist for ωL (d0 , c) < ω < ωH (d0 , c), where ωL (d0 , c) = ωc2 (N) and ωH (d0 , c) = ωc1 (1). This is the case shown in Figure 9. Furthermore, there exists a critical point of d0 = dc when ωc1 (1) = ωc2 (N), where only one light-heavy gradon appears, which, however, is across the whole system like an extended mode. Both the band boundaries and the transition frequencies [ωL (d0 , c) and ωH (d0 , c)] agree well with the numerical data. The two transition frequencies are represented by the vertical dashed lines in Figure 9(b), where they meet with the singularities in the DoS curve (line with circles). As there are various plasmon modes sustained by GPCs, we construct a phase diagram as Figure 10, which shows not only the case of c = 1.0 [Figure 10(b)] but also the case of c = 0.5 [Figure 10(a)]. Nevertheless, let us focus on Figure 10(b) for the discussion. Figure 10(b) contains four curves of ωc1 (1), ωc1 (N), ωc1 (1), and ωc2 (N) as functions of d0 . As we have mentioned, there indeed exists a point d0 = dc (vertical dashed line) when ωc1 (1)

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0.50 0.45

(a) c = 0.5

ω / ωp

0.40

ω1 ωΝ

0.35 0.30

ωc1(1) ωc1(N)

0.25

ωc2(1) ωc2(N)

dc = 4.27 a

0.20 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.50 (b) c = 1.0 light gradon 0.45 light heavy gradon

ω / ωp

0.40

ω1

0.35 0.30 0.25

ωΝ heavy gradon dc = 3.47 a

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0.20 Extended mode 2.0 2.5 3.0 3.5 4.0 4.5 5.0 d0/a Figure 10. Phase diagram showing the variation of the four characteristic frequencies as functions of d0 for (a) c = 0.5 and (b) c = 1.0, in the case of ε` = 3.0. The critical interparticle distance dc is represented by the vertical dashed lines, whereas the two horizontal lines represent ω1 and ωN , respectively. and ωc2 (N) cross. Also we notice that ωc1 (1) = ωc2 (1) → ω1 and ωc1 (N) = ωc2 (N) → ωN as d0 → ∞, which are clearly marked by the two horizontal dashed lines of ω1 = 0.378ω p and ωN = 0.333ω p , respectively. In consistent with the previous results, we now discuss the four shaded regions partitioned by the four curves in Figure 10(b). Specifically, (I) extended mode is possible only when ωc1 (1) < ω < ωc2 (N), and when d0 < dc = 3.47a corresponding to the left panels in Figures 11; (II) light-heavy gradons emerge when ωc2 (N) < ω < ωc1 (1) for d0 > dc, corresponding to the right panels in Figures 11 and 9; (III) the lower black region indicates a phase of heavy gradons which have relatively low frequencies; (IV) the upper dotted region indicates a phase of light gradons which have relatively high frequencies. Similarly, the same phase regions exist in Figure 10(a), where a reduced c = 0.5 results in an increased dc = 4.27a, but the discussions on the four phases through (I)–(IV) are still applicable. In fact, the critical dc is determined by ωc1 (1) = ωL (d0 , c) = ωH (d0 , c) = ωc2 (N) which yields   ε2 (x1 ) 1 ε2 (xN ) 1 − = 6λK0 × + . (52) 1 + 2ε2 (x1 ) 1 + 2ε2 (xN ) [1 + 2ε2 (x1 )]2 [1 + 2ε2 (xN )]2

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Graded Plasmonic Structures and Their Properties

Re [p] (arb. unit)

Re [p] (arb. unit)

d0 = 3a 4 (a) 2 0 -2 -4 4 (b) 2 0 -2 -4 4 (c) 2 0 -2 -4 2 (g) 1

d0 = 4a 6 (d) 3 0 -3 -6 6 (e) 3 0 -3 -6 6 (f) 3 0 -3 -6 4 (h) 2

0.318ωp

0.347ωp

0.386ωp 0.347ωp

0

0

-1

-2

-2

1

N/2 Site index n

311

N

-4

1

0.341ωp

0.352ωp

0.366ωp 0.352ωp

N/2 Site index n

N

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Figure 11. Real part of the induced dipole moment for longitudinal excitation at n = N/2 in a chain of of metallic spheres spaced by d0 = 3a (left panels) and d0 = 4a (right panels). The damping parameter Γ = 0.001ω p through (a)–(f) and Γ = 0.01ω p in (g) and (f). Frequencies are indicated in each panels.

Up to now, we have already interpreted the mode transitions in view of the equivalent coupled harmonic oscillators of graded masses and on-site potentials. From another perspective, we further examine the relationship between the site Mie resonance of isolated nanoparticle, the resonant band of an infinite plasmonic waveguiding chain, and the resonant band of an infinite GPC with infinitesimal gradient. In Figure 9 we have already plotted the corresponding results for the chains in homogeneous host of ε2 = ε` (dash-dotted lines) and ε2 = ε` + c (dashed lines), i.e., results for homogeneous PWs. It is known that infinite periodic plasmonic chains have resonant bands around respective Mie resonance ωn as ω2 = ω2n − 2λγ2n cos(kd0 ) cosh(τd0 ) ,

(53)

where k is the wave number of the plasmon wave, τ the attenuation coefficient. Here the nearest-neighboring electrodynamic coupling strength γ2n = γ20n K0 , where γ0n is hostdependent. The second-order correction due to dissipations is typically cosh (τd0 ) ∼ 1.001 for silver, which is negligible. Let us break the GPC into a large number of infinite segments of plasmonic chains, each of which approximated by homogeneous host ε2 (di ). For each of these segments we have Eq. (53), which means a series of bands located at different

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Mie resonance ωn . Overlapping of these bands can be used as guideline to determine mode types, as used in the vibrational lattice. Increased d0 defies the overlapping of these bands, therefore extended modes can not show up. For fixed d0 = 4a, decreasing the gradient coefficient to c = 0.5 will overlap all these bands again, resulting in extended modes. In this way, the critical interparticle distance dc is determined by the condition that the lower band edge of Eq. (53) for ε2 = ε2 (d1 ) equals to the upper band edge for ε2 = ε2 (dN), i.e., ω21 − 2λγ21 = ω2N + 2λγ2N which simplifies to ! γ201 γ20N 1 1 − = λK0 + , (54) 1 + 2ε2 (d1 ) 1 + 2ε2 (dN ) ω2p ω2p where γ0i more explicitly depends on the resonant frequency ωi , the optical effective electron mass m∗i , and the magnitude of the oscillating charge Q. For example, one can write γ20i = Qe/m∗i ε2 (di ), where e is the electron charge. Note that ω2p = Qe/m∗0 , we therefore get γ20i =

3ω2i ε2 (di ) 1 ≡ , 1 + 2ε2 (di) Mi λ

(55)

[1 + 2ε2 (di)]2 . 3ε22 (di)

(56)

and

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m∗i = m∗0

Here m∗0 denotes the optical effective electron mass of bulk metal in vacuum. While a free electron mass m0 = 9.1095 × 10−28 g, typically m∗0 = 8.7 × 10−28 g for Ag. Equation (56) does not account for size and quantum effects in the nanoparticle, however, it is a scale relation which indicates that the optical effective electron mass increases from left to the right in the GPCs. This is consistent with our terminologies of light, heavy, and light-heavy gradons. Also it is easy to show that Eq. (54) is exactly the same as Eq. (52).

2.4.

Graded Plasmonic Nanoshelled Chains

In this section, we consider a chain of equally spaced identical metal-dielectric nanoshells. The metallic shell is modelled with the modified Drude function εs (ω) = εa − (εb − εa )ω2p /ω(ω + iΓ), where εa , εb are 5.45 and 6.18, and the plasmon resonant frequency ω p = 1.72 × 1016 rad/s [44]. This nanoshell chain is immersed in the same graded host as discussed in the preceding section. When the particles are nearly touched, one needs to consider the multipole field interactions, however, the dipole fields could be adequate to qualitatively describe the surface plasmons in the graded nanoshell chains we consider. The nth nanoshell reads a quasistatic dipole polarizability αi = εh (di)a3 [µεc −εh (di )]/[µεc + 2εh (di )], where a is the outer radius of the nanoshell. Here the product µεc is the effective permittivity of the nanoshell derived from the classical electrodynamics, in which εc is the permittivity of the core material, and the coefficient µ reads µ = [2ζs(1 − s) + s(1 + 2s)]/[ζ(s − 1) + (1 + 2s)] ,

(57)

where ζ = (b/a)3 and s = εs (ω)/εc represent the core-shell radius ratio and their permittivity contrast, respectively. Therefore, we utilize the self-consistent coupled dipole equations Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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313

of the local fields |Ei = T · |Ei + |E(0)i. Here, |E(0)i represents the external driven fields, and the dipolar couplings between nanoshells are given by Ti, j = λ/εh (di )di,3 j , in which di, j represents the distance between the i-th and the j-th nanoshells, and the coefficient λ depends on the polarization: λ = 2(−1) for the longitudinal (transverse) modes. Since we seek the eigenproblem of excitation plasmon in the graded nanoshell chain, the external field |E(0) i is chosen to be zero. The eigenfrequencies and excitations of plasmon (eigenvectors) of this nanoshell chain can be in principle calculated by solving a secular equation when the total number N of the nanoshells is small. However, the complexity of the calculation drastically increases with increasing N, and with involvement of both retardation and dissipations. Thus, it is particularly hard to solve the eigenproblem of a chain with a large number of nanoshells. In this section, we introduce some approximations to simplify the equations, namely, we would neglect any dissipation in the system, i.e., simply set Γ = 0. We then use the Taylor expansion around the resonant frequency of the nth nanoshell to linearize the operator M(ω2 ), where Mi, j = −1/αi and Mi, j = Ti, j , M(ω2 ) = M(ω2i ) + M0 (ω2i )(ω2 − ω2i ) + O (ω4 ) .

(58)

The first two terms on the right-hand side of Eq. (58) are kept and the rest are considered to be small and therefore omitted. After some manipulations, The eigenequations for dipole column vectos |pi become

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ω2 pi = ω2i pi −

Biεc ω2i N Ti, j p j , εa − Ai εc i6∑ =j

(59)

where ωi is the resonant frequency of the ith nanoshell and the coefficients Ai and Bi are respectively √ −[(2ζ + 1)εc + 2(ζ + 2)εh (di )] ± Xi Ai = , (60) 4(1 − ζ)εc 3(ζ + 2)εh (di) 1 √ Bi = ∓ 4(1 − ζ)εc 4(1 − ζ)εc Xi  × ζ2 [6εh (di )2 − 6εc ] + ζ[24εh (di )2 + 39εc ] + [24εh (di )2 − 6εc ] , (61) where Xi = 4[εc − εh (di )]2 ζ2 + [4ε2c + 52εc εh (di) + 16εh (di )2 ]ζ + [εc − 4εh (di)]2 . Here the “±" in Ai (“∓" in Bi ) correspond to the higher and lower resonant frequencies of the ith nanoshell. Then, one can easily calculate the eigenvalues ω2 and eigenvectors e of Eq. (59), which characterizes the eigenfrequencies and the corresponding excitations of dipole plasmon in the graded nanoshell chain. In this part we examine only the lossless case by highlighting the essential physics, so Γ = 0. We choose the dielectric host to be εleft = 3, dielectric gradient coefficient c = 1. Thus εright = 4 [25,41]. The parameters of the nanoshell chain are ζ = 0.25, di,i±1 ≡ d = 3a, N = 200, and the permittivity of the nanoshell cores is set to εc = 20, which is a typical value in semiconductors. Since the longitudinal polarization can be fairly well investigated through the non-retarded method, we just present the longitudinal results. The transverse case may be strongly affected by including both retardation and radiation. In Figure 12(a), the two bands are plotted, together with the band in periodic nanoshell chain within the

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Jun Jun Xiao, Kousuke Yakubo and Kin Wah Yu Nanoshell index i

(a)

0.31

(b)

0.31 0.30

0.30

ω/ωp

ω/ωp

IPR

0.29 0.15

εh = 3.0

0.14 0.4

0.29 0.15 0.14

εh = 4.0

1

N/2

0.01

N

0.10

(d)

(c)

0.31

ω/ωp

DoS

0.3 0.2

0.30

0.1 0.0 0.29

0.30

1

N/2

0.15

ω/ωp

0.4

0.29 N

(f)

(e)

0.6 DoS

0.31

0.2 0.0

0.14

ω/ωp

0.15

1

0.14 N/2 N Nanoshell index i

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Figure 12. The two plasmon bands of the nanoshell chain. (b) IPR, and (c), (e) DoS’s of the two bands. (d) and (f) The excitations of plasmon in different parts of eigenfrequency region for the upper band and the lower band, respectively. Note that the amplitude of the mode pattern is normalized by a constant to increase visibility.

extreme hosts of permittivity εleft or εright as marked by the dashed (red) and dash-dotted (blue) lines, respectively. With the aid of these dispersion curves, one can divide the eigenfrequencies of each band into three parts as separated by the boundaries of the four additional dispersions, which correspond to three kinds of localization, similarly as in graded nanoparticle chains [25, 26]. Comparing these to the band edges of the graded nanosphere chain with the same metal dielectric function and nanodot radius [41], it is found that one can get higher eigenfrequencies than the unshelled nanosphere chain by introducing the nanoshelled structure, and the optical pulse will encounter lower loss in it, which has been proven with the linearization method but is not shown here. In order to see the degree of the localization of the eigenmodes, the IPRs are calculated and shown in Figure 12(b). In this figure, the extended modes with the IPR values being proportional to 1 /LD (D is the dimension of the system) correspond to the low-IPR modes; the high-IPR modes are the localized ones, whose IPR values are proportional to 1 /ξD , where ξ denotes the localization length. The DoS of the two bands are plotted in Figures 12(c) and 12(e), respectively. There exist two DoS peaks denoted by vertical dotted lines which just correspond to the two band edges in the dispersion curves of the periodic counterparts, as seen in Figure 12(a). The plasmon eigenmodes at the three regimes boundaried by these vertical dotted lines in

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Figures 12(c) and 12(e) are presented in Figures 12(d) and 12(f), respectively. We note that similar results can also be observed in graded nanosphere chains [24–26], however, in the present case, we have one more band and the eigenfrequencies extend to both the higher frequency and the lower frequency regimes due to strong inter-dot couplings between the hybridized plasmons inside the nanoshells. It should be mentioned that different spacings between the nanoshells lead to a change in this inter-dot coupling and therefore different localized modes, which is an efficient way to tune the eigenfrequencies of the nanoshell chain, and the essential way for the nanosphere chain [24–26]. We employ the diagrammatic band overlapping approach to investigate the localization of the eigenmodes [13,14,20,22]. For each band, one can map the graded plasmonic nanoshell chain to a one-dimensional coupled oscillators [24–26], by neglecting the far-field interactions and truncating the summation in Eq. (59) up to the nearest neighbors. The resulting system can also be mapped to a graded coupled harmonic oscillator chain with additional on-site harmonic potentials as discussed in Section 2.1.2.. The effective particle mass mi , and the strength of the additional spring Ui are, respectively mi = Ui =

ε a − Ai ε c , λBi εc ω2i ε a − Ai ε c − 2K0 , λBi εc

(62) (63)

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where K0 = (a/d)3 is a force constant between the adjacent oscillators which depends on the interparticle spacing and the nanoshell’s outer radius a. The edges of each localization phase can be determined by the eight characteristic frequencies p p = (U1 + 4K0 )/m1 , ωmin U1 /m1 , (64) ωmax 1 1 = (with the host dielectric constant εh = 3) and p p (UN + 4K0 )/mN , ωmin UN /mN , ωmax N = N =

(65)

(with the host dielectric constant εh = 4). It is noteworthy that apparently the nanoshell chain is analogous to a diatomic elastic chain [20], however, we do not achieve a direct mapping between these two problems. The global phase diagram of the present graded nanoshell chains with interparticle spacing d varying from 2.5a to 4a is shown in Figure 13. With increasing d, the band widths of both the two branches decrease, and the widths of the extended mode regions shrink even down to zero at around d = 3.5a. Beyond this a new kind of localization modes appears. In the inset of Figure 13, a typical excitation pattern of this kind of eigenmodes is plotted. It differs from all the localized modes and extended modes mentioned above, which is a new kind of localized modes discovered recently [41]. When the spacing d increases asymptotically to infinity, the two bands reduce to the resonant frequencies of single nanoshell in the graded host, and the corresponding excited modes degenerate to the discrete excitation modes of each nanoshell. This can be easily understood in view of the fact that when the spacing increases, the interactions between nanoshells diminish, therefore the eigenproblem reduces to the case of isolated single nanoshell at last.

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0.30

0.28

ei

ω/ωp

0.32

ω/ωp

0.16

0.2 0.1 0.0 -0.1 -0.2 1

N/2 Site index

N

0.15 0.14 2.5

3.0

3.5

4.0

d/a

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Figure 13. Global phase diagrams with (a) εc varying from 1 to 40 and with (b) ζ changing from 0.001 to 0.9. In both figures, other parameters are the same as in Figure 12. The two horizontal dotted lines represent band boundaries of corresponding graded nanosphere chains (see text).

In fact, the nanoshells enable one to control the eigenfrequencies through additional ways as compared with the unshelled nanoparticle chains, e.g., via changing the core material (εc ) or tuning the geometry of the nanoshell ( ζ). For better illustration, the global phase diagrams with varying εc and ζ are plotted in Figures 14(a) and 14(b), respectively. With small εc , the upper band reaches the highest frequency, possessing a relatively narrow band width. Meanwhile, the lower band is quite broad. On the other hand, as εc increases, both the two bands decline, while the upper band becomes broader and the lower band turns to be narrower. However, most of the eigenfrequencies of the upper band still remain higher than their counterparts (between the horizontal dotted lines) in the nanosphere chain. For varying ζ [Figure 14(b)], we can see that the two bands touch and appear inside the bands of nanosphere chain with small ζ. When ζ increases, the gap between the two bands expands, simultaneously the upper band ascends and the lower band descends in terms of frequency, with both the band widths shrink. Specifically, in this case the optical communication band (∼ 1550 nm) is achievable at the lower band just around ζ ≈ 0.8.

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317

0.4

ω/ωp

0.3 (b)

(a)

0.2 0.1 1

15

εc

30

0

0.2

0.4 ζ

0.6

0.8

Figure 14. Global phase diagrams with (a) εc varying from 1 to 40 and with (b) ζ changing from 0.001 to 0.9. In both figures, other parameters are the same as in Figure 12. The two horizontal dotted lines represent band boundaries of corresponding graded nanosphere chains (see text).

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3.

Wavepacket Dynamics in Graded Plasmonic Crystals

The normal and quasi-normal mode analysis shown in the preceding sections already gives deep interpretation of gradon physics, however, to clearly demonstrate the strong plasmonic wave manipulation capabilities of graded plasmonic crystals, we present below the study on wavepacket dynamics which further vividly reveals the more realistic wave behaviors in such structures.

3.1. Bloch Oscillation, Breathing-like Oscillation and Others Prompted by the experimental verification of gradons in these systems, we propose to study Bloch oscillation (BO) in the graded plasmonic nanoparticle chain [18,23]. To see if gradon localization is originated by Stark localization, we make connection of Bloch oscillation to gradon modes that we have identified. We also study dynamics of Bloch oscillations both in the coordinate and momentum space, and figure out experimentally observable signatures that can characterize gradons. Firstly, we perform analysis in which the initial conditions are selected according to an ensemble described by a Gaussian distribution. Preliminary results show that the mean displacement undergoes oscillation with Bloch frequency. More surprisingly, we also obtain Bloch oscillation in the width of displacement under appropriate conditions. We may call this oscillation the breathing gradon mode, as opposed to the usual breathing mode. Secondly, we confirm the semiclassical solutions by studying the dynamics of BO via timedomain simulation of propagation of wavepackets. The simulation results are compared with the semiclassical analytic solutions. As we obtained a variety of gradons in plasmonic chains, we extend the study to a variety of gradon localization. Moreover, the relation be-

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tween gradon localization and Stark localization (being the consequence of BO) will be elaborated by the band overlapping picture. It turns out that some gradons cannot be interpreted by the usual Stark localization. Finally, we should mention that the results offer great potential applications in the management of uncharged particles and controlling wave propagations by means of graded materials and graded systems. The details are as following. Once the complete set of gradon eigenmodes |φi of Eq. (45) has been obtained, we can construct an initial state to perform time domain simulations on the “wavepacket” dynamics. Let us take an initial wavefunction,   1 (x − x0 )2 −ik0 x φ(x, 0) = exp − e , (66) 4σ2x (2πσ2x )1/4 where k0 is the vacuum wavenumber determining the central frequency and the direction of propagation of the wavepacket. The intensity profile |φ(x, 0)|2 has a Gaussian distribution centered at x0 with spatial width σx . We then expand the initial wavefunction in terms of |φn i as |φ(0)i = ∑ An |φn i ,

(67)

n

where An = hφn | φ(0)i is the constituent component of the initial wavepacket. The intensities |An |2 of the various initial wavepackets have been shown as insets in Figure 15. The peak of |An |2 corresponds to the contribution from a dominant component of frequency ωn , which is in agreement with the central frequency obtained from the dispersion relation ω(k) at k0 . Thus the subsequent wavefunction at time t is |φ(t)i = ∑ An |φn ieiωnt .

(68)

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n

When the size N becomes large, one can replace the sum by an integral over the spectral components in Eqs. (67) and (68). The evolution of the wavepacket intensity |φ(x,t)|2 can be used to illustrate the various oscillations, including BO, breathing-wave-like oscillations and other kinds of motions. We show that combination of different localized modes (gradons) lead to different dynamic evolutions of the wavepacket. Here a few cases are addressed to illustrate the correspondence between the various gradons and the oscillation of wavepackets. As light-heavy gradons are localized in the middle part of the chain, the condition for the occurrence of BO is that the initial wavepacket only consists of light-heavy gradons. In the light-heavy gradon region, BO can occur. Depending on the initial wavepacket, breathing-wave-like (BW) oscillations can also occur. The differences are that the initial wavepacket that undergoes BO has a larger spatial width (narrow bandwidth) while the initial wavepacket that undergoes breathing-wave-like has a relatively smaller spatial width (broad bandwidth). In the heavy gradon region, the initial wavepackets consisting of red gradons must be reflected at the right (red) end of the chain, such kind of dynamic evolution is denoted by RR (right end reflection). While in the BG region, left (blue) end reflection denoted by LR occurs. Finally, for the U region, an initially extended wavepacket can be reflected by both ends of the chain, which is denoted by LRR (left and right ends reflection).

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Position x

Figure 15. Dynamics of various wavepackets shown by the contour plots of |φ(x,t)|2 on the x-t domain. (a) Bloch oscillation (BO) (d0 = 5.5a, σx = 5, k0 = 0.8π/d0, and TB = 11.8 ps), (b) breathing-wave-like oscillation (d0 = 5.0a, σx = 0.2, k0 = 0.8π/d0, and TB = 11.8 ps), and (c) reflection from the left (blue) end (d0 = 3.5a, σx = 5, k0 = 0.9π/d0, and TR = 7.3 ps), (d) reflection from the right (red) end (d0 = 4.0a, σx = 5, k0 = 0, and TR = 7.3 ps). To show more clearly the above analysis regarding gradon localization and the occurrence of BO, we compose the contour plots of the intensity profile |φ(x,t)|2 in the positiontime (i.e., x-t) domain for various initial wavepackets. The white color and red color indicate the very strong and relatively strong intensity, respectively, and the black color means the intensity is weak or zero. Figure 15(a) shows the evolution of |φ(x,t)|2 for d0 = 5.5a, σx = 5, and k0 = 0.8π/d0. The wavepacket exhibits an oscillatory motion: the mean position shows a periodic time-dependence while the width is nearly constant. This is a typical plasmonic BO process. More interestingly, in Figure 15(b), we show the case of d0 = 5.0a, σx = 0.2, and k0 = 0.8π/d0. Now the wavepacket’s width shows a periodic time-dependence but the mean position is nearly fixed at the initial place. This is a breathing-wave-like (BW) oscillation. Both initial wavepackets for BO and BW oscillation are only formed by light-heavy gradons which are normal modes localized in the middle part of the graded chain. However, the spatial widths of the two initial wavepackets are significantly different. For BO, the spatial width is larger than that of breathing-wave-like oscillation: the spatial width is σx = 5 for BO and σx = 0.2 for BW. We can estimate the period of Bloch oscillation. In semi-classical theory [18], the

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equation of motion of the Bloch wavenumber k reads k˙ = −∂ω/∂x = f , where f is almost constant for BO. The time taken for k to change by 2π/d 0 is defined as the period of BO, denoted as

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TB =

2π 2πN(2εl + c + ξ)3/2 ≈ . √ f c ηω p

(69)

Here the simplification is obtained by the fact that the average of cos k vanishes in a period. The plasmon resonant frequency [44] is ω p = 1.72 × 1016 rad/s. Thus the period of BO in Figure 15(a) is about 11.8 ps. This is in agreement with the previous studies for optical BO. For non-Bloch oscillations, the time between two neighboring reflections is defined as the period TR . Despite of the oscillation dynamics, we have also investigated the non-Bloch oscillations, e.g., wave reflection from the left end, the right end, or from both ends. Figure 15(c) and Figure 15(d) show the corresponding situations for reflection from the left and right end, respectively. In Figure 15(c) (d0 = 3.5a, σx = 5, k0 = 0.9π/d0), the initial wavepacket is constructed by linear combination of blue gradons only, which are localized modes residing at the left hand side of the chain, this wavepacket can be reflected by the left end of the chain and can not reach the right end. In Figure 15(d) (d0 = 4.0a, σx = 5, k0 = 0), the initial wavepacket is only formed by red gradons which are modes confined at the right hand side of the chain, the wavepacket is reflected at the right end of the chain and can not reach the left end. If the components of initial wavepacket all fall into the extended modes region, the wavepacket can reach both ends and be reflected in multiple fashion before eventually spreading across the whole chain (the contour plot is not shown here). The different features of various oscillations can also be demonstrated in the contour plots of |φ(x,t)|2 in reciprocal position-time (i.e., k-t) domain. For BO or BW, k varies periodically in the range [−π, π], while for other kinds of oscillations, k varies only in part of the range [−π, π]. There are two reasons why we only show the contour plots of |φ(x,t)|2 in real spatial position-time (i.e., x-t) domain. One is the contour plots in real space can show the obvious features of BO, BW, or other kinds of oscillations, but those in k space only show the periodic variation of k with time, which is sensitive to the initial value of k. The second reason is that the difference between BO and BW is more obvious in real space than that in k space.

3.2. Dissipation Effects In realistic metallic structures, there are unavoidable dissipations like radiation and Ohmic loss. The absorption characteristics of graded structure is quite different with that in homogenous structure, this can be demonstrated in damped vibrational lattices [19]. In a one-dimensional graded lattice, the energy damping effect is not similar to that in a homogenous lattice by an argument of perturbation theory [19], which is similar to the approach by Raman and Fan [45]. It is known that damped harmonic oscillation has been studied as a basic textbook problem. Assume u the displacement with respect to the equilibrium position, u˙ is the velocity of the oscillator. If we adopt the damping force of the form −bu, ˙ where b is a damping constant, the equation of motion reads M u¨ = −Ku − bu˙ which admits ˜ ˜ is the complex , where u0 is the amplitude of the displacement, ω a solution of u = u0 e−ßωt

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˜ can be exfrequency, m is the mass, and qK is the force constant. The complex frequency ω

˜ = ω20 − b2 /4M 2 −ßb/2M ≈ ω0 −ßb/2M −b2 /8M 2 ω0 +O(b3 ), pressed in terms of b as ω p √ where ω0 = K/M. In the under-damped case (0 ≤ b/2 KM < 1), the imaginary ˜ gives a factor exp(−bt/2M) to the displacement, and leads to a damping rate part of ω ˜ implies a shift in the normal frequency. Next we conγ0 = b/2M. The real part of ω sider a damped homogeneous harmonic-oscillator chain, the complex frequency have been q ˜ = ω20 − b2 /4M 2 − ßb/2M ≈ solved exactly and can also be expressed in terms of b as ω p ω0 −ßb/2M −b2 /8M 2 ω0 +O(b3 ), where ω0 = 2K(1 − cos k)/M, M and K represents the mass and force pconstant of each oscillator, k is the wave number. For under-damped oscillation (0 ≤ b/2 2KM(1 − cos k) < 1), the damping rate is still b/2M, which is independent of frequency. The frequency shift is approximately −b2 /8M 2 ω0 . In what follows, we aim at solving the relaxation problem in damped graded mass DGSL and show that the relaxation behavior of graded 2D system is different from that of homogeneous system. By using the Bloch theorem, the 2D equations of motion can be reduced to effective 1D equations of motion, After incorporating a damping term, we obtain the following equation of motion: mq u¨q (t) = ∑ Krq [ur (t) − uq (t)] + ψq uq (t) − bq u˙q (t) ,

(70)

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r

where {uq , mq , ψq , bq } denote the displacement about the equilibrium position, mass, onsite potential, and damping constant of particle at site q which runs over q = 1, 2, ..., N in longitudinal sites [14]. Here r denotes all sites of nearest neighbors to the site q in effective 1D chain for a 2D lattice. Note that Krq is force constant of the spring connecting particle q and its nearest neighbor particle r. In this Chapter, we consider the equal damping on all the sites, i.e. bq = b. Therefore the boundary conditions have no obvious effect on the relaxation. Here we impose the fixed boundary conditions on both ends in the gradient direction of 2D lattices. The effective on-site potential resulting from the transverse Bloch wave number ks reads ψq (ks ) = 2KqT (1 − cos ks ) ,

(71)

where KqT and ks are the spring constant and Bloch wave number in the transverse direction. The displacement solutions of Eq. (70) have the form |u(t)i = ∑α |Uießω˜ αt , where |u(t)i = {u1 (t), u2(t), ...uN(t)} and |Ui = {U1,U2 , ...UN } are N dimensional column vectors ˜ is the complex frequency. The velocity vector has of displacements and amplitudes, and ω the form |v(t)i = d(|u(t)i)/dt = ∑α |Vießω˜ αt . Therefore, Eq. (70) can be written in matrix form as      V V −B0 −K0 − Z0 ˜ , (72) iω = I 0 U U where I is identity matrix, B0 = M−1 B, K0 = M−1 K, and Z0 = M−1 Z represent massweighted damping, interparticle interactions, and on-site potential matrix, respectively. Here M−1 is the inverse of mass matrix with diagonal elements Mqq0 = mq δqq0 , where mq is defined as the gradient mass and δ is the Kronecker delta. In the above equations, the other matrices are defined as Bqq0 = bq δqq0 , Zqq0 = ψq δqq0 , and Kqq0 = 2Kq δqq0 − Kq δq,q0 −1 −

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Analytic (Cm=0.50) Numerical (Cm=0.50, b=0.001)

1

Normalized Relaxation Rate

0.8

0.6 ωc2

ωc1 2

1.5 (b) 2

3

2.5 Frequency ω0

3.5

Analytic (Cm=0.75) Numerical (Cm=0.75, b=0.001)

1.5

1 ωc2

0.5 1

2

ωc1 3 Frequency ω0

4

5

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Figure 16. Relaxation rate spectrum in damped 2D graded mass OGSL (N = 1000, m1 = 1.0, ks = 0.5π). (a) For Cm = 0.50, comparison of the analytic result (solid line) with weak damping (b = 0.001) numerical result (dashed line). (b) For Cm = 0.75, comparison of the analytic result (solid line)with weak damping ( b = 0.001) numerical result (dotted line). Critical frequencies ωc1 and ωc2 are presented in Sect. 2.1.2.. Kq−1 δq,q0 +1 , which represents the nearest-neighbor interaction matrix [13, 14]. For bq = 0 ˜ is real, dewe recover the lossless harmonic lattices. In this case, the eigen-frequency ω noted as ω0 . In the weak damping limit, we adopt the Rayleigh-Schrödinger perturbation method to calculate the relaxation rate. The Hamiltonian is defined as H = H0 + bW ,

(73)

where H0 = {{0, −K0 − Z0 }, {I, 0}} is the unperturbed Hamiltonian and bW = {{B0 , 0}, {0, 0}} is the perturbation. For small damping (b  1), we use a perturbation ˜ in terms of b around ω0 approach by expanding the complex frequency ω ˜ = ω0 + ibω(1) + b2 ω(2) + O(b3 ) , ω

(74)

where the first term ω0 is the unperturbed vibrational frequency, the absolute value of the second term |bω(1)| is the relaxation rate, the third term b2 ω(2) approximately measures frequency shift. p Here we consider the under-damped case, the condition 0 ≤ b/2Mω0 < 1, where ω0 = 2K(2 − cos k − cos ks )/M should be satisfied. Thus, the normalized relax-

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0.0 0.08 0.0 0.08 0.0 0.08

0.0 0.0 2.0 2.0 4.0 4.0

323

0.0 0.08 0.0 0.08 0.0 0.08

Figure 17. Comparison of the evolution of wavepackets of a undamped BO ( Γ = 0, solid lines) in the ideal nanoparticle chain and of a damped BO ( Γ = 0.08ω p , dashed lines) in the silver nanoparticle chains (d0 = 5.5a, σx = 5, k0 = 0.8π/d0 ). ation rate (NRR), denoted as γ reads

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γ=

˜ hφ(0)|W |φ(0)i Im(ω) = , b hφ(0)|φ(0)i

(75)

where |φ(0)i is the wave function of the unperturbed Hamiltonian H0 . By directly diagnolizing the Hamiltonian matrix H defined in Eq. (73), we can obtain ˜ and the ˜ and −iω. ˜ Then we obtain the frequency ω0 = Re(ω) complex eigenvalue pairs iω ˜ normalized relaxation rate γ = Im(ω)/b. The numerical results are shown as the dashed lines in Figure 16(a) (Cm = 0.50, ks = 0.5π, b = 0.001), and Figure 16(b) (Cm = 0.75, ks = 0.5π, b = 0.001). We can see that the weak damping numerical results match very well with analytic results [19]. It indicates that it is reasonable to take the first order perturbation to calculate the relaxation rate. For GPCs, we can omit the radiation loss in the longitudinal case. However, the absorption effect is also dramatic, which, however, does not extinguish the gradon effect. Examples can be found in Figures 11(g) and 11(h). Note that the gradon effect and the absorption effect can become indistinguishable with respect to the mode decaying from the localization center. For wavepacket dynamics, we examine the damping effect by taking into account the loss in the metal nanoparticles. For the damping case ( Γ 6= 0), the equation of motion is modified by simply replacing ω2 with ω(ω+iΓ) that is used in Section 2.3.. Thus the eigen˜ = ω0n + iγn, where γn = Γ/2 and ω0n = ωn (1 − Γ2 /8ω2n ). values become complex-valued ω Thus the wavepacket is damped by an overall factor e−Γt/2 . It is also possible to encompass the Ohmic loss by solving the eigenvalue problem of a dynamic matrix (not necessarily Hermitean). We can obtain the damping spectrum by a simple perturbation theory [19]. Since the radiation damping is included in the third order time derivative of polarization P in the equations of motion, we must define additional auxiliary fields to handle plasmonic problems with radiation damping. Figure 17 (d0 = 5.5a, σx = 5, k0 = 0.8π/d0) shows the evolution of wavepackets for undamped (Γ = 0, solid lines) BO in an ideal chain and damped ( Γ = 0.08ω p , dashed lines)

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BO in silver nanoparticle chains. At t = t0 (black lines) and t = t0 + 0.5TB (blue lines), the wavepacket reaches at the right-most and left-most of the BO region. The damped wavepackets almost overlap the undamp ones. We can observe that the wavepacket at the left-most is higher and narrower than that at the right-most. This is due to the width of wavepackets also oscillates with time, which is similar to the photonic BO in electrically modulated photonic crystals. As time goes on, at t = t0 + 2.0TB (red lines) and t = t0 + 4.0TB (green lines), the intensity of damped BO becomes smaller than that of the corresponding undamp one, with the damping rate Γ/2. At t = t0 + 2.5TB (orange lines) and t = t0 + 4.5TB (light green lines), the wavepacket of damped BO is higher and narrower than that of corresponding undamp one. The amplitude of oscillation (i.e., distance between the leftmost peak and right-most peak) is smaller in damped case than that in undamp case.

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4.

Conclusion

In summary, in this Chapter, we review the most fundamental part of graded plasmonic structures and discuss some of the prominent aspects of plasmon modes in this kind of structures. A peculiar plasmon mode, called plasmonic gradon which has flexible and frequency selective spatial extension and confinement center, is identified in various graded plasmonic structures, such as one-dimensional graded nanoparticle chain and graded nanoparticle chain waveguides. Optical signal steering inside such structures are studied theoretically and demonstrated numerically. Interplay of gradon confinement and propagation of optical signals is outlined. The plasmonic gradon confinement is analyzed with the aid of a quantum analogy and modeled in graded elastic networks which also give deep insight to the change of absorption characteristics with respect to the homogenous counterparts. As gradon modes and gradon transitions are nonuniversal, they offer great potential for applications in management of uncharged particles or controlling wave propagation by graded materials. In fact, we have also studied planar graded optical waveguide arrays with linearly varying propagation constant which can be obtained by taking advantage of the electro-optic effect. The waveguide array takes a planar structure composed of individual waveguides, in each of which light propagates along the longitudinal axis of waveguide, i.e., the z direction. The evanescent fields leaking from nearby waveguides are coupled leading to a collective supermode. The EM modes in such structure is thus quasi-two-dimensional and the supermodes are laterally (or transversely) confined by the gradon mechanism [23]. A further exploration on the spatio-temporal manipulation of light on graded plasmonic structure would be interesting for both ultrafast and active plasmonics. The phase diagram can aid proper design of such plasmonic devices for various nanophotonic applications.

Acknowledgments This work was supported partially by NSFC (No. 11004043), RGC General Research Fund of the Hong Kong SAR Government, and a Grant-in-Aid for Scientific Research (No. 22560058) from Japan Society for the Promotion of Science. We thank all our colleagues and collaborators on this subject for various discussions over the past years. We

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have learned tremendously from them and enjoy and appreciate the joint research efforts to better understand the gradon physics in plasmonic structures.

References [1] Zayats, A. V.; Smolyaninov, I. I.; Maradudin, A. A. Phys. Rep. 2004, 408, 131-314. [2] Zouhdi, S.; Sihvola, A.; Vinogradov, A. P. Metamaterials and Plasmonics: Fundamentals, Modelling, Applications ; Springer: Berlin, 2008. [3] Maier, S. A. Plasmonics: Fundamentals and Applications ; Springer: Berlin, 2007. [4] Schuller, J. A.; Barnard, E. S.; Cai, W.; Jun, Y. C.; White, J. S.; Brongersma, M. Nat. Mater. 2010, 9, 193-204. [5] Gramotnev, D. K.; Bozhevolnyi, S. I. Nat. Photonics 2010, 4, 83-91. [6] Gopinath, A.; Boriskina, S. V.; R.Premasiri, W.; Ziegler, L.; Reinhard, B. M.; Negro, L. D. Nano Lett. 2009, 9, 3922-3929. [7] Yu, Z.; Veronis, G.; Wang, Z.; Fan, S. Phys. Rev. Lett. 2008, 100, 023902(1-4). [8] Shin, H.; Fan, S. Phys. Rev. Lett. 2006, 96, 073096(1-4). [9] Penninkhof, J. J.; Moroz, A.; van Blaaderen, A.; Polman, A. J. Phys. Chem. C 2008, 112, 4146-4150.

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[10] Atwater, H. A.; Polman, A. Nat. Mater. 2010, 9, 205-213. [11] Lai, Y.; Ng, J.; Chen, H.; Han, D. Z.; Xiao, J. J.; Zhang, Z. Q.; Chan, C. T. Phys. Rev. Lett. 2009, 102, 253902(1-4). [12] Xiao, J. J.; Zheng, H. H.; Sun, Y. X.; Yao, Y. Opt. Lett. 2010, 35, 962-964. [13] Xiao, J. J.; Yakubo, K.; Yu, K. W. Phys. Rev. B 2006, 73, 054201(1-7). [14] Xiao, J. J.; Yakubo, K.; Yu, K. W. Phys. Rev. B 2006, 73, 224201(1-9). [15] Huang, J. P.; Yu, K. W. Phys. Rep. 2006, 431, 87-172. [16] Longhi, S. Phys. Rev. E 2007, 75, 026606(1-5). [17] Yang, N.; Li, N.; Wang, L.; Li, B. Phys. Rev. B 2007, 76, 020301(1-4). [18] Zheng, M. J.; Xiao, J. J.; Yu, K. W. J. Appl. Phys. 2009, 106, 113307(1-5). [19] Zheng, M. J.; Xiao, J. J.; Yakubo, K.; Yu, K. W. J. Phys. Soc. Jpn. 2009, 78, 124603(16). [20] Xiao, J. J.; Yakubo, K.; Yu, K. W. J. Phys.: Condens. Matter 2007, 19, 026224(1-10). [21] Xiao, J. J.; Yakubo, K.; Yu, K. W. J. Phys. Soc. Jpn. 2007, 76, 024602(1-4). Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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[22] Yakubo, K.; Xiao, J. J.; Yu, K. W. Physica B2007, 394(2), 262-266. [23] Zheng, M. J.; Xiao, J. J.; Yu, K. W. Phys. Rev. A2010, 81, 033829(1-7). [24] Xiao, J. J.; Yakubo, K.; Yu, K. W. Appl. Phys. Lett. 2006, 88, 241111(1-3). [25] Xiao, J. J.; Yakubo, K.; Yu, K. W. Appl. Phys. Lett.2006, 89, 221503(1-3). [26] Xiao, J. J.; Yakubo, K.; Yu, K. W. Physica B 2007, 394(2), 208-212. [27] Malyshev, A. V.; Malyshev, V. A.; Knoester, J. Nano Lett. 2008, 8, 2369-2372. [28] Li, K.; Stockman, M. I.; Bergman, D. J. Phys. Rev. Lett. 2003, 91, 227402(1-4). [29] Lévêque, G.; Martin, O. J. F. Phys. Rev. Lett. 2008, 100, 117402(1-4). [30] Hofmann, H. F.; Kosako, T.; Kadoya, Y. New J. Phys. 2007, 9, 217 (1-12). [31] Stockman, M. I. In Plasmonic Nanoguides and Circuits ; Bozhevolny, S. I., Ed.; World Scientific Publishing: Singapore, 2008; Chapter Adiabatic Concentration and Coherent Control in Nanoplasmonic. [32] Tsakmakidis, K. L.; Boardman, A. D.; Hess1, O. Nature 2007, 450, 397-401. [33] Gan, Q.; Gao, Y.; Wagner, K.; Vezenov, D. V.; Ding, Y. J.; Bartoli, F. J. Experimental verification of the "rainbow" trapping effect in plasmonic graded gratings , arXiv:1003.4060, 2010. [34] Verslegers, L.; Catrysse, P. B.; Yu, S.; Fan, S. Phys. Rev. Lett. 2009, 103, 033902(1-4).

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[35] von Freymann, G.; John, S.; Kitaev, V.; Ozin, G. A. Adv. Mater. 2005, 17, 1273-1276. [36] Centeno, E.; Cassagne, D.; Albert, J. P. Phys. Rev. B 2006, 73, 235119(1-5). [37] Luo, D.; Alagappan, G.; Sun, X. W.; Raszewski, Z.; Ning, J. P. Opt. Commun. 2008, 282, 2329-332. [38] Kurt, H.; Colak, E.; Cakmak, O.; Caglayan, H.; Ozbay, E. Appl. Phys. Lett. 2008, 93, 171108(1-4). [39] Kurt, H.; Citrin, D. S. Opt. Express.2007, 15, 1240-1253. [40] Centeno, E.; Cassagne, D. Opt. Lett. 2005, 30, 2278-2280. [41] Wang, S. M.; Xiao, J. J.; Yu, K. W. Opt. Commun. 2007, 279, 384-389. [42] Ling, C. W.; Zheng, M. J.; Yu, K. W. Opt. Commun. 2010, 283, 1945-1949. [43] Yakubo, K.; Nakayama, T. Fractal concepts in condensed matter physics ; Springer: Berlin, 2003. [44] Xiao, J. J.; Huang, J. P.; Yu, K. W. Phys. Rev. B 2005, 71, 045404(1-8). [45] Raman, A.; Fan, S. Phys. Rev. Lett. 2010, 104, 087401(1-4). Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

In: Plasmons: Theory and Applications Editor: Kristina N. Helsey, pp.327-343

ISBN: 978-1-61761-306-7 c 2011 Nova Science Publishers, Inc.

Chapter 14

I NFRARED S URFACE P LASMON S PECTROSCOPY OF L IVING C ELLS M. Golosovsky ∗, V. Yashunsky, A. Zilberstein, T. Marciano, V. Lirtsman, D. Davidov, B. Aroeti The Racah Insitute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel Department of Cell and Animal Biology, The Alexander Silberman Institute of Life Sciences, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel

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Abstract We report a spectroscopic technique that combines the Fourier-Transform Infrared Spectroscopy with the Surface Plasmon Resonance. This tool enables sensitive infrared spectroscopy of liquid and solid objects in the attenuated total reflectance mode. The FTIR-SPR technique is similar to FTIR-ATR technique but has higher sensitivity due to resonance amplification of the surface electric field. The label-free FTIR-SPR technique is especially advantageous for living cell studies since it combines spectroscopic information inherent to FTIR with structural information provided by the Surface Plasmon Resonance. We discuss the FTIR-SPR technique for label-free studies of cell sedimentation and spreading on substrate and for surface plasmon spectroscopy. PACS 05.45-a, 52.35.Mw, 96.50.Fm. Keywords: surface plasmon, infrared, FTIR, biosensor, cell culture

1.

Introduction

Surface plasmon resonance (SPR) technique [1, 2, 3, 4] became an important research tool in biophysics. SPR measures refractive index and optical absorption with high sensitivity and is particularly advantageous for biosensing since it can monitor kinetics of biological processes on surfaces and in solutions in real time. Although most SPR applications focus ∗

E-mail address: [email protected]

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on the visible range, the SPR setups operating in the near-infrared have been reported as well [5, 6, 7, 8, 9]. The advantages of the near-infrared range are related to the reduced conductor losses and to the concomitant increase in sensitivity. Refs. [5, 6] pioneered the surface plasmon excitation using FTIR spectrometer. This resulted in the powerful near-infrared surface plasmon technique with wavelength interrogation that offers several advantages over conventional SPR techniques based on angular interrogation. Recently, we extended the FTIR-SPR technique into mid-infrared range by using metallic mirrors and ZnS prism [10, 11]. The FTIR-SPR technique is particularly useful for real-time and quantitative measurements of dynamic processes in living cells. Indeed, the label-free SPR technique in the visible range has been successfully used in cell culture studies [14, 15, 16, 17]. However, since in the visible range the penetration depth of the surface plasmon wave into the cell is only ∼ 0.1µm, the SPR technique using visible light is mostly sensitive to the shape/mass redistribution occurring in the vicinity of cell basal membrane. In other words, the surface plasmon in the visible range senses mostly cell-substrate contact sites and is less sensitive to the processes occurring deeper in the cell. Here, the infrared surface plasmon is more advantageous since its penetration depth of a few µm is comparable to the cell height and it can sense the intracellular processes as well. In the context of biosensing the FTIR-SPR technique has several attractive features: high sensitivity to refractive index changes, variable probing depth, and spectroscopic capabilities. So far we used this technique to study alterations in membrane cholesterol levels in HeLa cells [12], de novo formation of early endocytic transport vesicles in response to transferrin internalization by human melanoma cells [13], and for detection of glucose uptake by erythrocytes [11]. While spectroscopic use of the SPR technique in the visible and near-infrared ranges has been demonstrated [7, 18, 19, 20], the most useful spectroscopic information appears usually in the mid-IR range. So far, FTIR spectroscopy has been performed in the transmission/reflection mode or using ATR setup. This is an important challenge to combine mid-IR spectroscopy and SPR. While conventional FTIR techniques aim at measuring absorption, the SPR technique measures a complementary parameter, namely, refractive index. The FTIR-SPR technique can be particularly useful for living cell cell studies. So far, FTIR absorption spectroscopy of living cells has been performed using synchrotron radiation [21, 22], ATR [23, 24, 25, 26], or evanescent field of the infrared fiber [27]; while the surface plasmon has been mostly used for detecting structural modifications in cells. Combination of surface plasmon resonance with spectroscopy can be particularly useful in the study of drug penetration into cells, since it can measure not only the spectroscopic signature of drug but the cell reaction to this drug as well.

2.

Methodology

The surface plasmon resonance at the metal-dielectric interface is usually excited in the Kretschmann’s geometry (Figs. 1,2) that employs high-refractive index prism operating in the regime of attenuated total reflection (ATR). The prism is coated with a thin metal film which is in direct contact with an analyte (liquid, thin film, or living cell layer). The reso-

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nance appears when the surface plasmon wave vector satisfies the phase matching condition 0 ksp = kx = k0np sin Θ

(1)

where k0 is wave vector in free space, np is the refractive index of the prism, Θ is the incident angle at the prism-metal interface. The real and imaginary parts of the surface plasmon wave vector are 0 00 + iksp = k0 ksp



m d m + d

1/2

(2)

where d (λ) is the dielectric permittivity of the analyte and m (λ) is the (negative) dielectric permittivity of the metal. For fixed incident angle and for wavelength interrogation, the resonant condition reduces to: n2d = d =

|m |n2p sin2 Θ |m | + n2p sin2 Θ

(3)

The surface plasmon resonance appears as a reflectivity minimum at certain wavelength λ which is inexplicitly given by Eq. 3. The reflectivity at the resonance can be usually approximated by a Lorentzian [28] R = |rF |2

00 Γ 1 − ksp 0 )2 + (k 00 + Γ)2 (kx − ksp sp

(4)

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where rF is the Fresnel reflection coefficient at the prism-metal interface and Γ is the radiation (coupling) loss which appears when the surface plasmon propagates along a thin metal film rather than along a bulk metal surface. The minimal reflectivity k00 − Γ sp Rmin = rF 00 ksp + Γ

(5)

0 . For optimal coupling, which is achieved by proper occurs at resonance condition, kx ≈ ksp 00 and Rmin = 0. choice of the metal film thickness, Γ = ksp For inhomogeneous medium whose properties vary in the z-direction, the refractive index of the analyte sensed by the evanescent wave is f = δ −1 nef d

Z ∞ 0

z

nd (z)e− δ dz

(6)

where δ is the surface plasmon penetration depth, δz =

λ (|m | − n2d ) 4πnd

(7)

The surface plasmon in the infrared range is usually used for highly sensitive refractometry. Tiny variations of the analyte refractive index, ∆n, are found from the shift of the surface plasmon resonance ∆ν = S∆n (8)

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where ν is the resonance wave number and S is the bulk sensitivity which can be found from Eqs. 1,3. In comparison to its visible range counterpart, the FTIR-SPR technique allows wavelength interrogation in addition to conventional angular interrogation. However, the requirements to conducting film are more stringent than for the surface plasmon technique in the visible range. On the one hand, the infrared surface plasmon is very narrow (and is therefore more sensitive) due to decreased conducting losses. On the other hand, to get full advantage of this enhanced sensitivity, the conducting film should be very thin. In the context of living cells and bacteria studies, the spectroscopic capability of the FTIR-SPR technique can be used to pinpoint in real time and label-free manner the structural changes in cells and bacteria. However, this imposes additional limitations, in particular, the requirement of biological compatibility practically excludes all other conducting films besides Au. In what follows we describe this FTIR-SPR technique and show several applications.

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3.

Experimental Setup

Figures 1,2 show our experimental setup. The infrared beam is emitted from the external port of the Bruker Equinox 55 FTIR spectrometer and passes through the 1 mm diameter pinhole collimator. The beam diameter is 3-4 mm and beam divergence is 0.52 0. The collimated beam is polarized using the grid polarizer, it is reflected from the right-angle ZnS prism coated with 12-20 nm thick Au film, and is focused by the parabolic mirror onto the liquid-nitrogen-cooled MCT detector. A temperature-stabilized flow chamber with analyte solution is in contact with the Au-coated base of the prism. For each incident angle we measured the reflectivity of the s-polarized beam and used it as a background for further measurements. Then we measured the reflectivity spectrum of the p-polarized beam at the same angle and identified the dip corresponding to surface plasmon resonance. For spectroscopy of solid and liquid samples and for some living cells studies we used the vertical setup (Fig. 1) since it allows both wavelength and angular interrogation. Here the prism with the attached flow chamber and the detector are mounted on the Θ − 2Θ rotating stage. At each incident angle we analyzed the reflectivity spectra, identified the surface plasmon resonance, and determined real and imaginary part of the refractive index of the analyte solution using Fresnel reflectivity formulae. For living cell studies, we used the horizontal setup (Fig. 2) that operates at fixed incident angle and allows only wavelength interrogation. The flow chamber contains growth solution that sustains cell functioning during experiment. The prism- flow chamber assembly is mounted on the vertical translation stage. The living cells are either grown directly on prism or injected into flow chamber and allowed for sedimentation. By adding chemicals and drugs into the growth solution we triggered different cell processes and studied cell response to these processes using surface plasmon resonance. The rationale for the horizontal arrangement of the flow chamber is the compatibility with the optical microscopy. Indeed, synchronously with the surface plasmon measurements, we took the optical timelapsed images of the gold-coated prism surface and the cells on it. This was done using high magnification optical zoom lens and a digital camera.

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331

Surface Plasmon Spectroscopy

Since the wavelength λ does not appear explicitly in Eq. 3, the resonance condition at fixed angle can, in principle, be satisfied for several wavelengths or for some range of wavelengths. This allows surface plasmon spectroscopy. To be specific, we consider a water solution in contact with the gold-coated ZnS prism. Figure 3 shows the angle Θext (ν), corresponding to the surface plasmon resonance at water/Au/ZnS interface at given wave number ν, using 450 ZnS prism (the angle is defined as in Fig. 2). Since dielectric constant of the gold at infrared frequencies is very high, Au >> 1, the angular range where surface plasmon can be excited is very close to the critical angle i.e., sin Θsp ≈ nwater /nZnS . The resonant wave number can be found from the intercept of the horizontal line Θ = Θext with the dependence Θext (ν), as shown in the Fig. 3.

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For Θ = 30.50 the surface plasmon is excited only at one wavelength, ν = 2916 cm−1 . The resonance appears in the spectral region where Θ(ν) dependence is very steep, hence the width of the surface plasmon is quite modest, ∆ν = 200 cm− 1. For Θ = 22.50 the surface plasmon is excited at several wavelengths: ν = 1470 cm−1 , 1970 cm−1 and at 5173 cm−1 . The most important resonance at ν = 5173 cm−1 appears in the spectral range where the slope of Θ(ν) dependence is small hence the width of the resonance is extremely large - ∆ν = 2000 cm−1 . Both resonances are sufficiently wide to allow spectroscopy within the resonance curve even at one incident angle. Figure 4 shows the corresponding experimental results. For Θ = 300 there is indeed a single narrow dip at ν = 2916 cm−1 while for Θ = 220 there is a wide resonance at 5173 cm −1 .

Figure 1. Surface plasmon resonance in the Kretschmann’s geometry. Vertical configuration allows angular and wavelength interrogation and is useful for spectroscopy. Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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Contrary to common wisdom, the width of the surface plasmon resonance at wavelength interrogation is not directly related to losses, but is determined mostly by dispersion. The plasmon at ν = 5173 cm−1 is very wide because the water dispersion in the corresponding spectral range is low. The losses for the narrow, ν =2916 cm−1 , resonance are in fact higher than for the broad resonance at ν = 5173 cm−1 . The broad surface plasmon resonance allows highly sensitive spectroscopic measurements. Indeed, weak water absorption peaks, indicated by arrows in the Fig. 4, are well resolved even without processing and deconvolution. To use the surface plasmon resonance as a spectroscopic tool, the reflectivity under the surface plasmon regime should be converted into refractive index spectra. This requires modelling of the surface plasmon reflectivity. Figure 4 shows that quite an impressive fit can be achieved using a simple optical model based on Fresnel reflectivity expressions. The fit does not take into account the interface roughness but does account for beam divergence. The fitting parameters here are the optical constants of Au film. Once these are found the model prediction for other angles is calculated without fitting parameters.

Figure 2. Surface plasmon resonance in the Kretschmann’s geometry. Horizontal configuration allows wavelength interrogation and is favorable for living cell studies since it is easily combined with optical microscopy. At the next step we measured the surface plasmon reflectivity from an unknown analyte and determined its refractive index spectrum using a set of spectra at different angles. We used this approach to determine the refractive index of living cell monolayer grown directly on the gold-coated ZnS prism. Figure 5 shows the refractive index of the cells. It gener-

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Figure 3. Incident angle corresponding to the excitation of the surface plasmon resonance at the water/Au/ZnS interface using 45 0 ZnS prism. For Θ = 30.50, the surface plasmon is excited at a single wave number, 2916 cm −1 , and its with is only 200 cm −1 . For Θ = 22.50, the surface plasmon is excited at 5173 cm −1 and its width is 2000 cm −1 The angle, Θext , is defined as in Fig.2. ally exceeds the water refractive index since the cells contain up to 30 % organic substances whose refractive index is higher than that of the water. Figure 6 shows the exploded view of the spectral region corresponding to absorption of the aliphatic bonds. For better visibility we plot there the first derivative of the refractive index. Three characteristic peaks corresponding to absorption bands of the CHx groups are clearly visible.

5. 5.1.

Living Cell Studies Cell Culture

We studied several epithelial cell lines i.e., MDCK, HeLa, A431, human melanoma cells (MEL 1106). From the point of view of the physicist, all these cells are similar in composition (70% of water, 30% of organic substances and salts), in shape (the cell in solution is a sphere with the radius ∼6 µm), but they differ in their ability to spread on different substrates and to form continuous monolayers. Here we focus on MDCK (Madin-Darby canine kidney) epithelial cells that have a strong tendency to form continuous monolayer. In this work we study the MDCK cell sedimentation and spreading on substrate. The MDCK cells were cultured routinely in modified Eagle’s medium (MEM, Biologi-

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cal Industries, Israel), supplemented with 4.5 g/l D-glucose, 10 % antibiotics, and 10 % fetal calf serum. A confluent MDCK monolayer cultured on a 10-cm dish, was washed twice with pre-warmed PBS and the cells were detached from the dish with trypsin/EDTA (0.25 % Trypsin/EDTA in Puck’s saline A; Biological Industries, Israel). Following trypsinization, the cells were resuspended in 10 ml minimum essential medium (MEM) Hanks’ salts supplemented with 20 mM Hepes, (pH 7.5) and 10% fetal calf serum.

5.2.

Experimental Protocol

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(a) The flow chamber was attached to the prism and filled with growth medium. The prism-flow chamber assembly was incubated for 2 hours at 37 0 C. Figure 7a shows the optical micrograph of the Au-coated prism exposed to growth medium (prior to cell sedimentation).

Figure 4. Infrared reflectivity from the water/Au/ZnS interface under Kretschmann’s configuration at two incident angles. The pink curve corresponds to a narrow-band surface plasmon excited in the spectral range where water dispersion is strong. The blue curve corresponds to the surface plasmon excited in the spectral range where the the water dispersion is weak. Broad surface plasmon explicitly reveals absorption peaks of water (double arrows). Continuous lines indicate model prediction using known optical constants of water. Note excellent agreement to experimental data. (b) The growth medium with cell suspension ( ∼ 5 × 105 cells/ml) was injected at a rate of 200 µl/min. The injection continued for 15 minutes and another 5 minutes were allowed for cells in solution to sediment onto the substrate. Figure 7b shows that at the end of the sedimentation phase the gold substrate is covered by spherical cells that do not contact each other. The surface cell density at the end of the injection phase was Cs ∼ 1700 cells/mm2.

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(c) The growth medium without cells was injected under constant flow rate of 25 µl/min during ∼ 400 minutes. During this time the cells spread on the substrate, lose their spherical shape and become more flat in such a way that the cell projection grows (Fig. 7c). Then the cell-cell attachment occurs and the clusters of several cells are formed. Eventually the clusters merge and develop a loose percolating cluster. Finally, the cells form a continuous monolayer with few isolated voids that slowly close over the time. The cell surface density almost did not increase during the measurements since the duration of our experiment was too short in order that considerable cell proliferation occur.

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(d) The growth medium flow was stopped and 0.05 % Trypsin/EDTA solution was introduced. This results in disruption of the cell-substrate and cell-cell contacts. Figure 7d shows that the cells regained their spherical shape while they are still residing on the substrate.

Figure 5. Refractive index of the living MDCK cell monolayer.

5.3.

SPR Measurements of Cell Sedimentation and Spreading on Substrate

5.3.1. SPR spectra All these processes were monitored by the FTIR-SPR technique. Each FTIR spectrum took 25 sec. (averaging over 8 scans, 8 cm−1 resolution, incident angle Θext = 21.350) and was synchronized with the optical time-lapsed imaging of the flow chamber contents. Figure 8 shows reflectivity spectra from the growth medium/cells/Au/ZnS prism for different phases

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of cell sedimentation and spreading. The black open symbols show reflectivity from the gold-coated ZnS prism in contact with growth medium, before cell seeding (see Fig. 7a). The surface plasmon resonance at 4263 cm −1 is clearly seen. The feature at 5180-5310 cm−1 arises from the water absorption peak. Upon formation of cell monolayer (Fig. 7c), the surface plasmon resonance shifts from 4263 cm −1 to 3903 cm−1 . This 390 cm−1 shift corresponds to increased refractive index of the cells with respect to that of the growth medium (mostly water), as it is shown in Fig. 5. The dip at 4470 cm −1 arises from the guided mode resonance in the cell monolayer and signifies formation of continuous cell monolayer, as expected for epithelial cells.

2922 cm

2956 cm

2850 cm

1 10

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st

Refractive index, 1 derivativ e

0 10

2 10

3 10

cells water 4 10 2800

2850

2900

2950

3000

-1

Wavenumber (cm ) Figure 6. First derivative of the refractive index of the living MDCK cell monolayer. Note three peaks corresponding to CHx absorption bands. The red dashed curve shows corresponding data for pure water calculated using the data of Ref. [35].

Following trypsinization, the cell-cell and cell-substrate contacts are disrupted and the cells regain their spherical shape, while touching the substrate only at the apex of the sphere (Fig. 7d). Then, the surface plasmon propagates mostly along the growth medium-Au interface and almost does not feel the cells. In accordance with this scenario, the surface plasmon resonance shifted back to 4250 cm −1 and the guided mode resonance disappeared. The degree of the surface plasmon penetration into cells can be judged from the very small residual 13 cm−1 shift of the surface plasmon resonance (see inset). The presence of cells on substrate is also manifested by the strong decrease of reflectivity at 4400-4900 cm −1 . We believe that this feature is strongly sensitive to scattering on cells.

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Figure 7. Optical micrographs of the Au-coated ZnS prism and the MDCK cells thereon at different phases of sedimentation, spreading, and trypsinization. The schematic drawing indicates cell shape during these phases. 5.3.2. Which cell properties are measured by the surface plasmon resonance?

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We demonstrate here that the surface plasmon resonance senses the surface cell concentration Cs and cell projection area, A. For quantitative determination of these parameters we generally followed the model developed by Refs. [29, 30]. This model assumes that the cell initially attaching to the substrate has a spherical shape with radius r0. Upon cell spreading, its shape changes from the sphere with the volume V = 4πr03/3 to the spherical cap (see 2 2 Fig. 7c) with the volume, V = πh 6 (h + 3r ). Here, h is the cell height over the substrate and r is the curvature radius of the cell surface. Since the cell volume is generally preserved upon spreading, this results in an inexplicit relation between r and h. The surface plasmon shift upon cell sedimentation is determined by the effective refractive index of the cells immersed in growth medium [29]: ∆n = Cs

Z ∞

z

(ncell − nmedium )A(z)e− δ

0

dz δ

(9)

where Cs is the cell concentration, ncell and nmedium are refractive indices of the cell and of the growth medium, correspondingly, δ is the surface plasmon penetration depth, and A(z) is the cross-section of the cell at the height z above the substrate. Equation 9 suggests that during the sedimentation phase, the surface plasmon measures cell concentration, Cs . After sedimentation phase is over, the surface plasmon resonance shift measures the cell cross-section weighted over surface plasmon penetration depth ( δ ≈ 2 µm for our wavelength range). At the end of the spreading process the cells form monolayer with the height h ∼ 8 µm. Since h > δ, this means that at least at the advanced phases of cell spreading, Eq. 9 reduces to ∆n ≈ Cs (ncell − nmedium )A where A = πh(2r − h) is the cell projection area i.e., the cell cross-section as seen by the optical microscopy. This was verified by our simultaneous optical microscopy and surface plasmon measurements. The reflectivity under the surface plasmon resonance is determined mostly by scattering, since our range of wavelengths does not include the absorption bands of cell constituents. The scattering of surface plasmon waves has been extensively studied mostly for regime of small and strong scatterers [33]. The cell size considerably exceeds the wavelength,

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Figure 8. Infrared reflectivity spectra from the layer of MDCK cells on Au-coated ZnS prism. The open black circles indicate reflectivity from the bare Au-coated ZnS prism exposed to growth solution consisting mostly of water (Fig. 7a). Note the surface plasmon dip at 4263 cm −1 . The feature at 5173 cm−1 arises from the water absorption peak. The red filled circles indicate reflectivity from the prism with a confluent MDCK cell monolayer (Fig. 7c). The surface plasmon dip has been shifted to 3900 cm −1 and the guided mode dip at 4470 cm−1 has appeared. The filled blue circles indicate reflectivity from the cell layer on the prism immediately after adding a small quantity of trypsin (Fig. 7d). The trypsinization disrupts the cell-cell and cell-substrate contacts, the cells regain the spherical shape and their contact with the substrate is minimized. Note, that the SP dip has been shifted back to 4250 cm−1 and the guided mode peak disappeared. The reflectivity at 4400-4900 cm −1 has been decreased indicating increased SP scattering. The inset zooms into the SP dip. The small difference between the reflectivity of bare Au substrate and the substrate with spherical cells is visible, indicating that the cells were not washed out.

ncell /nmedium ∼ 1, hence the cell is a large and weak scatterer. In the absence of appropriate theoretical model describing surface plasmon scattering on such large and weak scatterers as cells, we restrict ourselves to qualitative treatment of the problem based on the Beer’s law for surface waves. This law states that the imaginary part of the wave vector is 00 = C l proportional to the concentration of scatterers, ksp s cell , where lcell is the scattering cross-section of a single cell. Since the cells are large scatterers whose size exceeds the surface plasmon wavelength by an order of magnitude, it is natural to assume that the scattering

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occurs at cell-growth medium interface. Then lcell ∝ P where P is the cell perimeter. This reasoning reduces to conjecture that the surface plasmon reflectivity measures the length of the cell-growth medium interfaces per unit area. 5.3.3. Kinetics of cell spreading

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Time dependence of the surface plasmon shift ∆ν and reflectivity Rmin , yields kinetics of cell sedimentation and spreading. Figure 9 shows how the surface plasmon resonance has been shifted with time. The SPR shift varies fast during cell spreading phase and achieves saturation after 50 min. At this phase ∆ν ∝ Cs since the cells are still spheres and A is constant. After cell sedimentation is over, ∆ν grows slowly and this growth is associated with cell spreading i.e., increase of A. This ∆ν growth is not a steady process but goes through several phases, representing different strategies of cell spreading.

Figure 9. Time dependence of the surface plasmon shift νmin (lower panel) and the reflectivity at the surface plasmon dip, Rmin (upper panel) during cell sedimentation and spreading. t = 0 corresponds to cell injection. Upon cell sedimentation and spreading, the cell mass which is in close contact with prism increases and the SP resonant wave number is constantly shifted to the red, indicating increasing refractive index sensed by the SP wave. The Rmin is mostly determined by the scattering on cells and varies non monotonously through the cell deposition process. After trypsinization, when the cells regain the spherical shape and are eventually washed out, the SP wave number and reflectivity almost reversibly return to their initial values before cell deposition.

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These phases are also seen in the reflectivity. Rmin grows fast upon cell sedimentation due to increasing cell concentration Cs and goes to saturation after sedimentation is over. However, after 80 minutes the reflectivity starts to grow again, passes through the maximum, and decreases. After trypsinization, the reflectivity and the surface plasmon shift vary in reverse order (not shown here) and almost regain their initial values before cell spreading. Basing on our simultaneous optical measurements, we validated the above scenario, determined the cell projection and found surface coverage, f = Cs A. Following the approach adopted in the context of thin film growth, we plot in Fig. 10the reflectivity (in other words, interface perimeter per unit area) versus surface coverage. Figure 10 shows different phases of cell spreading. The end of the cell sedimentation phase corresponds to f = 0.025. After sedimentation is over, the surface coverage grows due to spreading of individual cells. When f = 0.5 − 0.6 the cells already contact one another and their further spreading is impeded. This bears some resemblance to formation of percolating cluster.

Figure 10. Reflectivity at the dip of the surface plasmon resonance, Rmin , versus cell surface coverage, f , during cell spreading. The range below f = 0.6 indicates individual cell spreading followed by clusters formation. The range above f = 0.6 is probably related to another spreading regime where the cells seek the voids in the cell layer and try to fill them. This is associated with enhanced surface plasmon scattering. When f ≈ 0.6, the cells obviously switch to another spreading regime, as revealed by increase in reflectivity without marked increase in the surface plasmon shift. We believe

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that for f < 0.6 the cells spread and occupy the nearby areas of substrate. When f < 0.6 the cell layer becomes too crowded and the cells should explore the remote places as well. It is known that to spread the cells send thin lamellipodia [36] to explore the substrate and to find unoccupied areas. Using this knowledge the cells cover these areas in such a way that the continuous monolayer is formed. We believe that this spreading regime is associated with increased lamellipodia production. While the lamellipodia perimeter is large (hence they are sensed through surface plasmon reflectivity), their thickness ( 0.1−0.3 µm) is significantly smaller than the surface plasmon penetration depth, hence the surface plasmon shift is less affected. The reflectivity sharply drops when the filling factor exceeds f = 0.75. Incidentally, this threshold is very close to that of random close packing of hard disks, fRCP = 0.82 [34]. The reason for the reflectivity drop is the decrease of scattering associated with void shrinking in cell monolayer.

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6.

Conclusions

We demonstrated here that the infrared surface plasmon can be used as a spectroscopic tool to measure infrared spectra in a thin layer. There are two distinctive features that make surface plasmon so different from conventional infrared spectroscopy techniques, such as attenuated total reflection and optical fibers. First, the surface plasmon measures refractive index in such a way that it is a complementary method to ATR that measures absorption. We developed an optical model of the reflectivity from multilayers in the regime of the surface plasmon resonance. This model allows quantitative determination of the refractive index spectrum of the analyte. Second, the surface plasmon has larger penetration depth ( ∼ 1 µm as compared to 0.5-0.7 µm in ATR). This is especially advantageous for living cell studies since surface plasmon probes the cell interior. The potential of the surface plasmon resonance for quantitative real-time and label-free studies of living cells on substrate has been demonstrated.

7.

Acknowledgments

This research has been supported by the Israeli Ministry of Industry and Trade through the NOFAR program.

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[5] A. G. Frutos, S. C. Weibel, R. M. Corn, Anal. Chem. 71, 3935 (1999). [6] B. P. Nelson, A. G. Frutos, J. M. Brockman, R. M. Corn, Anal. Chem. 71, 3928 (1999). [7] A. Ikehata, T. Roh, Y. Ozaki, Anal. Chem. 76, 6461 (2004). [8] S. Patskovsky, A. V. Kabashin, M. Meunier, J. H.T. Luong, Sensors and Actuators B 97, 409 (2004). [9] J.F. Masson, Y.C. Kim, A. Loius, L.A. Obando, W. Peng, K. S. Booksch, Appl. Spectroscopy 60, 1241 (2006). [10] V. Lirtsman, M. Golosovsky, D.Davidov, J. Appl. Phys. 103, 014702 (2008). [11] M. Golosovsky, V. Lirtsman, V. Yashunsky, D. Davidov, B. Aroeti, J. Appl. Phys. 105, 102036 (2009) [12] R. Ziblat, V. Lirtsman, D. Davidov, B. Aroeti, Biophys. J. 91, 776 (2006). [13] V. Yashunsky, S. Shimron, V. Lirtsman, A.M. Weiss, N. Melamed-Book, M. Golosovsky, D. Davidov, B. Aroeti, Biophys. J. 97, 1003 (2009). [14] K.F. Giebel, C. Bechinger, S. Herminghaus, M. Riedel, P. Leiderer, U. Weiland, M. Bastmeyer, Biophys. J. 76, 509 (1999). [15] Y. Fang, A. M. Ferrie, N. H. Fontaine, J. Mauro, J. Balakrishnan, Biophys. J. 91, 1925 (2006).

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[16] Y. Yanase, H. Suzuki, T. Tsutsui, I.Uechi, T. Hiragun, S. Mihara, M. Hide, Biosensors and Bioelectronics 22, 1081 (2007). [17] C.M. Cuerrier, V. Chabot, S. Vigneux, V. Aimez, E. Escher, F. Gobeil Jr., P.G. Charette, M. Grandbois, Cellular and Molecular Bioengineering, 1, 229 (2008). [18] A.A. Kolomenskii, P. D. Gershon, H. A. Schuessler, Appl. Optics 39, 3314 (2000). [19] M. Zangeneh, N.Doan, E.Sambriski, R. H. Terrill, Appl. Spectroscopy 58, 10 (2004). [20] J.V. Coe, K.R. Rodriguez, S. Teeters-Kennedy, K. Cilwa, J. Heer, H. Tian, S.M. Williams, J. Phys. Chem. 111, 17459 (2007). [21] L.M. Miller, P. Dumas, Biochimica et Biophysica Acta 1758, 846 (2006). [22] M.K. Kuimova, K.L.A. Chan, S.G.Kazarian, Appl. Spectroscopy 63, 164. (2009) [23] U. Zelig, J.Kapelushnik, R. Moreh, S.Mordechai, I. Nathan, Biophys. J. 97, 2107 (2009). [24] G. Hastings, R. Wang, P. Krug, D. Katz, J. Hilliard, Biopolymers 89, 921 (2008) [25] A. Hirano-Iwata, R. Yamaguchi, K.Miyamoto, Y.Kimura, M. Niwano, J. Appl. Phys. 105, 102039 (2009). Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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[26] H.Y. N.Holman, M.C. Martin, W.R. McKinney, Spectroscopy 17 139 (2003). [27] Y.Raichlin, A.Katzir, Appl. Spectroscopy 62, 55a (2008). [28] K. Johansen, H. Arwin, I. Lundstrom, B. Liedberg, Rev. Sci. Instr. 71, 3530 (2000). [29] J.J.Ramsden, S.Y. Li, E. Heinzle, and J. E. Prenosil, Cytometry 19, 97 (1995). [30] J.J. Ramsden, R. Horvath, J. Receptors and Signal Transduction, 29, 211 (2009). [31] R. Horvath, K. Cottier, H.C. Pedersen, J.J. Ramsden, Biosensors and Bioelectronics, 24, 799 (2008). [32] T.S. Hug, J. E. Prenosil, M. Morbidelli, Biosensors and Bioelectronics 16, 865 (2001). [33] T.A.Leskova, A.A. Maradudin, I.V. Novikov, J. Opt. Soc. Am. 17a, 1288 (2000). [34] D. Bideau, A.Gervois, L. Oger, J.P. Troadec, J. de Physique 47, 1697 (1986). [35] J.E. Bertie, Z. Lan, Appl. Spectroscopy 50, 1047 (1996).

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[36] P.K. Mattila, P. Lappalainen, Nature reviews, Molecular cell biology 9, 446 (2008).

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In: Plasmons: Theory and Applications Editor: Kristina N. Helsey, pp. 345-380

ISBN: 978-1-61761-306-7 c 2011 Nova Science Publishers, Inc.

Chapter 15

B ASIC P ROPERTIES OF P LASMONS J.T.Mendonc¸a IPFN, Instituto Superior T´ecnico 1049-001, Lisboa, Portugal

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Abstract We describe the basic properties of electron plasma waves or plasmons, giving particular emphasis to the wave and particle aspects of their behavior. We discuss the plasmon dispersion relations, including the cases of relativistic and quantum plasmas. Finite plasmas are also discussed, including Mie oscillations and Tonks-Dattner resonances in cylindrical and spherical geometries, and Trivelpiece-Gould modes. We then consider resonant kinetic processes associated with Landau damping of electron plasma waves by both electrons and photons. Nonlinear processes such as solitons and harmonic generation are considered. In what concerns the particle-like properties of electron plasma oscillations, we discuss the plasmon effective mass, as well as the plasmon effective charge. We also discuss plasmon processes in a time varying plasma, such as time refraction and plasmon acceleration. We then consider plasmon kinetic instabilities, and in particular plasmon beam effects. Finally, we describe the properties of plasmon states with orbital angular momentum, and their relevance to Raman scattering.

1

Introduction

Plasmons are the elementary quantum excitations associated with electron plasma waves, or in other words, are their corresponding quasi-particles. These waves were first studied by Langmuir [1], and for that reason they are sometimes called Langmuir waves. They play an important role in Plasma Physics, Space Science and Astrophysics, but also in Condensed Matter Physics. Plasma is an ionized gas composed of free particles with negative and positive electric charges, such that the net electric charge of the medium is equal to zero. Cold plasmas, produced in the laboratory or occurring naturally in space, are usually composed of free electrons and positive ions or protons. Plasmas composed of an equal number of free electrons

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and positrons are also relevant to Astrophysics, such as the plasma clouds in the vicinity of neutron stars, the relativistic fireballs eventually responsible for gamma-ray bursts, or the active galactic nuclei. In such a gaseous medium, the long range electromagnetic interactions between the individual particles create a collective field. The eigen-frequencies of this collective field characterize the mode resonances of the plasma medium. Extension of the plasma concept to the extreme matter conditions of the early Universe, or the dense matter produced by collisions between energetic ions, is called a quark-gluon plasma, where the electromagnetic interactions are replaced by the strong interactions, and the mediating bosons such as the photons and the plasmons are replaced by gluons. Apart from the difference in the nature of the force, many analogies remain between the electromagnetic and the quark-gluon plasma, which have not yet been completely explored. We will focus here on the usual electromagnetic plasmas [2], but a brief reference will also be made to the quark-gluon plasmas [3]. For Astrophysical applications of Plasma Physics, see reference [4]. In the absence of any external magnetic field, the plasma is an isotropic medium, where we can identify a single collective mode frequency, called the plasma frequency, ωp . In a magnetized plasma this simple picture is lost and we can find several kinds of basic mode frequencies, some associated with the response of the individual charged particles to the external field, such as the cyclotron frequencies, others associated with the collective excitations, such as the hybrid frequencies. Here we review some of the wave and particle properties of plasmons. First of all, we revisit plasmon dispersion relations. Finite plasmas are also discussed, including surface plasmon oscillations and internal Tonks-Dattner plasmon modes. Waves with transverse structure can propagate in cylindrical plasma columns, and usually called Trivelpiece-Gould modes. They can be seen as special versions of plasmons. We then consider resonant kinetic processes associated with Landau damping of electron plasma waves by electrons. For relativistic phase velocities, plasmons cannot interact resonantly with electrons, but can suffer Landau damping by photons. Nonlinear processes such as solitons and harmonic generation are also considered. Harmonic generation can lead to nonlinear plasmon oscillations which display recurrence phenomena. On the other hand, solitons can be seen as single plasmon objects which behave like gigantic quasi-particles. Other nonlinear processes are related with relativistic corrections, which occur for very large amplitude waves. In what concerns other particle-like properties of electron plasma oscillations, we discuss the concepts of plasmon effective mass, and plasmon effective charge, comparing them with the case of photons. We also discuss plasmon processes in a time varying plasma, such as time refraction and plasmon acceleration. Similar effects have been first introduced for the case of photons, and more recently extended to plasmons. It is interesting to note that time refraction always implies the excitation of counter-propagating waves, for both photons and plasmons. In what concerns plasmon acceleration, the associated frequency shifts can induce a significant change of phase velocity and eventually lead to efficient wave absorption. We then consider plasmon kinetic effects, and in particular plasmon beam instabilities. These effects are quite similar to those associated with charged particle beams, such as electron of ion beams. Therefore, in many respects, plasmons seem to behave as any other particle in a plasma, displaying effective electric change and mass, and being able to dissipate its kinetic energy in some other form of plasma oscillations, such as ion acoustic

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waves. Finally, we describe the recently proposed property of plasmon angular momentum. This is very similar to photon angular momentum, the main difference being associated with the fact that plasmons have zero spin and as a result their total angular momentum coincides with their orbital angular momentum. It has been suggested that such plasmons states could eventually be excited by stimulated Raman scattering. Given the huge amount of work published in this area, we have not attempted to give here an exhaustive list of references on plasmons. We have limited ourselves to quote, apart from the historical papers and some general plasma physics references, the papers published by the author which directly provided the support for the present review.

2

Dispersion relations

The theory of electron plasma waves started with the work of Vlasov [ ?], Landau [6], Bohm and Gross [7] and others [8, 25, 10] and is described in standard textbooks. We start by discussing the basic forms of linear dispersion relations, assuming isotropic and homogeneous plasmas. The simplest possible description of electron plasma oscillations is based on the electron fluid equations, which determine the mean electron density n and velocity v, as ∂v e + v · ∇v = − E + Se2 ∇ ln n (1) ∂t me √ p where e and me are the electron charge and mass, Se = 3vthe , where vthe = Te /me is the electron thermal velocity, and Te the electron temperature. For simplicity, electronion collisions which contribute to wave damping, will be ignored. The electrostatic field E = −∇V is determined by Poisson’s equation

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∂n + ∇ · vn = 0 , ∂t

∇2 V =

e (n − ni ) 0

(2)

where V is the potential and 0 is the permittivity of vacuum. For fast processes, the ions can be considered as imobile, and their mean density constant ni = n0 . The plasma equilibrium corresponds to charge neutrality n = n0 and v = E = 0. Let us assume that this ˜ , where n ˜ is equilibrium is slightly perturbed, and the electron density becomes n = n0 + n a small perturbation. By linearizing the above equations with respect to such perturbations, we can easily derive the evolution equation !

∂2 2 − Se2∇2 − ωpe n ˜=0 ∂t2

(3)

where ωpe is the electron plasma frequency, as defined by ωpe =

e2 n0 0 me

!1/2

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(4)

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J.T.Mendonc¸a

This is the characteristic frequency of the medium. Perturbations with frequency ω and wavevector k, evolving in space and time as n ˜ (r, t) ∝ exp(ik · r − iωt), will satisfy the plasmon dispersion relation 2 ω 2 = ωpe + k2Se2 (5) This shows that electron plasma waves, or their particle counterparts, the plasmons, have a cutoff frequency equal to the electron plasma frequency ωpe , below which they cannot exist. The same occurs for transverse photons, which is not surprising because plasmons are a kind of photons with longitudinal polarization ( k iparallel to E). But, in contrast to transverse photons which can exist with arbitrarily high frequency, the plasmon frequency is always close to its cut-off frequency, because electron Landau damping implies that k2 λ2De  1. Here we have used the electron Debye length λDe = Se /ωpe , which determines the plasma characteristic scale length. In the language of field theory, we can say that plasmons are quasi-particles associated with the plasma collective field. Each plasmon has energy ¯ hω, close to ¯ hωpe , where ¯ h is Plank’s constant divided by 2π. The above dispersion relation can also be written as (ω, k) = 0, where the longitudinal dielectric constant of the plasma is defined by (ω, k) = 1 + χ(ω, k) ,

χ(ω, k) = −

2   ωpe 1 + k2 λ2De 2 ω

(6)

The quantity χ(ω, k) is usually called the electron susceptibility. We can then say that any external perturbation can excite plasma oscillations, or plasmons, which then propagate in the medium, with group and phase velocities, vg and vph , given by

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vk ≡

∂ω vph = 2 , ∂k Se

vph ≡

ω Se =q k 1 − (ωpe /ω)2

(7)

The group velocity vk can be identified as the velocity at which the plasmons evolve in the medium. We clearly see that the phase velocity is always larger than Se , attaining relativistic velocities when k tends to zero. In contrast, the plasmon group velocity is always smaller than Se , and tends to zero in the same limit. Such a result has very important consequences, in the laboratory and in space, leading to possible mechanisms for the production of high energy cosmic rays, and possible use of relativistic plasma waves to built plasma accelerators, more efficient and compact than the present day particle accelerators. The large efficiency foreseen for plasma accelerators can be illustrated with the maximum accelerating field produced by a relativistic electron plasma wave, as given by ~ =− ∇·E

q e ne ⇒ eE0 = me c ωpe = 0.96 ne (cm−3) [eV /cm] 0

(8)

This means that E0 can attain values as high as 1GeV /cm, for typical values of the electron plasma density ne = 1019 cm−3 . This is four orders of magnitude larger than the accelerator fields in present day linacs, which are limited by the breakdown of the RF cavities. At SLAC, for instance, the maximum values of E0 are around 25M eV /m. For a recent account of the progress made in the field of plasma accelerators, see for instance the review article [12].

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In the above description, the ions were considered as immobile, providing of neutralizing background for the electron motion. This is generally valid, due to the large mass ratio. But the positive charges also give a small contribution to the plasma oscillations and, if the ions motion is taken into account, we obtain a new expression for the longitudinal plasma permittivity X ωp2 2 (ω) = 1 − 2 , ωp2 = ωpα (9) ω α=e,i where now both the electron and the ion plasma frequencies have to be defined 2 ωpi =

e2 n0 me 2 = ω 0 mi mi pe

(10)

Because of the large mass ratio, mi  me , the ion contribution is very small, and the resulting plasma frequency remains close to the electron plasma frequency, ωp ' ωpe . In contrast, the particular case of an electron-positron plasma, which is relevant to astrophysical applications, corresponds to mi = me , leading to (ω) = 1 − 2

2 ωpe ω2

(11)

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√ This means that the electron-positron plasma oscillates with a frequency ωp = 2ωpe . Finally, let us briefly mention the case of quark-gluon plasmas. Here, the above electromagnetic fields are replaced by the color fields, but the structure of plasma oscillations is maintained. For instance, the plasma frequency is given by a similar expression, with the following replacement e2 → g 2(2N + Nf )/2, valid for an SU(N) gauge theory, where Nf is number of flavors for quarks [3]. This represents the change from the electromagnetic coupling to the strong coupling. The resulting expression for the plasma frequency is ωp2 = (2N + Nf )

g 2T 2 6

(12)

The understanding of quark-gluon plasmas has considerable extended in the recent years, including refined nonlinear versions of the Landau damping. This completes our summary of plasmon dispersion relations, valid for an infinite and isotropic plasma. The influence of boundary conditions is discussed next.

3

Plasmons in a cylindrical geometry

Plasmons change their character when they are excited inside a cylindrical plasma column. The resulting plasmon modes are sometimes called Trivelpiece-Gould (TG) modes [13]. Let us consider a plasma column of radius a. We also assume that there is a static axial magnetic field, B0 = B0 ez . For a strong field confining field, such that the electron Larmor radius is much smaller than the plasma radius, we can use a one-dimensional model, where only the electron motion along ez is relevant. In that case, we can split the Laplace operator

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into the longitudinal and transversal components, ∇2 = ∇2⊥ + ∂z2 , in the wave equation (3). It is useful to expand the transverse part of the electrostatic potential V in a basis of Bessel functions, such that V (r, θ, z; t) =

∞ X

V`m (z; t)J`,m(k⊥`m r)eimθ

(13)

`,m=0

where V`m are the amplitudes of the different Bessel components. We assume that the plasma column is surrounded by a conductive wall. Because the electric field vanishes outside the column, we expect that wave propagation along the cylinder will correspond to the mode quantization in the transverse direction, such that Jm (k⊥`m a) = 0 ,

k⊥`m =

α`m a

(14)

where α`,m stands for the `th zero of the Bessel function of order m, and yields the following condition for the transverse potential profile 



2 ∇2⊥ + k⊥`,m Jm (k⊥`,m r)eimθ = 0

(15)

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Now, assuming wave propagation along the z axis, such that φ`m (z; t) ∝ exp(ikz z − iωt), we can derive the modified dispersion relation for plasmons with frequency ω = ω`m , such that " # 2 ωpe 2 2 ω`m = 2 + Se kz2 (16) 2 kz + k⊥`,m where kz is the longitudinal wave number. This result has been confirmed in several experimental conditions[13, 17]. An important difference with respect to the infinite plasma result is the absence of a lower cut-off. This means that, in principle, these wave modes can propagate for ω < ωpe , and can have arbitrarily low frequencies. Thus, in a sense, they look very much like acoustic waves. However, it should be noticed that, in specific experimental conditions, the longitudinal wave number kz is limited by the value of π/L, where L is the length of the plasma discharge, which implies a lower cut-off. We should also keep in mind that the ion motion has to be retained for frequencies comparable to the ion plasma frequency.

4

Mie and Tonks-Dattner resonances

Here we deal with internal resonances of bounded plasmas associated with plasmon states. Surface oscillations are usually called Mie oscillations, and bulk oscillations are known as Tonks-Dattner resonances [14]. Let us assume a finite plasma volume with a given geometry. It is possible to write the equation of motion for its centre of mass R as d2 R 2 + ωM R=0, dt2

1 R= N

Z

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rn(r)dr V ol

(17)

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!2/!pe2 0.01 0.1 1

k2"De2

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Figure 1: Dispersion relation for TG modes, for three different values of the transverse 2 2 λDe . momentum, k⊥ where N is the total number of electrons in the plasma volume. For a spherical plasma with radius a, and density profile n(r) only depending on the radius, we can define the Mie frequency ωM as 2 ωM

2 ωpe = n0 a3

Z a

n(r)r2dr

(18)

0

density profile such that where n0 is the mean electron plasma density. For a constant √ n(r) = n0 , we get the well known result ωM = ωpe / 3. That such centre of mass oscillations are related with surface plasmon oscillations can be seen by studying the potential perturbations at the plasma boundary. Assuming a uniform plasma with sharp plasma boundaries, sometimes referred as the water-bag model, we can use 2 ωpe 1− 2 ω

!

∇2 Vp = 0 ,

∇2Vv = 0

(19)

where Vp and Vv is the potential inside and outside the plasma. These potentials have to match at the plasma surface. We can see that two kinds of solutions are possible. First, we have a solution ω = ωpe , which corresponds to density perturbations which create no potential outside the plasma. In a spherical plasma, this is the case of oscillation of the plasma radius, of the form a(t) = a0 + cos(ωt), where   a0 is the oscillating amplitude, which correspond to a breading mode. The other kind of solutions satisfy ∇2Vp = 0, and

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have been solved for the case of a spheroid. The corresponding frequencies are [18] ω2 =

2 ωpe 0 m m m0 1 − (Qm l Pl /Ql Pl )

(20)

m m m are Legendre functions, with integer mode numbers where Qm l ≡ Ql (z) and Pl ≡ Pl (z) 0 m0 l and m such that√l ≥ |m|, and Ql and Plm are their derivatives with respect to the argument z = α/ α2 − 1. The parameter α is the aspect ratio of the spheroid and, when α → 1 we obtain a plasma sphere, where the above expression is reduced to 2 ω 2 = ωpe

l 2l + 1

(21)

We can see that, for l = 1 this coincides with the Mie frequency ωM defined above. Let us now turn to the bulk plasma oscillations. Density perturbations n ˜ , with frequency ω, will be described by the equations h

i

∇2 + k2 (r) n ˜=−

eE · ∇n0 + ∇ ln n0 · ∇˜ n, mSe2

∇·E= −

e n ˜ 0

(22)

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where thermal effects are retained, and the space dependent wavenumber k(r) is defined 2 (r)]/S 2. In the simple case of a uniform plasma slab, we have by k2 (r) = [ω 2 − ωpe e ∇n0 = 0 everywhere except at the plasma boundaries x = 0 and x = L. For unperturbed plasma boundaries, such that n ˜ (0) = n ˜ (L) = 0, the following dispersion relations are easily obtained (   ) λDe 2 2 2 (23) ωn = ωpe 1 + 2π(2n + 1) L where λDe = Se /ωpe is the electron Debye length, and the quantum number n can be zero or an integer. This shows the existence of a series of internal plasmon modes with an integer number of half wavelengths, usually known as Tonks-Dattner modes. Similar modes can be considered for cylindrical [15] and spherical geometries [16]. In the spherical case, the above dispersion relation is replaced by 2 2 = ωpe + α2n` ωn`

Se2 a2

(24)

where a is the plasma radius, and αn` are the n-th zeros of the Bessel functions J`+1/2 (z). The internal plasmon modes depend now on two quantum numbers n and `.

5

Plasma wave solitons

Nonlinear effects can modify the structure and properties of plasmon dispersion. This can occur for large amplitude oscillations. where linearization of plasma equations cannot be justified. Important examples of nonlinear wave phenomena associated with plasmons, are soliton solutions and harmonic cascades, to be described in this and the next section.

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Figure 2: Radial profile of internal plasmon modes (Tonks-Dattner modes) in a spherical plasma, for mode numbers n = 0, 1, 2 and ` = 0, 1, 2.

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Another source of plasmon nonlinearity is related with relativistic effects, to be described later. Electron plasma wave solitons, also called Langmuir solitons, were first discussed by Rudakov [19], Zakharov [20], and many others [21, 22].In order to study electron plasma wave solitons, we can go back to equations (1) and (3), and assume that the electron density n, velocity v and electrostatic potential φ (or the electric field E, can be divided in two terms, a fast term which evolves on a time scale of the order of 1/ωpe , and another term which evolves on a much slower time scale. For the one-dimensional problem, we can then write ˜ ˜ , v = vs + v˜ , E = Es + E (25) n = ns + n where the suffix s pertains for the slow terms. Taking appropriate time averages, we can then split the equations in two parts, leading to the following equation for the fast electric field ! 2 ∂2 n ∂ ∂ s 2 ˜=0 + ωpe − Se2 2 E (26) ∂x ∂t2 n0 ∂x In what concerns the slow quantities, we can write for the velocity 1 ∂ ∂vs =− < v˜2 > ∂t 2 ∂x

(27)

where the angle brackets denote a time average over the fast time scale. The nonlinear term on the left is clearly a ponderomotive force term Using the linear expression for v˜ ' ˜ we can write this equation as −i(e/mωpe )E, me

∂vs ∂ = − Φp , ∂t ∂x

Φp =

0 ˜ 2 |E| 2n0

(28)

The quantity Φp can be understood as a pseudo-potential. We can then assume that the slow density component ns , is in Boltzmann equilibrium in such as potential, allowing us to write   ˜ t)|2 Φp (x, t) 0 |E(x, '− (29) ns (x, t) = n0 exp − Te 2Te

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where the approximate expression is valid for moderate field intensities such that Φp  Te . We realize from here that the soliton will produce a local rarefarction of the plasma mean density, inside of which the fast electron plasma oscillations will be trapped. This allows us ˜ from equation (26). This equation can be solved by using to obtain a closed equation for E ˜ t) = E0(x, t) exp(−iω0 t) E(x,

(30)

where E0(x, t) is a slowly varying amplitude, and ω0 ' ωpe . Renormalizing the variables in such a way that ωpe t, τ= 2

ωpe z= x, Se

A=

r

0 E0 2n0Te

(31)

we can write the evolution equation for the field amplitude in the form of the well known nonlinear Schroedinger equation i

∂A ∂ 2A + + |A|2A = 0 ∂τ ∂z 2

(32)

Exact and approximate methods of solution, which can be found elsewhere [23] then lead to the following physically meaningful result 



˜ t) = E0sech (x − ut) exp(−iω0 t + iψ) E(x, ∆

(33)

where E0 is now a constant amplitude, u ' Se (0 |E0|2/2n0Te ) is the soliton velocity, and the soliton width ∆ is determined by

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∆=

Se q 2n0 Te /0 E0ωpe

(34)

We find here the well known soliton properties according to which larger the soliton amplitude E0 will have a smaller width, and will move faster. Going back to equation (29), we can then state the slow density perturbation associated with this envelope soliton solutions, in the form   0 |E0|2 (x − ut) sech2 (35) ns (x, t) = − eTe ∆ This shows the formation of a density cavitation, associated with the ponderomotive force of the high frequency fields. p However, such a cavitation can only exist for subsonic velocities, u < vac , where vac ' Te /mi is the ion acoustic velocity, and mi is the mass of the ions. If the ion motion is taken into account, it will lead to a multiplying factor of vac /(vac − u) in this last expression.

6

Harmonic cascades

Another interesting aspect of nonlinear plasmon properties is related with the possible excitation of harmonic cascades. Here we describe these cascades associated with TG modes Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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in bounded plasmas. The bounded plasma case is particularly interesting, not only because it contains the case of infinite plasmas as an appropriate limit, but also because it leads to the occurrence of recurrence phenomena, as discussed below. Indeed, harmonic cascades in cylindrical plasmas lead to the formation of nonlinear plasma oscillations, which display spatial and temporal recurrence. This was observed in radio-frequency discharges with a cylindrical geometry, where nonlinear TG modes were excited by an external source [17]. In order to describe such harmonic cascades, we consider the one-dimensional version of equations (1)-(2), and assume a superposition of standing waves in the axial direction n(z, t) = n0 1 +

∞ X

ikm z

nm (t)e

!

+ c.c.

,

v(z, t) =

∞ X

m

vm (t)eikm z + c.c.

(36)

m

where m is an integer and km = mπ/L, and L is the length of the plasma column. For the temporal evolution we can use vm (t) = am (t) exp −iωm t ,

ωm = ωpe km

1 2 2 /k 2 + λDe 1 + k⊥ m

!1/2

(37)

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where am (t) are slowly varying amplitudes. The corresponding amplitudes for the density perturbations will be approximately given by the linear expression nm = km vm /ωm . 2  k12 , and the fundamental mode freIn typical experimental conditions, we have k⊥ quency is nearly equal to ω1 = ωpe (π/k⊥L), where is the plasma radius. For moderate 2 2 values of m, we can still have k⊥  km , and ωm ' mω1 . Assuming that the fundamental mode is driven by an external antenna, which keeps its amplitude constant, and neglecting the nonlinear terms which do not contain the dominant mode v1 , we can determine the amplitude of the harmonics using the following coupled equations ∂am = −2iβkm (a1am−1 + a∗1 am+1 ) ∂t

(38)

where the geometric factor β is defined by 2 β= 2 a J1 (z10)

Z a 0

J03 (k⊥ r)rdr

(39)

Here J0 and J1 are Bessel functions of the first kind, z10 ' 2.4 is the first zero of J0 , and k⊥ = (z10/a). This system of coupled equations can be easily solved in terms of Bessel functions which, in the limit of small arguments (corresponding to week coupling or short interaction times) leads to am (t) =

r



2m − 1 2 cos Ωt − π 2 πm t 4



(40)

where Ω is a coupling frequency defined by Ω = 4βk1|a1|. Notice that this frequency is very similar to the Raby frequency for radiative coupling between two quantum states, and these oscillations are similar to the well known Rabi oscillations. The present classical problem is also very similar to the coupling between the equidistant quantum states of the harmonic oscillator, due to the presence of a resonant external field. It is also interesting to

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notice that the above solutions have been well confirmed in radio-frequency plasmas. They accurately describe the slow temporal oscillations of the different harmonics, the amplitude decay as 1/m, and the π/2 phase shift between consecutive harmonics. Last but not the least, harmonic superposition is able to explain the observed spacial recurrence of the shape of the wave signal [17]. Therefore, harmonic plasmon cascades provide a simple but accurate example of recurrence wave phenomena.

7

Plasmons in relativistic plasmas

Another source of nonlinearity can be found in the relativistic effects, as mentioned. They become relevant for very large amplitude plasma waves. Relativistic plasma waves were first considered Akhiezer and Polovin [24], and then by Dawson [25], Silin [26] and many others [27, 28, 29]. In order to describe such waves, let us assume propagation along the z-direction. The continuity and momentum conservation equations can be stated as 

∂ u ∂n +c n ∂t ∂z γ



=0,

u ∂u e ∂u +c =− E ∂t γ ∂z mc

(41)

√ where γ = 1 + u2 , and u = γv/c The electric field E is determined by Poisson’s equation, written in the form !  2  n mc 2 ~ = −k − 1 (42) ∇·E p e n0

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~ leads to where n0 is the equilibrium electron density, an kp = ωpe /c. Elimination of E ωp2 u ∂ u ∂u ∂ 2u + c = − n ∂t2 ∂t γ ∂z n0 γ

(43)

Let us now assume that the quantities u, n and E only depend on the space a time coordinates through the combination ξ = z−cβφ t, where βφ can be identified with the normalized wave phase ω/kc. We can then obtain a closed evolution equation for the electron velocity in a longitudinal wave, in the form u ∂2 (βφu − γ) = −kp2 2 ∂ξ (βφγ − u)

(44)

√ and γ = 1 + u2. This equation can easily be solved in two opposite limits. First, we consider the low energy case, such that γ ' 1, where it reduces to u 1 ∂ 2u ' −kp2 2 (1 − u2) 2 ∂ξ βφ 2

(45)

where u  βφ is assumed. Note that the differential operator, when applied to a wave of the form u(ξ) = u0 exp(kξ), generates a factor −(ω/cβφ)2. This leads to the dispersion relation ! a20 (46) ω ' ωpe 1 − 4

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where we use u0 = a0 = eE0 /mωc. This is in excellent agreement with a more accurate derivation, where the factor 1/4 is replaced by 3/16 [24, 27]. We then consider the high energy limit, such that γ  1. In this case we can take γ ' a0. Noting that βφ  1 for small enough wavenumbers k, we can replace equation (44) by kp2 u ∂ 2u ' − ∂ξ 2 βφ2 a0

(47)

This leads to ω 2 ' ωp2 a0, and finally to ω ' ωp



mc eE0

1/2

(48)

A more accurate derivation would lead to an additional multiplicative factor of π/2. Our short discussion shows that the relativistic plasmon frequency decreases with increasing wave amplitude. This is true for both the low and the high energy limits, and is associated with the relativistic mass increase of the electrons oscillating in very intense fields. The exact solution for the longitudinal wave frequency can be found in the original paper [24] (see also [28]).

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8

Electron Landau damping

Until now, we have only used the fluid plasma description. It is time to consider another important aspect of plasmon behavior, which is related to resonant wave-particle interactions. The discussion of these kinetic wave processes imply the use of a more refined description of electron plasma response, based on particle kinetic equations. We will see later that such a kinetic description can even be extended to the plasmon themselves. The fundamental kinetic equation for a plasma medium is the Vlasov equation, which determines the particle distribution function fα , with α = e, i, in the presence of a collective electromagnetic field. This equation is valid for plasma conditions such that the number of electrons inside the Debye sphere is much larger than one [11]. The Vlasov equation for the electron population is 



∂ eE ∂ + ~v · ∇ − · fe (r, v, t) = 0 ∂t me ∂v

(49)

where fe (r, v, t) is the single particle distribution function. This is a kind of collisionless Boltzmann equation, where the long range particle interactions (or long range collisions) included in the plasma mean field, as determined by the self-consistent electric field ~ =−e ∇·E 0

Z

fe (r, v, t)dv

(50)

dominate over the binary collisions and short range interactions. Linearization with respect to its equilibrium value ( fe = f0 + f˜) leads to a new expression for the electron susceptibility, given by Z e2 k · (∂f0/∂v) dv (51) χe (ω, k) = − 2 0 k ω−k·v

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The existence of a resonance in the denominator, associated electron velocities such that ω = k · v, implies that the susceptibility is complex. Writing it as χe = χ0e + iχe ”, we have for the real part Z ~ e2 k · (∂f0 /∂v) 0 dv (52) χe = − 2 P 0 k ω−k·v where P denotes the principal part of the integral. On the other hand, the real part reads e2 χe ” = −iπ 0 k2



∂f0 ∂v



δ(ω − k · v)

(53)

vk =ω/k

This last expression leads to Landau damping, for negative derivative of f0 (~v) at resonance vk = ω/k. This results from a statistical balance between Cherenkov emission and absorption of plasma waves by the nearly resonant particles. Strong wave-particle coupling occurs at resonance, and the wave can be damped, even in the absence of any collisional mechanism of wave dissipation. It is useful to consider the parallel and perpendicular velocity components, such that v = uk/k + v⊥ . Integration over the perpendicular velocity components leads to F0 (u) = R f0 (v)dv⊥. For a Maxwellian plasma with electron temperature Te , we can use 1 u2 exp − 2 F0e (u) = √ 2vthe 2πvthe

!

(54)

2 = T /m . The principal part of integral in equation (50), will lead to the same with vthe e e dispersion relation as the fluid theory. On the other hand, the imaginary part of the susceptibility will determine the non-collisional wave damping, as given by

r

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γ=−

π ωpe 1 exp − 2 2 −3 8 (kλD ) 2k λD

!

(55)

This damping coefficient tends to zero in the limit of very large wavelengths, or k → 0. Wave damping increases rapidly with k, and becomes very large for kλD ∼ 1. For kλD = 0.5, we get γ = 0.93ωpe, and the electron plasma waves are strongly damped. These waves can only propagate for wavelengths much larger than the electron Debye length, or kλD  1. We can then conclude that the frequency of electron plasma waves is always nearly equal to ωpe , otherwise they would be strongly attenuated by electron Landau damping. This effect was first observed by Malmberg and Wharton [30] The above kinetic description gives an oversimplified view of the kinetic processes associated with electron plasma waves. In particular, it excludes electron trapping [10] which can sometimes perturb locally the distribution function, in such a way that its derivative could locally vanish. This leads to the possible existence of low amplitude undamped modes [31, 32].

9

Photon Landau damping

We have just seen plasmon with very small wavenumber k, or very large phase velocity ω/k ∼ c, are not Landau damped by the electrons. However, these plasmons with relativisPlasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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tic phase velocity can be Landau damped by photons, as discovered in recent years [33], and discussed next. This new plasmon property has very important practical consequences. According to the photon kinetic theory [34], we can describe an arbitrary radiation spectrum with the help of a Wigner function which, in the classical limit is nothing but the photon occupation mumber. This Wigner function is a straightforward generalization of the concept introduced in the 30’s years of the last century, to describe quantum phenomena in a classical phase space. It can be defined as a Fourier transformation of the field autocorrelation. Our discussion here is limited to geometric optics. A more exact formulation can be found elsewhere [35, 36]. One of the basic ingredients of the theory of photon Landau damping is the kinetic equation for photons, which describes the space and time evolution of the photon number density N (r, k, t). This quantity is the classical limit of the photon Wigner function. Notice that here k refers to the wavevector or momentum of the photons, and ωk is the corresponding photon frequency. In the geometric optics approximation, the photon kinetic equation simply states the invariance of photon number density in a slowly varying background, as d N (r, k, t) = 0 (56) dt In this approximation, the photon number density can be defined as the electromagnetic spectral energy density W (r, k, t), divided by the single photon energy ¯ hωk . This equation can then be coupled to the electron plasma equations via the ponderomotive force. It will then lead to a self-consistent description, where the photon field can exchange energy with the plasmons, in the same way as the usual kinetic plasma theory discussed above describes the energy exchange between electrons and plasmons. The photons will then be seen as a quasi-particle, with dynamical and statistical properties very similar to the electrons. By developing the total derivative in equation (56), we can easily recognize that the photon kinetic equation takes the form of a Vlasov equation, and can be written as [34] 



∂ ∂ + vk · ∇ + Fk · N =0 ∂t ∂k

(57)

~k is the force acting on the photons, as deterwhere v is the photon group velocity, and F mined by 2 ωp0 ∂ωk , Fk = − ∇ n(r, t) (58) v= ∂k 2n0 ωk 2 + k2 c2 )1/2, is determined by the photon dispersion relation. In the simple where ωk = (ωpe case where the electron plasma density is constant in time, and is only space dependent, the force Fk will introduce a change in the photon momentum (or wavenumber), which is nothing but the well known refraction. In contrast, if there is time dependent, this same force will be responsible for a frequency shift (or energy shift), also known as photon acceleration [34]. We should keep in mind that this Vlasov description of the photon field stays valid as long as the electron plasma density varies slowly in space and time, with respect to the photon space and time scales 1/k and 1/ωk . It can therefore be used to describe interaction

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of an high frequency electromagnetic wave spectrum (or photon spectrum) with a given electron plasma wave (or plasmon mode). It will be shown below that the dominant photonplasmon coupling takes the form of a resonant wave-particle interaction of the Landau type. Let us now turn to the description of of an electron plasma wave submitted to an arbitrary photon environement. Retaining the ponderomotive force terms, which couple the electron density perturbations with the photon radiation field, we can derive our evolution equation in the form !

∂2 e2 n0 2 2 2 2 − S ∇ n ˜ + ω n ˜ = ∇ I(r, t) e p0 ∂t2 2m2

(59)

where we have defined the radiation intensity as the integral over the photon spectrum I(r, t) =

Z

N (r, k, t)

dk (2π)3

(60)

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The term on the right hand side of (59) represents the ponderomotive force, or the photon radiation pressure, acting on the electrons. This term is responsible for the coupling between the electrostatic plasmon oscillations n ˜ , and the transverse electromagnetic wave spectrum. The result of such a coupling can be studied by using a perturbative analysis. In this analysis we adopt the following notation: (ω, k) will refer to electrostatic waves and (ω 0 , k0) to the electromagnetic or photon spectrum. The dispersion relation of electron plasma waves with frequency ω and wavevector k, evolving in the presence of an arbitrary photon spectrum will then take the form (61) 1 + χe + χph = 0 where χe is the electron susceptibility described above, and χph the photon susceptibility, as determined by Z e2 k2 (∂Gp/∂p) dp (62) χph = 2m2e (p − p0 ) where p = k0 · k/k is the parallel photon momentum, and G0 is the parallel quasiprobability, as defined by G0 (p) =

Z

N0(p, k0⊥) dk0⊥ (∂u/∂p)0 (2π)2

(63)

where u is the parallel photon group velocity. The derivative in the denominator is calculated for the resonant situation of p = p0, such that the parallel photon velocity equals the phase velocity of the electron plasma wave u(p0) = ω/k. The existence of such a resonance in the denominator inside the integral of χph leads to the possible occurrence of photon Landau damping of the electron plasma waves, as determined by [33] e2 k2 π γ = ωp0 8 m2e



∂G0 ∂p



(64)

0

Such a result shows that, in the absence of electron Landau damping, electron plasma waves with relativistic phase velocities can still suffer Landau damping due to their resonant interaction with photons. For an equilibrium plasma state the derivative of the unperturbed Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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photon distribution N0 will always be negative. This will lead to wave damping. In contrast, for a plasma out of equilibrium, an inversion of photon population can occur as described by a positive derivative of N0 around the resonant value p = p0 . A negative photon Landau damping, or wave instability will take place, leading to the exponential increase of the number of plasmons. A photon beam, such as that associated with a laser pulser, will always create an instability region. This can be described as a modulational instability of the photon beam. These results were originally derived in the geometric optics approximation [33, 34], but a more exact solution, where photon recoil is retained, was also established [36]. By photon recoil we mean the reduction (or increase) of photon momentum, due to the emission (or absorption) of a plasmon. Photon Landau damping of electron plasma waves can then be seen as the result of unbalanced emission and absorption of plasmons by the photon field in a plasma.

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10

Laser wakefields

As a consequence of the photon Landau damping mechanism, conditions for coherent emission of plasmons by a photon beam can exist, leading the the excitation of large amplitude electron plasma wakefields by intense laser pulses. This coherent process is crucial to the mechanism of plasma acceleration. In order to understand it, we consider the following simplified model. Let us now consider the case of a short laser pulse propagating in the under-dense plasma, such that the pulse duration is smaller than the period of the electron plasma wave, 2π/ωpe. We assume a relativistic laser pulse with frequency ω0 , much larger the electron plasma frequency, such that the laser group velocity is close to the speed of light, vg ∼ c  Se . In order to simplify our discussion, we can also neglect wakefield dissipation and thermal effects. Finally, we neglect the gradients in the perpendicular direction, such that 2 |∂ 2/∂z 2|  k⊥ . This is valid for pancake-shape laser pulses. We are then reduced to a onedimensional problem. The equation describing the electron plasma oscillations associated with the laser wakefield can then be reduced to [34] !

∂2 + kp2 n ˜ (r⊥ , η) = F0 (r⊥, η) ∂η 2

(65)

Here, the space variable is η = (z − vg t). The ponderomotive force term F0 can be written in terms of the number of photons carried by the laser N (η), as F0 (~r⊥, η) =

2¯ hkp2 ∂ 2 N (η) me ω0 ∂η 2

(66)

The photon number density is obviously proportional to the laser pulse energy W (η), according to W (η) = h ¯ ω0 N (η) where ω0 is the laser frequency. An analytical solution of this laser wakefield equation can be found, in the form ρ(~r⊥, η) = Qpho N (η) − Qpho kp

Z η

N (~r⊥, η 0) sin[kp(η − η 0)]dη 0

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where we have used the electron charge density ρ = −e˜ n, and Qpho = −e¯ h(ekp2 /mω0 ) is the photon equivalent charge. The first term in this solution represents the total equivalent charge of the laser pulse itself. This charge is negative because the photons repel the plasma electrons, due to the radiation pressure or ponderomotive force. The second term is the wakefield excited by the photon pulse. We can see that it survives the driving pulse and stays well behind the pulse for a very long time. Such a wakefield structure moves with a very fast velocity, of the order of the laser group velocitty, and can be used to accelerate the electrons [12]. The photons carried inside the laser pulse envelope can also be accelerated by the wakefield. This leads to a significant spectral broadening of the laser radiation. Such a spectral broadening has recently been measured, for accelerator relevant laser parameters [37], and is well understood theoretically [34]. More accurate nonlinear models have also been developed, and very sophisticated numerical simulations using relativistic particle-in-cell codes [38], which corroborate this simple model and give information on the wavefield structure. It is important to notice that relativistic electron beams can also excite similar plasmon wakefields. Both laser and particle beam wakefields have been used to accelerate particles up to the GeV range on a centimeter scale-length.

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11

Plasmon kinetics

We have shown that photons can be described with a kinetic equation, and that such a descriptions opens the way to a new phenomenon called the photon Landau damping. But the plasmons themselves can also be described by similar wave kinetic equations [39, 40], as discussed here. Let us consider an isotropic and uniform plasma, in the absence of electromagnetic radiation. The electron plasma waves can be described by the following equation for the scalar potential V (r, t), as given by 1 ∂2 ∇2 − 2 2 Se ∂t

!

V =

ωp2 (r, t) V c2

(68)

The space and time variation of the electron plasma frequency can be due for instance to ion acoustic waves, any other low frequency mode oscillations. For a stationary plasma, we have n(~r, t) = n0 , and, for potential perturbations of the form V (r, t) = V0 exp(ik·r−iωt), 2 + k2S 2 . this equation simply leads to the dispersion relation ω 2 = ωp0 e But, in more general plasma conditions, we can have a broad spectrum of electron plasma fluctuations, and it is more convenient to replace the wave equation by a new kind of description based on field correlations. Let us then introduce the auto-correlation function C(r, t; s, τ ) = V ∗(r − s/2, t − τ /2) V ∗ (r + s/2, t + τ /2)

(69)

and define the plasmon Wigner function, by the Fourier transformation F (ω 0, k0; r, t) =

Z

ds

Z

dτ C(r, t; s, τ ) exp(−ik0 · s + iω 0 τ )

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For convenience, we use here k0 and ω 0 for the plasmon field. The notations k and ω, will be reserved to much lower frequency oscillations of the electron plasma density, such as ion acoustic waves. We can derive from (68), an exact evolution equation for this Wigner function, as iω

0

!

2 ωp0 ∂ S 2k0 + e0 ·∇ F = ∂t ω n0

Z

dk (2π)3

Z

dω n ˜ (ω, k) [F− −F+ ] exp(ik·r−iωt) (71) 2π

where n(ω, k) are the Fourier components of the electron density fluctuations, as determined by the Fourier decomposition of the electron plasma density. The quantities F± are defined by F± = F (ω 0 ± ω/2, k0 ± k/2) (72) This evolution equation looks quite impressive, but it can be simplified and solved in many interesting situations. First, assuming that the plasmon frequency and wavevector ω 0 and 0 0 2 k0 are not arbitrary, but related through the linear dispersion relation ω 2 = k 2 Se2 + ωpe , even if the quantities involved vary in space and time. This assumption allows us to write the Wigner function in reduced form, where the explicit dependence on the frequency disappears. It is also useful to consider the case in which a single ion acoustic wave is perturbing the background plasma and modulating its plasma density. In such a case, the wave kinetic equation (71) is reduced to

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iω

0





2 ωp0 ∂ + v0 · ∇ F = ∂t n0

Z

n ˜ (k) [F− − F+ ] exp(ik · r − iωt)

dk (2π)3

(73)

where v0 = Se2 k0/ω 0 is the plasmon velocity, and F± = F (k0 ± k/2, r, t). It should be noticed that, when the length scale of the plasma density perturbations is much larger than that of the electron plasma waves, |k|  |k0|, and the quantities (72) can be developed as F± ' F (k0) ±

k ∂F · 2 ∂k0

(74)

Equation (71) is now reduced to a Vlasov equation for plasmons, similar to that previously used for photons. The interesting thing about this kinetic description of electron plasma waves is that it shows that plasmons behave as a quasi-particle in the medium, thus connecting the classical plasma description to the general concepts of field theory. Electron plasma turbulence can then be described by a (quasi) distribution function F , and seen as equivalent to a gas of plasmons. If such gas in not in thermal equilibrium with the background medium, it will eventually lead to instabilities, as shown next.

12

Plasmon instabilities

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of laser fusion, where the incident laser beam excites a relativistic electron beam, which penetrates in the overdense plasma region and excites a plasmon spectrum. These plasmons can then decay into ion acoustic waves which are heavily damped, thus conveying part of the incident laser energy into the ions [41]. Starting with the fluid equations for the ion and electron populations (α = e, i), it is then possible to derive an equation for the density perturbations, as [40] 



X D D˜ nα qα n0α qβ n ˜ β − Sα2 ∇2 n ˜ α = Rα + να + Dt Dt mα β

(75)

where Sα2 = γαTα/mα , γe = 3, γi = 1, qi = Ze is the ion charge, and να the collision frequencies. We have also used the notation D = Dt





∂ + Uα · ∇ ∂t

(76)

where Uα are the mean velocity of the different particle species, and retained the nonlinear terms   n0 α 2 D Rα = ∇ |vα|2 − + να ∇ · (˜ nα vα) (77) 2 Dt Assuming perturbations of the form exp(ik · r − iωt), this leads to the following dispersion relation 2 ω2 ωpe pi 2 ZQi = Qe + ωpe (78) Ω2e − 2 Ωi Ω2i

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nα , and we have defined the auxiliary quantities where we have defined Qα = Rα/˜ 2 Ω2α = ωpα (1 + k2 λ2α) − ωα2

,

ωα2 = (ω − k · Uα )[(ω − k · Uα) + iνα ]

(79)

usually the term in Re in the dispersion relation (77) is larger by a factor of (mi/me ) than the term in Ri . This means the electron nonlinearities are dominant, and that the ion nonlinear term Ri can be ignored. After integration over the high frequency spectrum, we get   Z e2 2 2 [F0− − F0+ ] k0 2 dk0 Qe = − 2 k ωpe (80) me (2π)3 (ω − ~k · ~v 0) ω 0 From this we can derive an explicit form of the dispersion relation, as a function of a the unperturbed plasmon distribution function F0 . The dispersion relation is then 2 2 ωpi = Qe Ω2i Ω2e Ω2i − ωpe

(81)

As an illustration of this general result, let us consider the case of a low temperature plasmon beam (where temperature refers here to the dispersion of the plasmon distribution around a given mean value of the plasmon momentum k00. We will then assume the following mono-kinetic plasmon distribution F0 (k0) = (2π)3F0 δ(k0 − k00)

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0

Noting that (k0 /ω 0)2 = v 2/Se4, this leads to a simple expression for the nonlinear term Qe . The result is " # 0 0 2 v−2 v+2 e2 k2 ωpe F0 (83) Qe = − 2 0 ) − (ω − k · v 0 ) me Se4 (ω − k · v− + 0 = v 0 (k0 ± k/2). The dispersion relation can be then where we have used the notation v± 0 written in the following form

ω(ω + iνi ) −

2ω 2 k2 vac e2 F0 k2 ωpi pe = α(k) 2 2 2 4 2 (1 + k λe ) me Se (1 + k λ2e )

(84)

2 = λ2 ω 2 is the ion acoustic velocity, and we have defined the quantity where vac e pi

"

0

0

v−2 v+2 − α(k) = 0 ) 0 ) (ω − k · v− (ω − k · v+

#

(85)

In the absence of plasmons F0 = 0, this dispersion relation reduces to the well known ion acoustic dispersion relation. This result stays valid outside the quasi-classical limit (or geometric optics approximation), when the wavenumbers of the two types of waves, ion acoustic and electron plasma waves (or plasmons) become comparable k ∼ k0 . Therefore, they can be used to explore new areas of stability of ion acoustic oscillations driven by intense plasmon beams. In particular, we notice that the existence of the plasmon distribution F0 leads to a modified or renormalized ion acoustic velocity vren as defined by

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2 2 = vac vren

"

#

3 k2 ωpe e2 F0 1− α(k) m2e Se6(1 + k2 λ2e )

(86)

In order to show that the new dispersion relation contains unstable ion-acoustic mode solutions, we consider ω = ωr + iγ. The most unstable modes will have to maximise the nonlinear term α(k). From this we get, for the mode frequency and the mode growth rate, as determined by ωr2

2 k2 vac ' (1 + k2λ2e )

"

νi γ=− 1± 2

,

where we have

s

C(k) 1− 2 νi

#

(87)

0

2 k2 k02 Se2 e2 F0 ωpi (88) C(k) = − 2 me ωpe (ωr ωpe /Se2 − k · k00) We can see that instability occurs for C(k) < 0, which nearly corresponds to the case where v00 cos θ ≥ vac , where θ is the angle between k and k00, or equivalently, between the direction of ion acoustic wave propagation and the plasmon beam velocity. In this case we can write for the growth rate

"

νi 1− γ=− 2

s

#

|C(k)| |C(k)| 1+ ' νi2 4νi

(89)

We have therefore demonstrated that the ion acoustic waves can be driven unstable by a cold plasmon beam, even when the geometric optics approximation is not valid and the ion acoustic wave number k becomes comparable to the value of the plasmon wavenumber k00 . The details of this derivation are shown elsewhere [40].

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Plasmon effective mass

The plasmon effective mass can be easily understood in the context of field quantization in a plasma [42]. In contrast with the case of vacuum, where the only two orthogonal polarization states are possible for the transverse photon field, in a plasma we have three possible polarization states, due to the fact that the photon is a boson with spin one. The spin states ±1 correspond to the two transverse photons which survive in vacuum, and the sate 0 is that of a longitudinal photon, or plasmon. Here we focus on the longitudinal photon case. The quantization of the electrostatic field in a plasma can be carried out in a simple way, starting from the wave equations for both the scalar potentials, V , and the vector potential A, in the Coulomb gauge, which can be written as 1 ∂2 ∇2 − 2 2 S2 ∂t

!

ωp2 V = 2V , Se

1 ∂2 ∇2 − 2 2 c ∂t

!

A=

ωp2 A c2

(90)

2 . This shows the existence of the three distinct photon states, with two with Se2 = 3vthe transverse states (λ = 1, 2, for dressed photons) and to the longitudinal polarization ( λ = 3, for plasmons). The energy operator can then be written as

W =

3 Z X

w(k, λ)

λ=1

d~k (2π)3

(91)

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with the energy density operator for the different photon and plasmon states given by 



w(k, λ) = ¯ hωk a+ (k, λ)a(k, λ) + 1/2

(92)

where the photon frequencies ωk (λ) are determined by 

ωk (λ) = k2 c2λ + ωp2



(93)

with cλ = c, for λ = 1, 2, the two transverse photon states associated with the vector potentialA, and cλ = Se , for λ = 3, the plasmon state associated with the scalar potential V . From here we can recognize the existence of a photon mass vector, with components mλ =

¯ h ωp c2λ

(94)

This means that the plasmon mass will be much larger than the transverse photon mass, for non-relativistic plasmas, because c2  Se2. We conclude that the electrostatic field interactions mediated by plasmons have a much shorter range than the electromagnetic interactions mediated by photons, as expected.

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367

Plasmon effective charge

The concept of effective charge of a photon [43, 44, 34] was first proposed in order to explore the dynamical similarities between photons and electrons in a plasma, such as the possibility of both electron and photon beams to excite relativistic plasmon wakefields . This concept was recently extended to plasmons [45]. The effective charge of all these particle species is related with the ponderomotive force exerced by the respective fields on the plasma particles. The derivation of this quantity can easily be performed by assuming that the mean plasma quantities, such as electron mean velocity, density and potential, are the sum of three different terms: the unperturbed (zero order) quantities, ne = n0 , v = 0, V = 0, the first order perturbed quantities, associated with the electron plasma oscillations, (n1 , v1, V1) ∝ exp(ik0 · r − iω0t), where ω0 ' ωpe , and the second order quantities, related with the nonlinear ponderomotive force. It can easily be shown that the first order quantities n1 and v1 are given by n1 =

n0 k0 · v1 , ω0

v1 = −

ek0 ϕ1 me ω0

(95)

Following a standard perturbative analysis, we can then derive an evolution equation for the second order electron density perturbation, as !





n0 2 ∂ ∂2 2 − Se2 ∇2 n2 + ωpe n2 = ∇2 − k0 · ∇ |v1|2 2 ∂t 2 ω0 ∂t

(96)

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After a Fourier transformation, we get n2 =



2 0 ωpe k2 k02 |ϕ1|2 2ω k · k0 1+ 2me ω 2 ω02 (1 + χe ) ω0 k2



(97)

Introducing the plasmon number density, or number of plasmons N , as given by N=

0 2 k |φ1|2 2¯ hω0 0

(98)

and defining the effective plasmon charge qpl through the identity −en2 ≡ qpl N

(99)

we can then obtain, for the effective plasmon charge 

2 e¯ h ωpe k2 2ω k · k0 1+ qpl = − 2 me ω0 ω (1 + χe ) ω0 k2



(100)

This result is formally very similar to the photon effective charge, with small differences mainly due to the existence of first order density perturbations n1 for plasmons, which don’t exist for the transverse photon. Notice that this is simply a Fourier component of the total effective charge. But its physical reality is demonstrated by equation (99), where we can see that a given plasmon density creates a total charge distribution in the plasma, in the

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same way as a given electron density with charge −e would do, therefore creating a similar electrostatic potential. For very low frequency second order oscillations, ω  ωpe , the influence of the ion motion cannot be ignored. We can now define the effective plasmon charge with the help of a generalized charge identity X

qα n2α ≡ −en2e + Zen2i = qpl N

(101)

α

where we have the particle charges, qe = −e and qi = Ze, and the equilibrium densities n0e = n0 and n0i = n0 /Z. The final result will then be 

2 e¯ h ωpe k2 2ω k · k0 qpl = − 1+ 2 me ω0 ω (ω) ω0 k2

"

2 ωpe 1 1+ 2 ω (1 + χi )

#

(102)

where we have defined the dielectric function, (ω), and the particle susceptibilities as (ω) = (1 + χe )(1 + χi ) −

2 2 ωpe ωpi , ω4

χα = −

 ωpα  2 2 1 + k λ Dα ω2

(103)

Comparing with equation (100) we can see here two kinds of ion contributions. One is due to the ion contribution to the second order electron density perturbation n2e, now included in (ω). The other is due to the existence of a second order ion density perturbation n2i , which leads to the second term inside the square brackets.

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15

Plasmons in a time-varying plasma

Let us consider the case of electron plasma waves in a time varying plasma, where both the plasma density and the electron effective mass can evolve in time. This problem is important in many relevant physical conditions. The ions are assumed at rest, providing charge neutralization background, and the electrons are described by the modified electron fluid equations ∂n + ∇ · (nv) = F (t) ∂t

(104)

3kB T d [me (t)v] + eE + ∇n = 0, (105) dt n Here F (t) is a possible ionization or recombination rate, kB is Boltzmann constant, and T the electron plasma temperature, assumed constant. The electric field E is determined by Poisson’s equation ~ = e [n0 (t) − n], (106) ∇·E 0 where n0 (t) = n0 f (t) is the neutralizing ion density. We now consider the electron mean ˜ , where the last term describes a wave perturbation. If density of the form n = n0 (t) + n

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we linearize the above equations with respect to the perturbation and assume that the wave propagation takes place along the z-axis, we obtain the following wave equation [ ?] 



∂ 2n ˜ ∂ 2n ˜ ∂n ˜ 2 − S (t) + ωp2 (t)˜ n = ν(t) − F (t) + F 0 (t), e 2 2 ∂t ∂z ∂t

(107)

p

where Se (t) = 3kB T /me (t) a time dependent electron thermal velocity, F 0 (t) = dF/dt, and ωp (t) is the time dependent electron plasma frequency defined by "

#1/2

e2 n0 f (t) ωp (t) = 0 me (t)

.

(108)

In equation (107), the coefficient ν(t) determines the rate of change of the background electron plasma properties d h 2 i ln ωp (t) . (109) ν(t) = dt It should be emphasized that the wave equation (107) is exact and that no approximation, apart from linearization with respect to the wave perturbation, was introduced. In order to understand the physical meaning of this wave equation, and the main differences with respect to the stationary plasma, we first consider the case of a very slow variation in the medium, such that all the terms on the right hand side can be neglected, ν(t) ' 0, and F (t) ' 0. For a given wavenumber k, we have the simple solution of 

n ˜ k (z, t) = A±k exp ±ikz − i

Z t

0

0



ω(t )dt ,

(110)

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where A±k are constant wave amplitudes and the time varying frequency ω(t) obeys the time dependent dispersion relation ω 2(t) = ωp2(t) + Se2(t)k2 .

(111)

Let us now consider a general situation, where the electron plasma frequency changes continuously in time. We can assume a wave solution of the form n ˜ k (z, t) = Ak (t)eikz−iφ(t) + A−k e−ikz−iφ(t) + c.c.,

(112)

where A±k (t) are slowly varying amplitudes, such that |dA±k (t)/dt|  |ω(t)A±k (t)|, and φ(t) =

Z t

ω(t0 )dt0.

(113)

Replacing this solution in the wave equation (107), assuming that the instantaneous dispersion relation is always verified, and that |ν(t)|  ω(t), we arrive at the following two relations [?]   1 ω0 d Ak = − ν(t) A−k e2iφ(t) (114) dt 2 ω and   1 ω0 d A−k = − ν(t) Ak e−2iφ(t) , (115) dt 2 ω

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where ω 0 ≡ dω(t)/dt. It is interesting to solve these coupled equations in a situation where the wave propagating in the positive direction is dominant, |Ak (t)|  |A−k (t)|. We can then assume that Ak (t) ' Ak (0) = const. and A−k (0) = 0. A simple integration leads to A−k (t) = R(t)Ak (0)

(116)

where we have defined a temporal reflection coefficient R(t), such that 1 R(t) = 2

Z t  0  ω

ω

0



"

− ν(t ) exp −2i

Z t0

00

00

ω(t )dt

#

dt0 .

(117)

t0

This solution can also be derived by an alternative and quite different method [ ?], where the continuous variation of the medium is decomposed into a succession of infinitesimal steps at times tj = jτ , with j = 0, 1, 2, 3, ..., such that ωp (t) = ωp,j

(j − 1 < t/τ < j) ,

ωp,j = ωp,j−1 +

dωp τ dt

(τ → 0).

(118)

Each step can be considered as an elementary time refraction process [34]. Time refraction was first introduced as a temporal process exactly equivalent to the usual (space) refraction, where the background medium is assumed to change abruptly its dispersive properties, thus creating a temporal boundary between to different media. The continuity conditions for the plasmon density and velocity perturbations can be written as

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˜ (z, tj − 0) , n ˜ (z, tj + 0) = n

~v (z, tj + 0) = ~v(z, tj − 0).

(119)

From this analysis, the temporal reflection coefficient R(t) can be recovered. We conclude that the existence of reflected waves A.k is imposed by the continuity of the electron density and velocity perturbations. Such a conclusion, that temporal reflection implies the excitation of waves propagating in opposite direction, is also true for the case of transverse photons. It a consequence of total momentum conservation.

16

Plasmon acceleration

We can now push forward the quasi-particle concept of plasmons, and consider the motion of individual wavepackets, or quasi-particles. In general terms, we can say that the quasiparticle equations of motion take the form dr = v0 dt

,

dk0 = F0 ≡ −f 0 (k0)∇Φ(r, t), dt

(120)

where r is the position of the quasi-particles (average position of individual wavepackets), f 0 (k0) and the background potential Φ(r, t) depend on the type of quasi-particle and on the plasma perturbation. These equations coincide with the photon ray equations in a slowly ~ 0 can be due to the plasma density modulation varying plasma. For instance, the force F associated to the existence of an ion acoustic wave. If we assume the propagation of this

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slow wave along the direction Ox, as defined by Φ(~r, t) = Φ0 cos(kx−ωt), we can describe the plasmon parallel motion as 

ω dx = u− dt k



dp = −kf 0 (p)Φ0 sin(kx − ωt). dt

,

(121)

The perpendicular motion is trivially determined by the conservation of the perpendicular plasmon momentum, k0⊥ = constant. These equations show the existence of an elliptic fixed point at ω π u(p) = , x= (122) k k This means that, for quasi-particles such that u(p) ' ω/k, we have trapped oscillations at the bottom of the slow wave potential, with small amplitudes x ˜ = x − π/k around the fixed point. From eqs. (120) we can then derive ˜ ∂u d2 x = −k2 f 0(p) Φ0 x ˜, 2 dt ∂p

(123)

which is the equation for the harmonic oscillator with a frequency q

ωb = k f 0 (p)(∂u/∂p)Φ0,

(124)

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which is basically the bounce frequency for deeply trapped oscillations of quasi-particles in the slow wave potential Φ. The similarities with the electron bounce frequency are striking. We illustrate the above formalism, we assume an isotropic plasma with a broadband electron plasma wave turbulence. The explicit expressions for the velocity and force in the above quasi-particle equations, are 2 ~v 0 = 3vthe

~k0 ω0

,

1 e2 ∂ n ˜e F~ 0 = − 0 . 2ω 0 me ∂~r

(125)

Here, n ˜e is the perturbed electron plasma number density which, for the ion acoustic waves can be related with the potential perturbation by n ˜e = 0 Φ/(eλ2D ). This leads to a force, given by eq. (121) with 1 e , (126) f 0 (~k0 ) = 0 2ω me λ2D where ω 0 ' ωpe . From this we can determine, for example, the bounce frequency ωb of a trapped plasmon in the potential well of the ion acoustic waves. It will be given by ωpe ωb ' k 0 ω



eΦ0 2me

1/2

.

(127)

Comparing this with the electron bounce frequency ωbe = (eΦ0/me )1/2, we conclude that the plasmon (just like the photon) behaves in plasma as a particle, similar in many respects to an electron, or an ion. This important question of plasmon propagation in a space-time varying medium was recently considered in reference [47].

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17

Angular momentum of plasmons

The concept of photon orbital angular momentum has emerged in recent years in the literature, and only very recently it was discussed in the context of plasma physics. In a recent work [48], we have also proposed the existence of orbital angular momentum for plasmons and phonons, and studied the plasmon modes with some detail [49]. It should be stressed that, given the absence of spin, the orbital angular momentum for plasmons coincides with their total angular momentum. We discuss here the peculiarities of these plasmon states. We start from the linearlized wave equation for plasmons. But, instead of the usual plane wave solutions, which carry no angular momentum, we consider beam type of solutions of the form V (r, t) = V˜ (r) exp(ikz − iωt) where propagation is assumed along the Oz axis, and the amplitude V˜ (r) is a slowly function of the spatial coordinates describing the beam profile. Starting from the wave equation for plasmons, and taking the paraxial approximation, (|∂ 2V˜ /∂z 2 |  2|k∂ n ˜/∂z|) we can derive a paraxial equation for plasmons, of the form 

∇2⊥



∂ ˜ + 2ik V (r) = 0 ∂z

(128)

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where the longitudinal wave number k satisfies the linear dispersion relation k2 Se2 = ω 2 − 2 . Given the obvious analogy with the well known paraxial equations for photons, we can ωpe now discuss more general solutions of equation (128), represented in terms of LaguerreGauss (LG) functions. It is well known that the LG functions form a set of orthogonal and normalized functions in the perpendicular plane (r, ϕ). Each individual plasmon mode can be defined as (129) V˜ (r, ϕ, z) = V˜0 Fpl (r, z) exp(ilϕ) where p and l are the radial and angular mode numbers, respectively, ϕ is the azimuthal angle, and the Laguerre-Gauss function is defined by 

1 (l + p)! Fpl (r, z) = √ 2 π p!

1/2

X l Llp (X) exp(−X/2),

(130) |l|

where X = r2/w(z)2, and w(z) denotes the plasmon beam waist. Here Lp (X) are the associated Laguerre polynomials, as defined by the Rodrigues formula Llp (X) =

i exp(X)X −l dp h l+p X exp(−X) p! dX p

(131)

At this point it should be noticed that, if we assume l = 0, p = 0, and take R(z) = −ikw2 (z), a purely Gaussian solution will be recovered. Exploring further the analogy with photons, it is expected that these plasmon LG solutions will be able to describe electrostatic waves with a finite angular momentum. Having settled the electrostatic potential, we can determine the corresponding electric field E(r, t) = −∇V . This leads to electric field components given, in cylindrical coordinates, by ∂V 1 ∂Fpl ≡− Er = − ∂r Fpl ∂r

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!

V

(132)

Basic Properties of Plasmons

373

1 ∂V il ≡− V r ∂ϕ r ! ∂V 1 ∂Fpl Ez = − ≡ − ik + V ∂z Fpl ∂z Eϕ = −

(133)

This shows that electric field lines are not straight lines as in the usual plane wave solutions, but take the form of helical structures with radial and polar components. Here we should keep in mind that the axial field component along the Oz Ez ∼ −ikV is still the dominant one, in accordance with the paraxial approximation. It is obvious that this electric field can still be cast in the usual way, as E(r, t) = −ikef f V (r, ϕ, t)

(134)

where the new quantity kef f is an effective wavevector with components determined by equations (133). It is now important to show that these LG plasmon modes have orbital angular momentum. The angular momentum density can be defined by M(r) = (r × T) ,

T = W vg eef f

(135)

where T is the energy flux, vg the plasmon group velocity and eef f = kef f /|kef f | is the unit vector along the wave electric field. Here, W denotes the energy density associated with the plasmons, as defined as "

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2 2 2 0 2 (ω + k Se ) W = |E|2 1 + ωpe 4 (ω 2 − k2 Se2)

#

(136)

We should notice that the factor under square brackets is always of the order of 2. Of particular importance is the axial component of the angular momentum density M, which can be explicitly written as 0 kef f |V (r)|2 (137) Mz = l 2 vg This shows that the axial angular momentum for plasmons is proportional to the angular mode number l, which is in qualitative agreement with the case of transverse photon. It is useful to compare this result with the total angular momentum for photons [50] Mzph





0 σz r ∂ = l+ |(E(r)|2 2ω 2 ∂r

(138)

where σz = ±1 pertains for right and left-handed circular polarization, and ω is here the photon frequency. The additional term on σz corresponds to the spin part of the photon angular momentum, which is absent en the case of plasmons, where only the orbital component exists. For equal electric field amplitudes, and for linearly polarized transverse radiation σ 0, with emission of plasmons. It is interesting to consider this expression in the quasi-classical limit, such that k  p0 , when the parallel momentum of the particles is much larger than the wave momentum. In this case, we can expand the functions G0(p0 ±q/2) around the resonant momentum p = p0 , and obtain  1/2   ρ0 dG0 π . (150) γ= 4 2 Ae dp p=p0 We are then reduced to the familiar classical result.

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!2/! !p2 Quant.

Class.

k2"D2

Figure 4: Quantum versus classical plasmon dispersion relation.

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19

Conclusions

We have discussed several aspects of electron plasma waves, including not only classical results associated with their linear and nonlinear dispersion properties, such electron Landau damping, solitons and harmonic cascades, but also more recently discovered properties such as photon Landau damping of plasmons, the effective charge and mass of plasmons, and their orbital angular momentum. This provides a broad and consisted picture of the electron plasma waves, where both the wave and particle properties of plasmons are described and interrelated. This provides an illuminating example of a classical field theory which appeal for the use of quantum theory methods in a classical context. Second quantization has been used to study photons in a plasma, and to describe the photon change and mass [42]. A similar exercise could possibly be also made for the case of plasmons. We have also shown that electrostatic plasma waves can attain relativistic phase velocities. This is of major practical importance, because such relativistic waves can easily be excited by laser or energetic particle beams. Particle resonance interactions with such waves can then be used to built new types of plasma accelerators, more efficient and compact than the existing ones. A new approach to plasma turbulence has been explored in recent years, where quasi-particles (photons or plasmons) can resonantly interact with large scale plasma oscillations (such as electron plasma waves or ion acoustic waves), leading to quasi-particle Landau damping [57]. Examples of such quasi-particle resonant processes were considered here, and others have been recently explored. From this work emerges the concept of plasmon as an essential ingredient of plasma turbulence, a quasi-particle with properties

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usually associated with electrons or ions, such as mass, charge and acceleration, which can exchange energy with the other particles in the medium and contribute to the dispersion relation of other waves. Our recent proposal of a plasmon angular momentum completes this quasi-particle picture and can lead to stimulating new theoretical developments and experimental observations.

References [1] L. Tonks and I. Langmuir, Phys. Rev., 33, 195 (1929). [2] For a general description of plasma physics see, for instance, T.J.M. Boyd and J.J. Sanderson, The Physics of Plasmas, Cambridge University Press (2003). [3] E.V. Shuryak, Sov. Phys. JETP, 47, 212 (1978); H.A. Weldon, Phys. Rev. D, 26, 1394 (1983). [4] T. Tajima and K. Shibata, Plasma Astrophysics , Addison-Wesley, Reading (1997). [5] A. Vlasov, J. Phys., 9, 25 (1945). [6] L. Landau, J. Phys., 10, 25 (1946). [7] D. Bohm and E.P. Gross, Phys. Rev., 75, 1864 (1949). [8] N.G. Van Kampen, Physica, 21, 949 (1955).

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[9] J.M. Dawson, Phys. Fluids , 5, 517 (1962). [10] T.M. O’Neil, Phys. Fluids , 8, 2255 (1965). [11] D.R. Nicholson, Introduction to Plasma Theory , Wiley and Sons, Chichester (1983). [12] R. Bingham, J.T. Mendonc¸a and P.K. Shukla, Plasma Phys. and Controlled Fusion , 64, R-1 (2004). [13] A.W. Trivelpiece and E.W. Gould, J. Appl. Phys., 30, 1784 (1959). [14] L. Tonks, Phys. Rev. 38, 1458 (1931); A. Dattner, Phys. Rev. Lett., 10, 205 (1963) [15] J.V. Parker, J.C. Nickel and R.W. Gould, Phys. Fluids , 7, 1489 (1964). [16] J.T. Mendonc¸a, R. Kaiser, H. Terc¸as and J. Loureiro, Phys. Rev. A, 78, 013408 (2008). [17] J.A.C. Cabral, L.M. Lap˜ao and J.T. Mendonc¸a, Phys. Fluids B , 5, 787 (1993). [18] D.H.E. Dubin, Phys. Rev. E, 53, 5268 (1996). [19] L.I. Rudakov, Dokl. Akad. Nauk. SSSR, 207, 84 (1972). [20] V.E. Zakharov, Zh. Eksp. Teor. Fiz., 62, 1945 (1972). Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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[21] R. Stenzel, A.Y. Wong and H.C.P. Kim, Phys. Rev. Lett., 32, 654 (1974). [22] V.E. Zakharov, in Handbook of Plasma Physics , A.A. Galeev and R.N. Sudan editors, vol. II, North Holland, Amsterdam (1984). [23] R.K. Dodd, J.C. Eilbeck, J.D. Gibbon abd H.C. Morris, Solitons and Nonlinear Wave Equations , Academic Press, New York (1982). [24] A.I. Akhiezer and R.V. Polovin, Sov. Phys. JETP, 30, 915 (1956). [25] J.M. Dawson, Phys. Rev., 113, 383 (1959). [26] V.P. Silin, Sov. Phys. JETP, 11, 1136 (1960). [27] P. Kaw and J.M. Dawson, Phys. Fluids , 13, 472 (1970). [28] A. Decoster, Phys. Rep., 47, 285 (1978). [29] P.K. Shukla, N.N. Rao, M.Y. Yu and N.L. Tsintsadze, Phys. Rep., 138, 1 (1986). [30] J.H. Malmberg and C.B. Wharton, Phys. Rev. Lett., 13, 184 (1964). [31] J.P. Holloway and J.J. Dorning, Phys. Rev. A, 44, 3856 (1991). [32] H. Schamel, Phys. Plasmas, 7, 4831 (2000). [33] R. Bingham, J.T. Mendonc¸a and J.M. Dawson, Phys. Rev. Lett., 78, 247 (1997).

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[34] J.T. Mendonc¸a, Theory of Photon Acceleration , Institute of Physics Publishing, Bristol (2001). [35] J.T. Mendonc¸a, Phys. Scripta, 74, C61 (2006). [36] J.T. Mendonc¸a and A. Serbeto, Phys. Plasmas, 15, 113105 (2008). [37] C.D. Murphy et al., Phys. Plasmas, 13, 033108 (2006). [38] R. Fonseca et al., Lecture Notes in Computer Science, 2329, III-342, Springer-Verlag, Heidelberg (2002). [39] J.T. Mendonc¸a and R. Bingham, Phys. Plasmas, 9, 2604 (2002). [40] J.T. Mendonc¸a, A. Serbeto, J.R. Davies and R. Bingham, Plasma Phys. Control. Fusion, 50, 105009 (2008). [41] J.T. Mendonc¸a, P. Norreys, R. Bingham and J.R. Davies, Phys. Rev. Lett., 94, 245002 (2005). [42] J.T. Mendonc¸a, A.M. Martins and A. Guerreiro, Phys. Rev. E, 62, 2989 (2000). [43] J.T. Mendonc¸a, L.O. Silva, R. Bingham, N.L. Tsintsadze, P.K. Shukla and J.M. Dawson, Phys. Lett. A, 239, 373 (1998). Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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[44] N.L. Tsintsadze, J.T. Mendonc¸a and P.K. Shukla, Phys. Lett. A, 249, 110 (1998). [45] J.T. Mendonc¸a, A. Serbeto and S. Ali, J. Plasma Phys., accepted (2010). [46] J.T. Mendonc¸a, New J. Phys., 11, 013029 (2009). [47] I.Y. Dodin, V.I. Geyko and N.J. Fisch, Phys. Plasmas, 16, 112101 (2009). [48] J.T. Mendonc¸a, B. Thid´e and H. Then, Phys. Rev. Lett., 102, 185005 (2009). [49] J.T. Mendonc¸a, S. Ali and B. Thid´e, Phys. Plasmas, 16, 124024 (2009). [50] L. Allen, M.W. Beijersbergen, R.J.C. Spreeuw and J.P. Woerdman, Phys. Rev. A, 45, 8185 (1992). [51] H. Terc¸as, J.T. Mendonc¸a and P.K. Shukla, Phys. Plasmas, 15, 072109 (2008). [52] E.M. Lifshitz and L.P. Pitaevskii, Physical Kinetics , Butterworth-Heinemann, Oxford (1995). [53] P.K. Shukla and B. Eliasson, Phys. Uspekhi , 53, 51 (2010). [54] F. Haas, Phys. Plasmas, 12, 062117 (2005). [55] G. Manfredi, Fields Inst. Commun., 46, 263 (2005). [56] J.T. Mendonc¸a, E. Ribeiro and P.K. Shukla, J. Plasma Phys., 74, 91 (2008).

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[57] J.T. Mendonc¸a, R. Bingham and P.K. Shukla, Phys. Rev. E, 68, 016406 (2003).

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In: Plasmons: Theory and Applications Editor: Kristina N. Helsey, pp. 381-435

ISBN: 978-1-61761-306-7 c 2011 Nova Science Publishers, Inc.

Chapter 16

E XCITON -P LASMON I NTERACTIONS IN I NDIVIDUAL C ARBON N ANOTUBES I. V. Bondarev, L. M. Woods and A.Popescu∗ Department of Physics, North Carolina Central University, Durham, NC, USA Department of Physics, University of South Florida Tampa, FL, USA

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Abstract We use the macroscopic quantum electrodynamics approach suitable for absorbing and dispersing media to study the properties and role of collective surface excitations — excitons and plasmons — in single-wall and double-wall carbon nanotubes. We show that the interactions of excitonic states with surface electromagnetic modes in individual small-diameter ( . 1 nm) single-walled carbon nanotubes can result in strong excitonsurface-plasmon coupling. Optical response of individual nanotubes exhibits Rabi splitting ∼ 0.1 eV, both in the linear excitation regime and in the non-linear excitation regime with the photoinduced biexcitonic states formation, as the exciton energy is tuned to the nearest interband surface plasmon resonance of the nanotube. An electrostatic field applied perpendicular to the nanotube axis can be used to control the exciton-plasmon coupling. For double-wall carbon nanotubes, we show that at tube separations similar to their equilibrium distances interband surface plasmons have a profound effect on the inter-tube Casimir force. Strong overlapping plasmon resonances from both tubes warrant their stronger attraction. Nanotube chiralities possessing such collective excitation features will result in forming the most favorable inner-outer tube combination in double-wall carbon nanotubes. These results pave the way for the development of new generation of tunable optoelectronic and nano-electromechanical device applications with carbon nanotubes.

∗

E-mail address: [email protected]

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Keywords: Carbon nanotubes, Near-field effects, Excitons, Plasmons

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1

Introduction

Single-walled carbon nanotubes (CNs) are quasi-one-dimensional (1D) cylindrical wires consisting of graphene sheets rolled-up into cylinders with diameters ∼ 1 − 10 nm and lengths ∼ 1 − 104 µm [1, 2, 3, 4]. CNs are shown to be useful as miniaturized electronic, electromechanical, and chemical devices [5], scanning probe devices [6], and nanomaterials for macroscopic composites [7]. The area of their potential applications was recently expanded to nanophotonics [8, 9, 10, 11, 12, 13] after the demonstration of controllable single-atom incapsulation into CNs [14, 15, 16, 17], and even to quantum cryptography since the experimental evidence was reported for quantum correlations in the photoluminescence spectra of individual nanotubes [18]. For pristine (undoped) single-walled CNs, the numerical calculations predicting large exciton binding energies (∼ 0.3 − 0.6 eV) in semiconducting CNs [19, 20, 21] and even in some small-diameter (∼ 0.5 nm) metallic CNs [22], followed by the results of various exciton photoluminescence measurements [18, 23, 24, 25, 26, 27], have become available. These works, together with other reports investigating the role of effects such as intrinsic defects [25, 28], exciton-phonon interactions [26, 28, 29, 30, 31], biexciton formation [32, 33], exciton-surface-plasmon coupling [34, 35, 36, 37], external magnetic [38, 39] and electric fields [37, 40], reveal the variety and complexity of the intrinsic optical properties of CNs [41]. Carbon nanotubes combine advantages such as electrical conductivity, chemical stability, high surface area, and unique optoelectronic properties that make them excellent potential candidates for a variety of applications, including efficient solar energy conversion [7], energy storage [14], optical nanobiosensorics [42]. However, the quantum yield of individual CNs is normally very low. Nanotube composites of CN bundles and/or films could surpass this difficulty, opening up new paths for the development of high-yield, highperformance optoelectronics applications with CNs [43, 44]. Understanding the inter-tube interactions is important in order to be able to tailor properties of CN bundles and films, as well as properties of multi-wall CNs. This is also important for experimental realization of new effects and devices proposed recently, such as trapping of cold atoms [42, 45] and their entanglement [11] near single-walled CNs, surface profiling [6] and nanolithography applications [46] with double-wall CNs. Here, we use the macroscopic Quantum ElectroDynamics (QED) formalism developed earlier for absorbing and dispersive media [47, 48, 49, 9] and then successfully employed to study near-field EM effects in hybrid CN systems [10, 11, 45], to investigate the properties and role of collective surface excitations — excitons and plasmons — in single-wall and double-wall CNs. First, we show that, due to the presence of low-energy ( ∼ 0.5 − 2 eV) weakly-dispersive interband plasmon modes [50] and large exciton excitation energies in the same energy domain [51, 52], the excitons can form strongly coupled mixed excitonplasmon excitations in individual small-diameter ( . 1 nm) semiconducting single-walled CNs. The exciton-plasmon coupling (and the exciton emission accordingly) can be controlled by an external electrostatic field applied perpendicular to the CN axis (the quantum confined Stark effect). The optical response of individual CNs exhibits the Rabi splitting of

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∼ 0.1 eV, both in the linear excitation regime and in the non-linear excitation regime with the photoinduced biexcitonic states formation, as the exciton energy is tuned to the nearest interband surface plasmon resonance of the CN. Previous studies of the exciton-plasmon coupling have been focused on artificially fabricated hybrid plasmonic nanostructures, such as dye molecules in organic polymers deposited on metallic films [53], semiconductor quantum dots coupled to metallic nanoparticles [54], or nanowires [55], where semiconductor material carries the exciton and metal carries the plasmon. Our results are particularly interesting since they reveal the fundamental electromagnetic (EM) phenomenon — the strong exciton-plasmon coupling — in an individual quasi-1D nanostructure, a carbon nanotube, as well as its tunability feature by means of the quantum confined Stark effect. We expect these results to open up new paths for the development of tunable optoelectronic device applications with optically excited carbon nanotubes, including the strong excitation regime with optical non-linearities. Next, we turn to the double-wall carbon nanotubes to investigate the effect of collective surface excitations on the inter-tube Casimir interaction in these systems. The Casimir interaction is a paradigm for a force induced by quantum EM fluctuations. The fundamental nature of this force has been studied extensively ever since the prediction of the existence of an attraction between neutral metallic mirrors in vacuum [49, 56]. In recent years, the Casimir effect has acquired a much broader impact due to its importance for nanostructured materials and devices. The development and operation of micro- and nanoelectromechanical systems are limited due to unwanted effects, such as stiction, friction, and adhesion, originating from the Casimir force [57]. This interaction is also an important component for the stability of nanomaterials. Here, we show that at tube separations similar to their equilibrium distances interband surface plasmons have a profound effect on the inter-tube Casimir force. Strong overlapping plasmon resonances from both tubes warrant their stronger attraction. Nanotube chiralities possessing such collective excitation features will result in forming the most favorable inner-outer tube combination in doublewall carbon nanotubes. This theoretical understanding is important for the development of nano-electromechanical devices with CNs. This Chapter is organized as follows. Section 2 introduces the general Hamiltonian of the exciton interaction with vacuum-type quantized surface EM modes of a single-walled CN. No external EM field is assumed to be applied. The vacuum–type–field we consider is created by CN surface EM fluctuations. Section 3 explains how the interaction introduced results in the coupling of the excitonic states to the nanotube’s surface plasmon modes. Here we derive and discuss the characteristics of the coupled exciton–plasmon excitations, such as the dispersion relation, the plasmon density of states (DOS), and the optical response functions, for particular semiconducting CNs of different diameters. We also analyze how the electrostatic field applied perpendicular to the CN axis affects the CN band gap, the exciton binding energy, and the surface plasmon energy, to explore the tunability of the exciton-surface-plasmon coupling in CNs. Section 4 derives and analyzes the Casimir interaction between two concentric cylindrical graphene sheets comprising a double-wall CN. The summary and conclusions of the work are given in Sec. 5. All the technical details about the construction and diagonalization of the exciton–field Hamiltonian, the EM field Green tensor derivation, the perpendicular electrostatic field effect, are presented in the Appendices in order not to interrupt the flow of the arguments and results.

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Figure 1: The geometry of the problem.

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2

Exciton-electromagnetic-field interaction on the nanotube surface

We consider the vacuum-type EM interaction of an exciton with the quantized surface electromagnetic fluctuations of a single-walled semiconducting CN by using our recently developed Green function formalism to quantize the EM field in the presence of quasi-1D absorbing bodies [58, 59, 60, 61, 62, 9]. No external EM field is assumed to be applied. The nanotube is modelled by an infinitely thin, infinitely long, anisotropically conducting cylinder with its surface conductivity obtained from the realistic band structure of a particular CN. Since the problem has the cylindrical symmetry, the orthonormal cylindrical basis {er , eϕ, ez } is used with the vector ez directed along the nanotube axis as shown in Fig. 1. Only the axial conductivity, σzz , is taken into account, whereas the azimuthal one, σϕϕ , being strongly suppressed by the transverse depolarization effect [63, 64, 65, 66, 67, 68], is neglected. The total Hamiltonian of the coupled exciton-photon system on the nanotube surface is of the form ˆ ex + H ˆ int , ˆ =H ˆF + H (1) H where the three terms represent the free (medium-assisted) EM field, the free (noninteracting) exciton, and their interaction, respectively. More explicitly, the second quantized field Hamiltonian is XZ ∞ ˆ ω), ˆ dω ~ω fˆ† (n, ω)f(n, (2) HF = n

0

where the scalar bosonic field operators fˆ† (n, ω) and fˆ(n, ω) create and annihilate, respectively, the surface EM excitation of frequency ω at an arbitrary point n = Rn = {RCN , ϕn , zn } associated with a carbon atom (representing a lattice site – Fig. 1) on the Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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surface of the CN of radius RCN . The summation is made over all the carbon atoms, and in the following it is replaced by the integration over the entire nanotube surface according to the rule Z Z 2π Z ∞ X 1 1 ...= dRn . . . = dϕn RCN dzn . . . , (3) S0 S0 0 −∞ n √ where S0 = (3 3/4)b2 is the area of an elementary equilateral triangle selected around ˚ is the each carbon atom in a way to cover the entire surface of the nanotube, b = 1.42 A carbon-carbon interatomic distance. The second quantized Hamiltonian of the free exciton (see, e.g., Ref. [69]) on the CN surface is of the form X X † † ˆ ex = Ef (n)Bn+m,f Bm,f = Ef (k)Bk,f Bk,f , (4) H n,m,f

k,f

† where the operators Bn,f and Bn,f create and annihilate, respectively, an exciton with the energy Ef (n) in the lattice site n of the CN surface. The index f (6= 0) refers to the internal degrees of freedom of the exciton. Alternatively,

1 X † ik·n † † =√ Bn,f e and Bk,f = (Bk,f )† Bk,f N n

(5)

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create and annihilate the f -internal-state exciton with the quasi-momentum k = {kϕ, kz }, where the azimuthal component is quantized due to the transverse confinement effect and the longitudinal one is continuous, N is the total number of the lattice sites (carbon atoms) on the CN surface. The exciton total energy is then written in the form ~2kz2 2Mex (kϕ)

(f ) (kϕ) + Ef (k) = Eexc

(6)

Here, the first term represents the excitation energy (f )

(f ) (kϕ ) = Eg (kϕ) + Eb (kϕ) Eexc

(7) (f )

of the f -internal-state exciton with the (negative) binding energy Eb , created via the interband transition with the band gap Eg (kϕ) = εe (kϕ) + εh (kϕ ),

(8)

where εe,h are transversely quantized azimuthal electron-hole subbands (see the schematic in Fig. 2). The second term in Eq. (6) represents the kinetic energy of the translational longitudinal movement of the exciton with the effective mass Mex = me + mh , where me and mh are the (subband-dependent) electron and hole effective masses, respectively. The two equivalent free-exciton Hamiltonian representations are related to one another via the obvious orthogonality relationships 1 X −i(n−m)·k 1 X −i(k−k0 )·n e = δkk0 , e = δnm N n N k

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(9)

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Figure 2: Schematic of the two transversely quantized azimuthal electron-hole subbands (left), and the first-interband ground-internal-state exciton energy ( right) in a smalldiameter semiconducting carbon nanotube. Subbands with indices j = 1 and 2 are shown, along with the optically allowed (exciton-related) interband transitions [67]. See text for notations. with the k-summation running over the first Brillouin zone of the nanotube. The bosonic ˆ F are transformed to the k-representation in the same way. field operators in H The most general (non-relativistic, electric dipole) exciton-photon interaction on the nanotube surface can be written in the form (we use the Gaussian system of units and the Coulomb gauge; see details in Appendix A) X Z ∞ (+) (−) † ˆ ˆ int = dω [ gf (n, m, ω)Bn,f − gf (n, m, ω)Bn,f ] f(m, ω) + h.c., (10) H n,m,f

0

where (±)

gf (n, m, ω) = g⊥ f (n, m, ω) ±

ω k g (n, m, ω) ωf f

(11)

with ⊥(k)

gf

(n, m, ω) = −i

4ωf p π~ω Re σzz (RCN , ω) (dfn )z ⊥(k) Gzz (n, m, ω) c2 (f )

(12)

being the interaction matrix element where the exciton with the energy Eexc = ~ωf is ˆ n)z |f i in the lattice site n by excited through the electric dipole transition (dfn )z = h0|(d the nanotube’s transversely (longitudinally) polarized surface EM modes. The modes are represented in the matrix element by the transverse (longitudinal) part of the Green tensor zz-component Gzz (n, m, ω) of the EM subsystem (Appendix B). This is the only Green tensor component we have to take into account. All the other components can be safely neglected as they are greatly suppressed by the strong transverse depolarization effect in

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CNs [63, 64, 65, 66, 67, 68]. As a consequence, only σzz (RCN , ω), the axial dynamic surface conductivity per unit length, is present in Eq.(12). Equations (1)–(12) form the complete set of equations describing the exciton-photon coupled system on the CN surface in terms of the EM field Green tensor and the CN surface axial conductivity.

3

Exciton-surface-plasmon coupling

For the following it is important to realize that the transversely polarized surface EM mode contribution to the interaction (10)–(12) is negligible compared to the longitudinally polarized surface EM mode contribution. As a matter of fact, ⊥ Gzz (n, m, ω) ≡ 0 in the model of an infinitely thin cylinder we use here (Appendix 5), thus yielding (±)

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g⊥ f (n, m, ω) ≡ 0, gf (n, m, ω) = ±

ω k g (n, m, ω) ωf f

(13)

in Eqs. (10)–(12). The point is that, because of the nanotube quasi-one-dimensionality, the exciton quasi-momentum vector and all the relevant vectorial matrix elements of the momentum and dipole moment operators are directed predominantly along the CN axis (the longitudinal exciton; see, however, Ref. [70]). This prevents the exciton from the electric dipole coupling to transversely polarized surface EM modes as they propagate predominantly along the CN axis with their electric vectors orthogonal to the propagation direction. The longitudinally polarized surface EM modes are generated by the electronic Coulomb potential (see, e.g., Ref. [71]), and therefore represent the CN surface plasmon excitations. These have their electric vectors directed along the propagation direction. They do couple to the longitudinal excitons on the CN surface. Such modes were observed in Ref. [50]. They occur in CNs both at high energies (well-known π-plasmon at ∼ 6 eV) and at comparatively low energies of ∼ 0.5 − 2 eV. The latter ones are related to the transversely quantized interband (inter-van Hove) electronic transitions. These weaklydispersive modes [50, 72] are similar to the intersubband plasmons in quantum wells [73]. They occur in the same energy range of ∼ 1 eV where the exciton excitation energies are located in small-diameter (. 1 nm) semiconducting CNs [51, 52]. In what follows we focus our consideration on the exciton interactions with these particular surface plasmon modes.

3.1

The dispersion relation

To obtain the dispersion relation of the coupled exciton-plasmon excitations, we transfer the total Hamiltonian (1)–(10) and (13) to the k-representation using Eqs. (5) and (9), and then diagonalize it exactly by means of Bogoliubov’s canonical transformation technique (see, e.g., Ref. [74]). The details of the procedure are given in Appendix 5. The Hamiltonian takes the form X ˆ = ~ωµ (k) ξˆ† (k)ξˆµ (k) + E0 . (14) H µ

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Here, the new operator ξˆµ (k) = +

Z

∞

Xh h

† u∗µ (k, ωf )Bk,f − vµ (k, ωf )B−k,f

i

f

ˆ ω) − v ∗ (k, ω)fˆ†(−k, ω) dω uµ (k, ω)f(k, µ

(15)

i

0

annihilates and ξˆµ† (k) = [ξˆµ (k)]† creates the exciton-plasmon excitation of branch µ, the quantities uµ and vµ are appropriately chosen canonical transformation coefficients. The ”vacuum” energy E0 represents the state with no exciton-plasmons excited in the system, and ~ωµ (k) is the exciton-plasmon energy given by the solution of the following (dimensionless) dispersion relation x2µ

Z

f

¯ (x)ρ(x) xΓ 0 = 0. x2µ − x2

(16)

~ωµ (k) Ef (k) ~ω , xµ = , εf = 2γ0 2γ0 2γ0

(17)

−

ε2f

2 − εf π

∞

dx 0

Here, x=

with γ0 = 2.7 eV being the carbon nearest neighbor overlap integral entering the CN surface axial conductivity σzz (RCN , ω). The function 4|dfz |2 x3 f ¯ Γ0 (x) = 3~c3

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with dfz =

P

ˆ

n h0|(dn)z |f i



2γ0 ~

2

(18)

represents the (dimensionless) spontaneous decay rate, and ρ(x) =

3S0 1 Re 2 σ ¯zz (x) 16παRCN

(19)

stands for the surface plasmon density of states (DOS) which is responsible for the exciton decay rate variation due to its coupling to the plasmon modes. Here, α = e2 /~c = 1/137 is the fine-structure constant and σ ¯zz = 2π~σzz /e2 is the dimensionless CN surface axial conductivity per unit length. Note that the conductivity factor in Eq. (19) equals   4αc 1 ~ 1 =− Im (20) Re σ ¯zz (x) RCN 2γ0x zz (x) − 1 in view of Eq. (17) and equation σzz (x) = −

iω [zz (x) − 1] 4πSρT

(21)

representing the Drude relation for CNs, where zz is the longitudinal (along the CN axis) dielectric function, S and ρT are the surface area of the tubule and the number of tubules per unit volume, respectively [59, 62, 64]. This relates very closely the surface plasmon Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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Figure 3: (a),(b) Calculated dimensionless (see text) axial surface conductivities for the (11,0) and (10,0) CNs. The dimensionless energy is defined as [Energy]/2γ0, according to Eq. (17).

DOS function (19) to the loss function −Im(1/) measured in Electron Energy Loss Spectroscopy (EELS) experiments to determine the properties of collective electronic excitations in solids [50]. σzz (x)] for the (11,0) Figure 3 shows the low-energy behaviors of σ ¯zz (x) and Re[1/¯ and (10,0) CNs (RCN = 0.43 nm and 0.39 nm, respectively) we study here. We obtained them numerically as follows. First, we adapt the nearest-neighbor non-orthogonal tightbinding approach [75] to determine the realistic band structure of each CN. Then, the roomtemperature longitudinal dielectric functions zz are calculated within the random-phase approximation [76, 77], which are then converted into the conductivities σ ¯zz by means of the Drude relation. Electronic dissipation processes are included in our calculations within the relaxation-time approximation (electron scattering length of 130RCN was used [30]). We did not include excitonic many-electron correlations, however, as they mostly affect the σzz ) which is responsible for the CN optical absorption [20, 22, 67], real conductivity Re(¯ whereas we are interested here in Re(1/¯ σzz ) representing the surface plasmon DOS according to Eq. (19). This function is only non-zero when the two conditions, Im [¯ σzz (x)] = 0 and Re[¯ σzz (x)] → 0, are fulfilled simultaneously [72, 73, 76]. These result in the peak structure of the function Re(1/¯ σzz ) as is seen in Fig. 3. It is also seen from the comparison of Fig. 3 (b) with Fig. 3 (a) that the peaks broaden as the CN diameter decreases. This is consistent with the stronger hybridization effects in smaller-diameter CNs [78]. Left panels in Figs. 4(a) and 4(b) show the lowest-energy plasmon DOS resonances calculated for the (11,0) and (10,0) CNs as given by the function ρ(x) in Eq. (19). Also shown σzz (x)]. In all there are the corresponding fragments of the functions Re [¯ σzz (x)] and Im[¯ graphs the lower dimensionless energy limits are set up to be equal to the lowest bright exci(11) ton excitation energy [Eexc = 1.21 eV (x = 0.224) and 1.00 eV (x = 0.185) for the (11,0) and (10,0) CN, respectively, as reported in Ref.[51] by directly solving the Bethe-Salpeter equation]. Peaks in ρ(x) are seen to coincide in energy with zeros of Im[¯σzz (x)] {or zeros of Re[zz (x)]}, clearly indicating the plasmonic nature of the CN surface excitations under

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Figure 4: (a),(b) Surface plasmon DOS and conductivities (left panels), and lowest bright exciton dispersion when coupled to plasmons (right panels) in (11,0) and (10,0) CNs, respectively. The dimensionless energy is defined as [ Energy]/2γ0, according to Eq. (17). See text for the dimensionless quasi-momentum. consideration [72, 79]. They describe the surface plasmon modes associated with the transversely quantized interband electronic transitions in CNs [72]. As is seen in Fig. 4 (and in Fig. 3), the interband plasmon excitations occur in CNs slightly above the first bright exciton excitation energy [67], in the frequency domain where the imaginary conductivity (or the real dielectric function) changes its sign. This is a unique feature of the complex dielectric response function, the consequence of the general Kramers-Kr¨onig relation [47]. We further take advantage of the sharp peak structure of ρ(x) and solve the dispersion equation (16) for xµ analytically using the Lorentzian approximation ρ(x) ≈

ρ(xp)∆x2p . (x − xp )2 + ∆x2p

(22)

Here, xp and ∆xp are, respectively, the position and the half-width-at-half-maximum of the plasmon resonance closest to the lowest bright exciton excitation energy in the same nanotube (as shown in the left panels of Fig. 4). The integral in Eq. (16) then simplifies to the form Z Z ¯ f (x)ρ(x) F (xp )∆x2p ∞ dx 2 ∞ xΓ 0 dx ≈ 2 2 2 2 π 0 xµ − x xµ − xp 0 (x − xp )2 + ∆x2p

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    xp F (xp )∆xp π arctan + = x2µ − x2p ∆xp 2 ¯ f (xp)ρ(xp)/π. This expression is valid for all xµ apart from those with F (xp ) = 2xpΓ 0 located in the narrow interval (xp −∆xp , xp+∆xp ) in the vicinity of the plasmon resonance, provided that the resonance is sharp enough. Then, the dispersion equation becomes the biquadratic equation for xµ with the following two positive solutions (the dispersion curves) of interest to us s ε2f + x2p 1 q ± (ε2f − x2p )2 + Fp εf . (23) x1,2 = 2 2 Here, Fp = 4F (xp )∆xp(π − ∆xp /xp) with the arctan-function expanded to linear terms in ∆xp/xp  1. The dispersion curves (23) are shown in the right panels in Figs. 4(a) and 4(b) as functions of the dimensionless longitudinal quasi-momentum. In these calculations, we ¯ f (xp) [Eq.(18)] from the equation estimated the interband transition matrix element in Γ 0 2 3 rad |df | = 3~λ /4τex according to Hanamura’s general theory of the exciton radiative decay rad is the exciton intrinsic radiative lifetime, and in spatially confined systems [80], where τex λ = 2πc~/E with E being the exciton total energy given in our case by Eq. (6). For zigzagtype CNs considered here, the first Brillouin zone of the longitudinal quasi-momentum is given by −2π~/3b ≤ ~kz ≤ 2π~/3b [1, 2]. The total energy of the ground-internal-state exciton can then be written as E = Eexc + (2π~/3b)2t2 /2Mex with −1 ≤ t ≤ 1 representing the dimensionless longitudinal quasi-momentum. In our calculations we used the (11) rad = 14.3 ps and 19.1 ps, lowest bright exciton parameters Eexc = 1.21 eV and 1.00 eV, τex Mex = 0.44m0 and 0.19m0 (m0 is the free-electron mass) for the (11,0) CN and (10,0) CN, respectively, as reported in Ref.[51] by directly solving the Bethe-Salpeter equation. Both graphs in the right panels in Fig. 4 are seen to demonstrate a clear anticrossing behavior with the (Rabi) energy splitting ∼ 0.1 eV. This indicates the formation of the strongly coupled surface plasmon-exciton excitations in the nanotubes under consideration. It is important to realize that here we deal with the strong exciton-plasmon interaction supported by an individual quasi-1D nanostructure — a single-walled (small-diameter) semiconducting carbon nanotube, as opposed to the artificially fabricated metal-semiconductor nanostructures studied previuosly [53, 54, 55] where the metallic component normally carries the plasmon and the semiconducting one carries the exciton. It is also important that the effect comes not only from the height but also from the width of the plasmon resonance as it is seen from the definition of the Fp factor in Eq. (23). In other words, as long as the plasmon resonance is sharp enough (which is always the case for interband plasmons), so that the Lorentzian approximation (22) applies, the effect is determined by the area under the plasmon peak in the DOS function (19) rather than by the peak height as one would expect. However, the formation of the strongly coupled exciton-plasmon states is only possible if the exciton total energy is in resonance with the energy of a surface plasmon mode. The exciton energy can be tuned to the nearest plasmon resonance in ways used for excitons in semiconductor quantum microcavities — thermally [81, 82, 83] (by elevating sample temperature), or/and electrostatically [84, 85, 86, 87] (via the quantum confined Stark effect with an external electrostatic field applied perpendicular to the CN axis). As is seen

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from Eqs. (6) and (7), the two possibilities influence the different degrees of freedom of the quasi-1D exciton — the (longitudinal) kinetic energy and the excitation energy, respectively. Below we study the (less trivial) electrostatic field effect on the exciton excitation energy in CNs.

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3.2

The perpendicular electrostatic field effect

The optical properties of semiconducting CNs in an external electrostatic field directed along the nanotube axis were studied theoretically in Ref. [40]. Strong oscillations in the band-to-band absorption and the quadratic Stark shift of the exciton absorption peaks with the field increase, as well as the strong field dependence of the exciton ionization rate, were predicted for CNs of different diameters and chiralities. Here, we focus on the perpendicular electrostatic field orientation. We study how the electrostatic field applied perpendicular to the CN axis affects the CN band gap, the exciton binding/excitation energy, and the interband surface plasmon energy, to explore the tunability of the strong exciton-plasmon coupling effect predicted above. The problem is similar to the well-known quantum confined Stark effect first observed for the excitons in semiconductor quantum wells [84, 85]. However, the cylindrical surface symmetry of the excitonic states brings new peculiarities to the quantum confined Stark effect in CNs. In what follows we will generally be interested only in the lowest internal energy (ground) excitonic state, and so the internal state index f in Eqs. (6) and (7) will be omitted for brevity. Because the nanotube is modelled by a continuous, infinitely thin, anisotropically conducting cylinder in our macroscopic QED approach, the actual local symmetry of the excitonic wave function resulted from the graphene Brillouin zone structure is disregarded in our model (see, e.g., reviews [41, 67]). The local symmetry is implicitly present in the surface axial conductivity though, which we calculate beforehand as described above. 1 We start with the Schr¨odinger equation for the electron and hole on the CN surface, located at re = {RCN , ϕe, ze } and rh = {RCN , ϕh, zh }, respectively. They interact with each other through the Coulomb potential V (re , rh ) = −e2 /|re − rh |, where  = zz (0). The external electrostatic field F = {F, 0, 0} is directed perpendicular to the CN axis (along the x-axis in Fig. 1). The Schr¨odinger equation is of the form i h ˆ h (F) + V (re , rh ) Ψ(re , rh ) = EΨ(re , rh ) ˆ e (F) + H (24) H with 2 ˆ e,h (F) = − ~ H 2me,h

1 R2CN

∂2 ∂2 + 2 ∂ϕ2e,h ∂ze,h

!

∓ ere,h · F

(25)

We further separate out the translational and relative degrees of freedom of the electronhole pair by transforming the longitudinal (along the CN axis) motion of the pair into its 1

In real CNs, the existence of two equivalent energy valleys in the 1st Brillouin zone, the K- and K 0-valleys with opposite electron helicities about the CN axis, results into dark and bright excitonic states in the lowest energy spin-singlet manifold [88]. Since the electric interaction does not involve spin variables, both K- and K 0-valleys are affected equally by the electrostatic field in our case, and the detailed structure of the exciton wave function multiplet is not important. This is opposite to the non-zero magnetostatic field case where the field affects the K- and K 0-valleys differently either to brighten the dark excitonic states [39], or to create Landau sublevels [67] for longitudinal and perpendicular orientation, respectively.

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center-of-mass coordinates given by Z = (me ze + mh zh )/Mex and z = ze − zh . The exciton wave function is approximated as follows Ψ(re , rh ) = eikz Z φex (z)ψe(ϕe )ψh (ϕh ).

(26)

The complex exponential describes the exciton center-of-mass motion with the longitudinal quasi-momentum kz along the CN axis. The function φex (z) represents the longitudinal relative motion of the electron and the hole inside the exciton. The functions ψe (ϕe ) and ψh (ϕh ) are the electron and hole subband wave functions, respectively, which represent their confined motion along the circumference of the cylindrical nanotube surface. Each of the functions is assumed to be normalized to unity. Equations (24) and (25) are then rewritten in view of Eqs. (6)–(8) to yield   ~2 ∂2 − − eRCN F cos(ϕe ) ψe (ϕe ) = εe ψe (ϕe ), (27) 2me R2CN ∂ϕ2e   ~2 ∂2 − + eRCN F cos(ϕh ) ψh (ϕh ) = εh ψh (ϕh ), (28) 2mh R2CN ∂ϕ2h  2 2  ~ ∂ − + V (z) φex (z) = Eb φex (z), (29) eff 2µ ∂z 2

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where µ = me mh /Mex is the exciton reduced mass, and Veff is the effective longitudinal electron-hole Coulomb interaction potential given by Z Z 2π e2 2π dϕe dϕh |ψe (ϕe )|2|ψh(ϕh )|2V (ϕe , ϕh, z) (30) Veff (z) = −  0 0 with V being the original electron-hole Coulomb potential written in the cylindrical coordinates as 1 . (31) V (ϕe, ϕh , z) = 2 {z + 4R2CN sin2[(ϕe − ϕh )/2]}1/2 The exciton problem is now reduced to the 1D equation (29), where the exciton binding energy does depend on the perpendicular electrostatic field through the electron and hole subband functions ψe,h given by the solutions of Eqs. (27) and (28) and entering the effective electron-hole Coulomb interaction potential (30). The set of Eqs. (27)-(31) is analyzed in Appendix 5. One of the main results obtained in there is that the effective Coulomb potential (30) can be approximated by an attractive cusp-type cutoff potential of the form Veff (z) ≈ −

e2 , [|z| + z0(j, F )]

(32)

where the cutoff parameter z0 depends on the perpendicular electrostatic field strength and on the electron-hole azimuthal transverse quantization index j = 1, 2, ... (excitons are created in interband transitions involving valence and conduction subbands with the same quantization index [67] as shown in Fig. 2). Specifically, z0 (j, F ) ≈ 2RCN

π − 2 ln 2 [1 − ∆j (F )] π + 2 ln 2 [1 − ∆j (F )]

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(33)

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with ∆j (F ) given to the second order approximation in the electric field by e2 R6CN wj2 2 F , ~4 θ(j −2) 1 wj = + , 1 − 2j 1 + 2j

∆j (F ) ≈ 2µMex

(34)

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where θ(x) is the unit step function. Approximation (32) is formally valid when z0 (j, F ) is much less than the exciton Bohr radius a∗B (= ~2 /µe2 ) which is estimated to be ∼ 10RCN for the first (j = 1 in our notations here) exciton in CNs [19]. As is seen from Eqs. (33) and (34), this is always the case for the first exciton for those fields where the perturbation theory applies, i. e. when ∆1(F ) < 1 in Eq. (34). Equation (29) with the potential (32) formally coincides with the one studied by Ogawa and Takagahara in their treatments of excitonic effects in 1D semiconductors with no external electrostatic field applied [89]. The only difference in our case is that our cutoff parameter (33) is field dependent. We therefore follow Ref. [89] and find the ground-state (11) binding energy Eb for the first exciton we are interested in here from the transcendental equation s   q (11) |Eb | 1 2z0(1, F ) (11) 2µ|Eb | + = 0. (35) ln ~ 2 Ry ∗ In doing so, we first find the exciton Rydberg energy, Ry ∗ (= µe4 /2~22 ), from this equation at F = 0. We use the diameter- and chirality-dependent electron and hole effective masses from Ref. [90], and the first bright exciton binding energy of 0.76 eV for both (11,0) and (10,0) CN as reported in Ref. [21] from ab initio calculations. We obtain Ry ∗ = 4.02 eV and 0.57 eV for the (11,0) tube and (10,0) tube, respectively. The difference of about one order of magnitude reflects the fact that these are the semiconducting CNs of different types — type-I and type-II, respectively, based on (2n + m) families [90]. The (11) parameters Ry ∗ thus obtained are then used to find |Eb | as functions of F by numerically solving Eq. (35) with z0 (1, F ) given by Eqs. (33) and (34). The calculated (negative) binding energies are shown in Fig. 5(a) by the solid lines. Also shown there by dashed lines are the functions (11)

Eb

(11)

(F ) ≈ Eb

[1 − ∆1(F )]

(36)

with ∆1 (F ) given by Eq. (34). They are seen to be fairly good analytical (quadratic in field) approximations to the numerical solutions of Eq. (35) in the range of not too large fields. The exciton binding energy decreases very rapidly in its absolute value as the field (11) increases. Fields of only ∼ 0.1 − 0.2 V/µm are required to decrease |Eb | by a factor of ∼ 2 for the CNs considered here. The reason is the perpendicular field shifts up the ”bottom” of the effective potential (32) as shown in Fig. 5(b) for the (11,0) CN. This makes the potential shallower and pushes bound excitonic levels up, thereby decreasing the exciton binding energy in its absolute value. As this takes place, the shape of the potential does not change, and the longitudinal relative electron-hole motion remains finite at all times. As a consequence, no tunnel exciton ionization occurs in the perpendicular field, as opposed to the longitudinal electrostatic field (Franz-Keldysh) effect studied in Ref. [40] where the

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Figure 5: (a) Calculated binding energies of the first bright exciton in the (11,0) and (10,0) CNs as functions of the perpendicular electrostatic field applied. Solid lines are the numerical solutions to Eq. (35), dashed lines are the quadratic approximations as given by Eq. (36). (b) Field dependence of the effective cutoff Coulomb potential (32) in the (11,0) CN. The dimensionless energy is defined as [Energy]/2γ0, according to Eq. (17).

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non-zero field creates the potential barrier separating out the regions of finite and infinite relative motion and the exciton becomes ionized as the electron tunnels to infinity. The binding energy is only the part of the exciton excitation energy (7). Another part comes from the band gap energy (8), where εe and εh are given by the solutions of Eqs. (27) and (28), respectively. Solving them to the leading (second) order perturbation theory approximation in the field (Appendix 5), one obtains Eg(jj)(F )

≈

Eg(jj)

  mh ∆j (F ) me ∆j (F ) − 1− , 2Mex j 2wj 2Mex j 2wj

(37)

where the electron and hole subband shifts are written separately. This, in view of Eq. (34), yields the first band gap field dependence in the form   3 Eg(11)(F ) ≈ Eg(11) 1 − ∆1 (F ) , 2

(38)

The bang gap decrease with the field in Eq. (38) is stronger than the opposite effect in the negative exciton binding energy given (to the same order approximation in field) by Eq. (36). Thus, the first exciton excitation energy (7) will be gradually decreasing as the perpendicular field increases, shifting the exciton absorption peak to the red. This is the basic feature of the quantum confined Stark effect observed previously in semiconductor nanomaterials [84, 85, 86, 87]. The field dependences of the higher interband transitions exciton excitation energies are suppressed by the rapidly (quadratically) increasing azimuthal quantization numbers in the denominators of Eqs. (34) and (37). Lastly, the perpendicular field dependence of the interband plasmon resonances can be obtained from the frequency dependence of the axial surface conductivity due to excitons

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Figure 6: (a),(b) Calculated dependences of the first bright exciton parameters in the (11,0) and (10,0) CNs, respectively, on the electrostatic field applied perpendicular to the nanotube axis. The dimensionless energy is defined as [Energy]/2γ0, according to Eq. (17). The energy is measured from the top of the first unperturbed hole subband. (see Ref. [67] and refs. therein). One has ex (ω) ∼ σzz

X

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(jj) 2 j=1,2,... [Eexc ] −

−i~ωfj (~ω)2 − 2i~2ω/τ

,

(39)

where fj and τ are the exciton oscillator strength and relaxation time, respectively. The ex (ω)] has maxima. Testing plasmon frequencies are those at which the function Re [1/σzz (11) (22) it for maximum in the domain Eexc < ~ω < Eexc , one finds the first interband plasmon resonance energy to be (in the limit τ → ∞) s (11) (22) [Eexc ]2 + [Eexc ]2 (11) . (40) Ep = 2 (11)

Using the field dependent Eexc given by Eqs. (7), (36) and (38), and neglecting the field (22) dependence of Eexc , one obtains to the second order approximation in the field # " (11) (11) /2E 1 +E g exc ∆1(F ) . (41) Ep(11)(F ) ≈ Ep(11) 1 − (22) (11) 1 +Eexc /Eexc Figure 6 shows the results of our calculations of the field dependences for the first bright exciton parameters in the (11,0) and (10,0) CNs. The energy is measured from the top of the first unperturbed hole subband (as shown in Fig. 2, right panel). The binding energy field dependence was calculated numerically from Eq. (35) as described above [shown in Fig. 5 (a)]. The band gap field dependence and the plasmon energy field dependence were calculated from Eqs. (37) and (41), respectively. The zero-field excitation energies and zero-field binding energies were taken to be those reported in Ref. [51] and in Ref. [21], respectively, and we used the diameter- and chirality-dependent electron and hole effective

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masses from Ref. [90]. As is seen in Fig. 6 (a) and (b), the exciton excitation energy and the interband plasmon energy experience red shift in both nanotubes as the field increases. However, the excitation energy red shift is very small (barely seen in the figures) due to (11) the negative field dependent contribution from the exciton binding energy. So, Eexc (F ) (11) and Ep (F ) approach each other as the field increases, thereby bringing the total exciton energy (6) in resonance with the surface plasmon mode due to the non-zero longitudinal kinetic energy term at finite temperature. 2 Thus, the electrostatic field applied perpendicular to the CN axis (the quantum confined Stark effect) may be used to tune the exciton energy to the nearest interband plasmon resonance, to put the exciton-surface plasmon interaction in small-diameter semiconducting CNs to the strong-coupling regime.

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3.3

The optical absorption

Here, we analyze the longitudinal exciton absorption line shape as its energy is tuned to the nearest interband surface plasmon resonance. Only longitudinal excitons (excited by light polarized along the CN axis) couple to the surface plasmon modes as discussed at the very beginning of this section (see Ref. [70] for the perpendicular light exciton absorption in CNs). We start with the linear (weak) excitation regime where only single-exciton states are excited, and follow the optical absorption/emission lineshape theory developed recently for atomically doped CNs [10]. (Obviously, the absorption line shape coincides with the emission line shape if the monochromatic incident light beam is used in the absorption experiment.) Then, the non-linear (strong) excitation regime is considered with the photonduced excitation of biexciton states. When the f -internal state exciton is excited and the nanotube’s surface EM field subsystem is in vacuum state, the time-dependent wave function of the whole system ”exciton+field” is of the form 3 X ˜ |ψ(t)i = Cf (k, t) e−iEf (k)t/~|{1f (k)}iex|{0}i (42) k,f

+

XZ k

∞

dω C(k, ω, t) e−iωt|{0}iex|{1(k, ω)}i. 0

Here, |{1f (k)}iex is the excited single-quantum Fock state with one exciton and |{1(k, ω)}i is that with one surface photon. The vacuum states are |{0}iex and |{0}i for the exciton subsystem and field subsystem, respectively. The coefficients Cf (k, t) and C(k, ω, t) stand for the population probability amplitudes of the respective states of the ˜f (k) = Ef (k)− i~/τ with Ef (k) given whole system. The exciton energy is of the form E by Eq. (6) and τ being the phenomenological exciton relaxation time constant [assumed to be such that ~/τ  Ef (k)] to account for other possible exciton relaxation processes. From the literature we have τph ∼ 30−100 fs for the exciton-phonon scattering [40], τd ∼ 50 ps 2

We are based on the zero-exciton-temperature approximation in here [91], which is well justified because of the exciton excitation energies much larger than kB T in CNs. The exciton Hamiltonian (4) does not require the thermal averaging over the exciton degrees of freedom then, yielding the temperature independent total exciton energy (6). One has to keep in mind, however, that the exciton excitation energy can be affected by the enviromental effect not under consideration in here (see Ref. [92]). 3 See the footnote on page 392 above.

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for the exciton scattering by defects [25, 28], and τrad ∼ 10 ps − 10 ns for the radiative decay of excitons [51]. Thus, the scattering by phonons is the most likely exciton relaxation mechanism. We transform the total Hamiltonian (1)–(10) to the k-representation using Eqs. (5) and (9) (see Appendix 5), and apply it to the wave function in Eq. (42). We obtain the following set of the two simultaneous differential equations for the coefficients Cf (k, t) and C(k, ω, t) from the time dependent Schr¨odinger equation Z i X ∞ (+) ˜f (k)t/~ −iE ˙ =− dω gf (k, k0, ω) C(k0, ω, t) e−iωt, (43) Cf (k, t) e ~ 0 0 k

i C˙ (k0, ω, t) e−iωtδkk0 = − ~

X

(+)

˜

[gf (k, k0, ω)]∗Cf (k, t) e−iEf (k)t/~ .

f

The δ-symbol on the left in Eq. (44) ensures that the momentum conservation is fulfilled in the exciton-photon transitions, so that the annihilating exciton creates the surface photon with the same momentum and vice versa. In terms of the probability amplitudes above, the exciton emission intensity distribution is given by the final state probability at very long times corresponding to the complete decay of all initially excited excitons, 1 X (+) |gf (k, k, ω)|2 ~2 f 2 ∞ ˜f (k)−~ω]t0/~ 0 0 −i[E dt Cf (k, t ) e .

I(ω) = |C(k, ω, t → ∞)|2 =

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Z ×

0

(44)

Here, the second equation is obtained by the formal integration of Eq. (44) over time under the initial condition C (k, ω, 0) = 0. The emission intensity distribution is thus related to the exciton population probability amplitude Cf (k, t) to be found from Eq. (43). The set of simultaneous equations (43) and (44) [and Eq. (44), respectively] contains no approximations except the (commonly used) neglect of many-particle excitations in the wave function (42). We now apply these equations to the exciton-surface-plasmon system in small-diameter semiconducting CNs. The interaction matrix element in Eqs. (43) and (44) is then given by the k-transform of Eq. (13), and has the following property (Appendix 5) 1 ¯f 1 (+) |g (k, k, ω)|2 = Γ (x)ρ(x) 2γ0~ f 2π 0

(45)

¯ f (x) and ρ(x) given by Eqs. (18) and (19), respectively. We further substitute the with Γ 0 result of the formal integration of Eq. (44) [with C (k, ω, 0) = 0] into Eq. (43), use Eq. (45) with ρ(x) approximated by the Lorentzian (22), calculate the integral over frequency analytically, and differentiate the result over time to obtain the following second order ordinary differential equation for the exciton probability amplitude [dimensionless variables, Eq. (17)] C¨ f (β) + [∆xp − ∆εf + i(xp − εf )]C˙ f (β) + (Xf /2)2Cf (β) = 0,

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¯ f (xp )]1/2 with Γ ¯ f (xp) = Γ ¯ f (xp )ρ(xp), ∆εf = ~/2γ0τ , β = where Xf = [2∆xpΓ 0 2γ0t/~ is the dimensionless time, and the k-dependence is omitted for brevity. When the total exciton energy is close to a plasmon resonance, εf ≈ xp , the solution of this equation is easily found to be     q 1 δx − δx− δx2 −Xf2 β/2 e Cf (β) ≈ 1+ q (46) 2 δx2 − Xf2     q 2 2 δx 1  e− δx+ δx −Xf β/2, + 1 − q 2 δx2 − X 2 f

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¯ f (εf )]1/2. This solution is valid when εf ≈ where δx = ∆xp − ∆εf > 0 and Xf = [2∆xp Γ xp regardless of the strength of the exciton-surface-plasmon coupling. It yields the exponen¯ f (εf )β], in the weak coupling tial decay of the excitons into plasmons, |Cf (β)|2 ≈ exp[−Γ 2 regime where the coupling parameter (Xf /δx)  1. If, on the other hand, (Xf /δx)2  1, then the strong coupling regime occurs, and the decay of the excitons into plasmons proceeds via damped Rabi oscillations, |Cf (β)|2 ≈ exp(−δxβ) cos2 (Xf β/2). This is very similar to what was earlier reported for an excited two-level atom near the nanotube surface [58, 59, 60, 9]. Note, however, that here we have the exciton-phonon scattering as well, which facilitates the strong exciton-plasmon coupling by decreasing δx in the coupling parameter. In other words, the phonon scattering broadens the (longitudinal) exciton momentum distribution [93], thus effectively increasing the fraction of the excitons with εf ≈ xp . In view of Eqs. (45) and (46), the exciton emission intensity (44) in the vicinity of the plasmon resonance takes the following (dimensionless) form ¯ I(x) ≈ I¯0 (εf )

2 X Z ∞ dβ Cf (β) ei(x−εf +i∆εf )β , f

(47)

0

¯ f (εf )/2π. After some algebra, this results in ¯ = 2γ0I(ω)/~ and I¯0 = Γ where I(x) ¯ I(x) ≈

I¯0 (εf ) [(x − εf )2 + ∆x2p ] , [(x − εf )2 − Xf2/4]2 + (x − εf )2(∆x2p + ∆ε2f )

(48)

where ∆x2p > ∆ε2f . The summation sign over the exciton internal states is omitted since only one internal state contributes to the emission intensity in the vicinity of the sharp plasmon resonance. The line shape in Eq. (48) is mainly determined by the coupling parameter (Xf /∆xp)2 . It is clearly seen to be of a symmetric two-peak structure in the strong coupling regime where (Xf /∆xp )2  1. Testing it for extremum, we obtain the peak frequencies to be

x1,2

vs u     Xf u ∆xp 2 ∆xp 2 t = εf ± 1+8 −4 2 Xf Xf

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[terms ∼ (∆xp)2(∆εf )2/Xf4 are neglected], with the Rabi splitting x1 − x2 ≈ Xf . In the weak coupling regime where (Xf /∆xp )2  1, the frequencies x1 and x2 become complex, indicating that there are no longer peaks at these frequencies. As this takes place, Eq. (48) is approximated with the weak coupling condition, the fact that x ∼ εf , and Xf2 = ¯ f (εf ), to yield the Lorentzian 2∆xpΓ

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˜ I(x) ≈

I¯0(εf )/[1 + (∆εf /∆xp)2] h i2 q ¯ f (εf )/2 1 + (∆ε /∆xp )2 (x − εf )2 + Γ f

peaked at x = εf , whose half-width-at-half-maximum is slightly narrower, however, than ¯ f (εf )/2 it should be if the exciton-plasmon relaxation were the only relaxation mechaΓ nism in the system. The reason is the competing phonon scattering takes excitons out of resonance with plasmons, thus decreasing the exciton-plasmon relaxation rate. We therefore conclude that the phonon scattering does not affect the exciton emission/absorption line shape when the exciton-plasmon coupling is strong (it facilitates the strong coupling regime to occur, however, as was noticed above), and it narrows the (Lorentzian) emission/absorption line when the exciton-plasmon coupling is weak. The non-linear optical susceptibility is proportional to the linear optical response function under resonant pumping conditions [94]. This allows us to use Eq. (48) to investigate the non-linear excitation regime with the photoinduced biexciton formation as the exciton energy is tuned to the nearest interband plasmon resonance. Under these conditions, the third-order longitudinal CN susceptibility takes the form [32, 94] # " 1 1 ˜ , − χ(3)(x) ≈ I(x) f x − εf + i(Γf/2 + ∆εf ) x − (εf − |εXX f |) + i(Γ /2 + ∆εf ) (49) XX where εf is the (negative) dimensionless binding energy of the biexciton composed of two f -internal state excitons, and χ0 is the frequency-independent constant. The first and second terms in the brackets represent bleaching due to the depopulation of the ground state and photoinduced absorption due to exciton-to-biexciton transitions, respectively. The binding energy of the biexciton in a small-diameter ( ∼ 1 nm) CN can be evaluated by the method pioneered by Landau [95], Gor’kov and Pitaevski [96], Holstein and Herring [97] — from the analysis of the asymptotic exchange coupling by perturbation on the configuration space wave function of the two ground-state one-dimensional (1D) excitons. Separating out circumferential and longitudinal degrees of freedom of each of the excitons by means of Eq. (26), one arrives at the biexciton Hamiltonian of the form [see Fig. 7 (a)]

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Figure 7: (a) Schematic (arbitrary units) of the exchange coupling of two ground-state 1D excitons to form a biexcitonic state. (b) The coupling occurs in the configuration space of the two independent longitudinal relative electron-hole motion coordinates, z1 and z2 , of each of the excitons, due to the tunneling of the system through the potential barriers formed by the two single-exciton cusp-type potentials [bottom, also in (a)], between equivalent states represented by the isolated two-exciton wave functions shown on the top.

ˆ 1 , z2, ∆Z) = − 1 H(z 2 −



∂2 ∂2 + ∂z1 ∂z2



(50)

  1 1 1 1 1 + + + 2 |z1 | + z0 |z2 + ∆Z| + z0 |z2 | + z0 |z1 − ∆Z| + z0 1 1 − − |(z1 + z2 )/2 + ∆Z| + z0 |(z1 + z2 )/2 − ∆Z| + z0 1 1 + + . |(z1 − z2 )/2 + ∆Z| + z0 |(z1 − z2 )/2 − ∆Z| + z0

Here, z1,2 = ze1,2 − zh1,2 is the electron-hole relative motion coordinates of the two 1D excitons, z0 is the cut-off parameter of the effective longitudinal electron-hole Coulomb potential (32), and ∆Z = Z2 − Z1 is the center-of-mass-to-center-of-mass inter-exciton separation distance. Equal electron and hole effective masses me,h are assumed [90] and ”atomic units” are used [95, 96, 97], whereby distance and energy are measured in units of the exciton Bohr radius a∗B and in units of the doubled ground-state-exciton binding energy

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2Eb = −2Ry ∗ /ν02 , respectively. The first two lines in Eq. (50) represent two isolated noninteracting 1D excitons [see Fig. 7 (a)]. The last two lines are their exchange Coulomb interactions — electron-hole and electron-electron + hole-hole, respectively. The Hamiltonian (50) is effectively two dimensional in the configuration space of the two independent relative motion coordinates, z1 and z2 . Figure 7 (b), bottom, shows schematically the potential energy surface of the two closely spaced non-interacting 1D excitons [line two in Eq. (50)] in the (z1 , z2) space. The surface has four symmetrical minima [representing equivalent isolated two-exciton states shown in Fig. 7 (b), top], separated by the potential barriers responsible for the tunnel exchange coupling between the two-exciton √ states in the configuration space. The coordinate transformation x = (z1 − z2 − ∆Z)/ 2, √ y = (z1 + z2)/ 2 places the origin of the new coordinate system into the intersection of the two tunnel channels between the respective potential minima [Fig. 7 (b)], whereby the exchange splitting formula of Refs. [95, 96, 97] takes the form Ug,u (∆Z) − 2Eb = ∓J(∆Z),

(51)

where Ug,u are the ground and excited state energies, respectively, of the two coupled excitons (the biexciton) as functions of their center-of-mass-to-center-of-mass separation, and

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2 J(∆Z) = 3!

  ∂ψ(x, y) √ dy ψ(x, y) ∂x −∆Z/ 2 x=0

Z

√ ∆Z/ 2

(52)

is the tunnel exchange coupling integral, where ψ(x, y) is the solution to the Schr¨odinger equation with the Hamiltonian (50) transformed to the (x, y) coordinates. The factor 2/3! comes from the fact that there are two equivalent tunnel channels in the problem, mixing three equivalent indistinguishable two-exciton states in the configuration space [one state is given by the two minima on the y-axis, and two more are represented by each of the minima on the x-axis — compare Fig. 7 (a) and (b)]. The function ψ(x, y) in Eq. (52) is sought in the form ψ(x, y) = ψ0 (x, y) exp[−S(x, y)] ,

(53)

where ψ0 = ν0−1 exp[−(|z1(x, y, ∆Z)| + |z2(x, y, ∆Z)|)/ν0] is the product of two singleexciton wave functions 4 representing the isolated two-exciton state centered at the minimum √ z1 = z2 = 0 (or x = −∆Z/ 2, y = 0) of the configuration space potential [Fig. 7 (b)], and S(x, y) is a slowly varying function to take into account the deviation of ψ from ψ0 due to the tunnel exchange coupling to another equivalent isolated two-exciton state centered at √ z1 = ∆Z, z2 = −∆Z (or x = ∆Z/ 2, y = 0). Substituting Eq. (53) into the Schr¨odinger equation with the Hamiltonian (50) pre-transformed to the (x, y) coordinates, one obtains in the region of interest   1 1 1 1 ∂S √ − √ + √ √ = ν0 − , ∂x x + 3∆Z/ 2 x − ∆Z/ 2 y − 2∆Z y + 2∆Z 4

This is an approximate solution to the Shr¨odinger equation with the Hamiltonin given by the first two lines in Eq. (50), where the cut-off parameter z0 is neglected [89]. This approximation greatly simplifies problem solving here, while still remaining adequate as only the long-distance tail of ψ0 is important for the tunnel exchange coupling. Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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up to (negligible) terms of the order of the inter-exciton van der Waals energy and up to second derivatives of S. This equation is to be solved with the boundary √ √ condition S(−∆Z/ 2, y) = 0 originating from the natural requirement ψ(−∆Z/ 2, y) = √ ψ0(−∆Z/ 2, y), to result in ! x+3∆Z/√2 2√2∆Z(x+∆Z/√2) √ + S(x, y) = ν0 ln . (54) x − ∆Z/ 2 y 2 − 2∆Z 2 After plugging Eqs. (54) and (53) into Eq. (52), and retaining only the leading term of the integral series expansion in powers of ν0 subject to ∆Z > 1, Eq. (51) becomes   2  e 2ν0 ∆Z e−2∆Z/ν0 . Ug,u (∆Z) ≈ 2Eb 1 ± 2 (55) 3ν0 3

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The ground state energy Ug of two coupled 1D excitons is now seen to go through the negative minimum (biexcitonic state) as the inter-exciton center-of-mass-to-center-of-mass separation ∆Z increases (Fig. 8). The minimum occurs at ∆Z0 = ν0 /2, whereby the biexciton binding energy is EXX ≈ (2Eb/9ν0)(e/3)2ν0−1 , or, expressing ν0 in terms of Eb and measuring the energy in units of Ry ∗,  2/√|Eb| − 1 2 ∗ 3/2 e EXX [in Ry ] ≈ − |Eb | . (56) 9 3 The energy EXX can be affected by the quantum confined Stark effect since |Eb| decreases quadratically with the perpendicular electrostatic field applied as shown in Fig. 5 (a). Since e/3 ∼ 1, the field dependence in Eq. (56) mainly comes from the pre-exponential factor. So, |EXX | will be decreasing quadratically with the field as well, for not too strong perpendicular fields. At the same time, the equilibrium inter-exciton separation in the biexciton, ∆Z0 = ν0 /2 ∼ |Eb|−1/2 , will be slowly increasing with the field consistently with the lowering of |EXX |. In the zero field, one has roughly EXX ∼ |Eb|3/2 ∼ R−0.9 CN for the biexciton binding energy versus the CN radius RCN (|Eb| ∼ R−0.6 as reported in Ref. [19] CN −1 from variational calculations), pretty consistent with the RCN dependence obtained numerically [32]. Interestingly, as RCN goes down, |EXX | goes up faster than |Eb | does. This is partly due to the fact that ∆Z0 slowly decreases as RCN goes down, — a theoretical argument in support of experimental evidence for increased exciton-exciton annihilation in small diameter CNs [98, 99, 100]. Figure 8 shows the ground state energy Ug (∆Z) of the coupled pair of the first bright excitons, calculated from Eq. (55) for the semiconducting (11,0) CN exposed to different perpendicular electrostatic fields. The inset shows the field dependences of EXX [as given by Eq. (56)] and of ∆Z0. All the curves are calculated using the field dependence of Eb obtained as described in the previous subsection (Figs. 5 and 6). They exhibit typical behaviors discussed above. Figure 9 compares the linear response lineshape (48) with the imaginary part of Eq. (49) representing the non-linear optical response function under resonant pumping, both calculated for the 1st bright exciton in the (11,0) CN as its energy is tuned (by means of the quantum confined Stark effect) to the nearest plasmon resonance (vertical dashed line in the figure). The biexciton binding energy in Eq. (49) was taken to be EXX ≈ 52 meV

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Figure 8: Calculated ground state energy Ug of the coupled pair of the first bright excitons in the (11,0) CN as a function of the center-of-mass-to-center-of-mass inter-exciton distance ∆Z and perpendicular electrostatic field applied. Inset shows the biexciton binding energy EXX and inter-exciton separation ∆Z0 (y- and x-coordinates, respectively, of the minima in the main figure) as functions of the field.

as given by Eq. (56) in the zero field. [Weak field dependence of EXX (inset in Fig. 8) plays no essential role here as |EXX |  |Eb | ≈ 0.76 eV regardless of the field strength.] The phonon relaxation time τph = 30 fs was used as reported in Ref. [29], since this is the shortest one out of possible exciton relaxation processes, including exciton-exciton annihilation (τee ∼ 1 ps [98]). Clear line (Rabi) splitting effect ∼ 0.1 eV is seen both in the linear and in non-linear excitation regime, indicating the strong exciton-plasmon coupling both in the single-exciton states and in the biexciton states as the exciton energy is tuned to the interband surface plasmon resonance. The splitting is not masked by the exciton-phonon scattering. This effect can be used for the development of new tunable optoelectronic device applications of optically excited small-diameter semiconducting CNs in areas such as nanophotonics, nanoplasmonics, and cavity quantum electrodynamics, including the strong excitation regime with optical non-linearities. In the latter case, the experimental observation of the non-linear absorption line splitting predicted here would help identify the presence and study the properties of biexcitonic states (including biexcitons formed by excitons of different subbands [33]) in individual single-walled CNs, due to the fact that when tuned close to a plasmon resonance the exciton relaxes into plasmons at a rate much greater than −1 −1 (  τee ), totally ruling out the role of the competing exciton-exciton annihilation τph process.

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Figure 9: [(a), (b), and (c)] Linear (top) and non-linear (bottom) response functions as given by Eq. (48) and by the imaginary part of Eq. (49), respectively, for the first bright exciton in the (11,0) CN as the exciton energy is tuned to the nearest interband plasmon resonance (vertical dashed line). Vertical lines marked as X and XX show the exciton energy and biexciton binding energy, respectively. The dimensionless energy is defined as [Energy]/2γ0, according to Eq. (17).

4

Casimir Interaction in Double-Wall Carbon Nanotubes

Here, we consider the Casimir interaction between two concentric cylindrical graphene sheets comprising a double-wall CN, using the macroscopic QED approach employed above to study the exciton-surface-plasmon interactions in single wall nanotubes. 5 The method is fully adequate in this case as the Casimir force is known to originate from quantum EM field fluctuations. The fundamental nature of this force has been studied for many years since the prediction of the attraction force between two neutral metallic plates in vacuum (see, Refs. [49, 56]). After the first report of observation of this spectacular effect [101], new measurements with improved accuracy have been done involving different geometries [102, 103, 104]. The Casimir force has also been considered theoretically with methods primarily based on the zero-point summation approach and Lifshitz theory [105, 106]. The Casimir effect has acquired a much broader impact recently due to its importance for nanostructured materials, including graphite and graphitic nanostructures [56] which 5

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can exist in different geometries and with various unique electronic properties. Moreover, the efficient development and operation of modern micro- and nano-electromechanical devices are limited due to effects such as stiction, friction, and adhesion, originating from or closely related to the Casimir effect [57]. The mechanisms governing the CN interactions still remain elusive. It is known that the system geometry [107, 108] and dielectric response [45, 62] have a profound effect on the interaction, in general, but their specific functionalities have not been qualitatively and quantitatively understood. Since CNs of virtually the same radial size can possess different electronic properties, investigating their Casimir interactions presents a unique opportunity to obtain insight into specific dielectric response features affecting the Casimir force between metallic and semiconducting cylindrical surfaces. This can also unveil the role of collective surface excitations in the energetic stability of multi-wall CNs of various chiral combinations. Since Lifshitz theory cannot be easily applied to geometries other than parallel plates, researchers have used the Proximity Force Approximation (PFA) to calculate the Casimir interaction between CNs [107, 109] (see also Ref. [56] for the latest review). The method is based on approximating the curved surfaces at very close distances by a series of parallel plates and summing their energies using the Lifshitz result. Thus, the PFA is inherently an additive approach, applicable to objects at very close separations (still to be greater than objects inter-atomic distances) under the assumption that the CN dielectric response is the same as the one for the plates. This last assumption is very questionable as the quasi-1D character of the electronic motion in CNTs is known to be of principal importance for the correct description of their electronic and optical properties [1, 59, 64]. We model the double-wall CN by two infinitely long, infinitely thin, continuous concentric cylinders with radii R1,2, immersed in vacuum. Each cylinder is characterized by the complex dynamic axial dielectric function zz (R1,2, ω) with the z-direction along the CN axis as shown in Fig. 10. The azimuthal and radial components of the complete CN dielectric tensor are neglected as they are known to be much less than zz for most CNs [64]. The QED quantization scheme in the presence of CNs [49, 62] generates the second-quantized Hamiltonian Z XZ ∞ ˆ i , ω) ˆ dω~ω dRi fˆ† (Ri , ω)f(R H= i=1,2 0

of the vacuum-type medium assisted EM field, with the bosonic operators fˆ† and fˆ creating and annihilating, respectively, surface EM excitations of frequency ω at points R1,2 = {R1,2, ϕ1,2, z1,2} of the double-wall CN system. The Fourier-domain electric field operator at an arbitrary point r = (r, ϕ, z) is given by XZ ˆ i , ω), ˆ ω) = iωµ0 dRi G(r, Ri, ω) · J(R E(r, i=1,2

where G(r, Ri, ω) is the dyadic EM field Green’s function (GF), and ˆ i , ω) = ω J(R µ0 c2

s

~ Im zz (Ri, ω) ˆ f (Ri , ω)ez πε0

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Figure 10: Schematic of the two concentric CNs in vacuum. The CN radii are R1 and R2 . The regions between the CN surfaces are denoted as (1), (2), and (3). is the surface current density operator selected in such a way as to ensure the correct QED equal-time commutation relations for the electric and magnetic field operators [49, 62]. Here, ez is the unit vector along the CN axis, ε0, µ0 , and c are the dielectric constant, magnetic permeability, and vacuum speed of light, respectively. The dyadic GF satisfies the wave equation ∇ × ∇ × G(r, r0, ω) −

ω2 G(r, r0, ω) = δ(r − r0) I c2

(57)

with I being the unit tensor. The GF can further be decomposed as follows (s,f )

G(s,f ) = G(0)δsf + Gscatt

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(s,f )

where G(0) and Gscatt represent the contributions of the direct and scattered waves, respectively [110, 111], with a point-like field source located in region s and the field registered in region f (see Fig. 10). The boundary conditions for Eq. (57) are obtained from those for the electric and magnetic field components on the CN surfaces [45, 59], which result in h i (58) er × G(r, r0, ω) R+ − G(r, r0, ω) R− = 0, 1,2

1,2

h i er ×∇× G(r, r0, ω) R+ − G(r, r0, ω) R− = iωµ0σ (1,2)(r, ω)· G(r, r0, ω) R 1,2

1,2

1,2

(59)

where er is the unit vector along the radial direction. The discontinuity in Eq. (59) results from the full account of the finite absorption and dispersion for both CNs by means of their conductivity tensors σ (1,2) approximated by their largest components (1,2) (R1,2, ω) = − σzz

iωε0 (1,2) [ (R1,2, ω) − 1] SρT zz

(60)

[compare with Eq. (21)]. (s,f ) Following the procedure described in Refs. [110, 111], we expand G(0) and Gscatt into series of even and odd vector cylindrical functions with unknown coefficients to be found from Eqs. (58) and (59). This splits the EM modes in the system into TE and TM polarizations, with Eqs. (58) and (59) yielding a set of 32 equations (16 for each polarization) with

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32 unknown coefficients. The unknown coefficients are found determining the dyadic GF in each region.6 Using the expressions for the electric and magnetic fields, the electromagnetic stress tensor is constructed [49, 112] T(r, r0) = T1 (r, r0) + T2 (r, r0) −

  1 I T r T1 (r, r0) + T2 (r, r0) 2

Z  ~ ∞ ω2  T1 (r, r ) = dω 2 Im G(r, r0, ω) π 0 c   Z ∞ ← ~ 0 0 0 dω Im ∇ × G(r, r , ω)× ∇ T2 (r, r ) = − π 0 0

(61) (62) (63)

We are interested in the radial component Trr which describes the radiation pressure of the virtual EM field on each CN surface in the system. The Casimir force per unit area exerted on the surfaces is then given by [49]  h i (i) 0 (i+1) 0 Trr (r, r ) − Trr (r, r ) , i = 1, 2 (64) Fi = lim lim 0

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r→Ri

r →r

The forces F1,2 calculated from Eq. (64) are of equal magnitude and opposite direction, indicating the attraction between the cylindrical surfaces. The Casimir force thus obtained accounts simultaneously for the geometrical curvature effects (through the GF tensor) and the finite absorption and dissipation of each CN [through their dielectric response functions (60)]. The dielectric response functions of particular CNs were calculated from the CN realistic band structure as described above, in Section 3. We decomposed them into the Drude contribution and the contribution originating from (transversely quantized) interband inter electronic transitions, zz = D zz +zz , in order to be able to see how much each individual contribution affects the inter-tube Casimir attraction. It is interesting to consider the case of infinitely conducting parallel plates first using (1,2) Eq. (64). This is obtaned by taking the limits σzz → ∞ and R1,2 → ∞ while keeping constant the inter-tube distance, R1 − R2 = d. We find ~c F =− 16π 2R41 ×



Z

∞

dx1 x1 0

∞ X n=0

(2 − δn0 ) In (x1 )Kn (x2) − In (x2)Kn (x1)

 
x21 Kn0 2 (x1) + n2 + x21 Kn2 (x1) In2(x1)Kn (x2 )/Kn(x1) − 2In (x1)In (x2 )
  − x21 In0 2(x1 ) + n2 + x21 In2(x1) Kn (x1)Kn (x2)
  − 2 x21 In0 (x1 )Kn0 (x1) + n2 + x21 In (x1)Kn (x1 ) In (x2 )Kn(x1 )

where x1,2 = xR1,2, In (x) and Kn (x) are the modified Bessel functions of the first and second kind, respectively. The above expression is obtained by making the transition to imaginary frequencies ω → iω, and using the Euclidean rotation technique as described in Refs. [112, 113]. This can further be evaluated by summing up the series over n using 6

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Figure 11: The Casimir force per unit area as a function of the inter-tube separation d, for different pairs of CNs. The inset shows force found with the full dielectric function and the Drude contribution only for the same CN pairs indicated in the figure. the large-order Bessel function expansions [114]. This results in F ∼ (−1/3)(~cπ 2/240d4) which is 1/3 of the well-known result for two parallel plates [49, 56]. This deviation originates from zz 6= 0 only and the remaining dielectric tensor components being zero in our model. Figure 11 presents results from the numerical calculations of F as a function of the inter-tube surface-to-surface distance for various pairs of CNs with their realistic chirality dependent dielectric responses taken into account. We have chosen the inner CN to be the achiral (12, 12) metallic nanotube, and to change the outer tubes. As R2 is varied, one can envision double wall CNs consisting of metal/metal or metal/semiconductor combinations of different chiralities but of similar radial dimensions. Figure 11 shows that F decreases in strength as the surface-to-surface distance increases. This dependence is monotonic for the zigzag (m, 0) and armchair (n, n) outer tubes, but it happens at different rates. The attraction is stronger if the outer CN is an armchair (n, n) one as compared to the attraction for the outer (m, 0) nanotubes. At the same time, for chiral tubes the Casimir force decreases as a function of d in a rather irregular fashion. It is seen that for relatively small d, the interaction force can be quite different. For example, the attraction between (27, 4)@(12, 12) and (21, 13)@(12, 12) differ by ∼ 20 % ˚ The differences in favor of the second pair, even though the radial difference is only 0.2 A. between the different CNs become smaller as their separation becomes larger, and they eventually become negligible as the Casimir force diminishes at large distances.

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Figure 12: The Casimir force per unit area as a function of the inter-tube separation d for selected CN pairs. The insets show the EELS spectra for several CNs. We also calculate the Casimir force using the D zz (ω) contribution alone in each dielectric function. The inset in Fig. 11 indicates that the attraction is stronger when the interband transitions are neglected. The decay of F as a function of d is monotonic. Including the inter zz (ω) term not only reduces the force, but also introduces non-linearities due to the chirality dependent optical excitations. At large surface-to-surface separations, the discrepancies between the force calculated with the full dielectric response, and those obtained ˚ this difference with the Drude term only become less significant. We find that for d ∼ 15 A, is less than 10 %. To investigate further the important functionalities originating from the cylindrical geometry and the CN dielectric response properties, F is calculated for different achiral inner/outer nanotube pairs. Studying zigzag and armchair CNs allows tracking generalities from (ω) in a more controlled manner. The results are presented in Fig. 12. We have chosen representatives of three inner CN types – metallic (12, 12), semi-metallic (21, 0), ˚ 8.22 A, ˚ and 7.83 A, ˚ and semiconducting (20, 0) tubules. They are of similar radii, 8.14 A, respectively. We see that depending on the outer nanotube types, the F versus d curves are positioned in three groups. The weakest interaction is found when there are two zigzag concentric CNs (top two curves). The fact that some of these are semi-metallic and others are semiconducting does not seem to influence the magnitude and monotonic decrease of the Casimir force. The attraction is stronger when there is a combination of an armchair and a zigzag CNT as compared to the previous case. The curves for (m, 0)@(12, 12), (n, n)@(21, 0), and

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(n, n)@(20, 0) are practically overlapping, meaning that the specific location of the zigzag and armchair tubes (inner or outer) is of no significance to the force. The small deviations can be attributed to the small differences in the inner CN radii. Finally, we see that the strongest interaction occurs between two armchair CNs (red curve). These functionalities are not unique just for the considered CNs. We have performed the same calculations for many different achiral tubes, and we always find that the strongest interaction occurs between two armchair CNs and the weakest — between two zigzag CNs (provided that their radial dimensions are similar).

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The results from these calculations are strongly suggestive that the CN collective excitation properties have a strong effect on their mutual interaction. This is particularly true for the relatively small distances of interest here, for which the dominant contribution of plasmonic modes to the Casimir interactions has been realized for planar [115] and linear [117] metallic systems. To elucidate this issue here, we calculate the EELS spectra, given by −Im[1/(ω)], and compare them for various inner and outer CNs combinations — Fig. 12 (inset). Considering F as a function of d and the specific form of the EELS spectra, it becomes clear from the inset in Fig. 12 that the low frequency plasmon excitations, given by peaks in −Im[1/(ω)], are key to the strength of the Casimir force. We always find that the strongest force is between the tubules with well pronounced overlapping low frequency plasmon excitations. This is consistent with the conclusion of Ref. [117] for generic 1Dplasmonic structures. However, in our case we deal with the interband plasmons originating from the space quantization of the transverse electronic motion, and, therefore, having quite a different frequency-momentum dispersion law (constant) as compared to that normally assumed (linear) for plasmons [50]. A weaker force is obtained if only one of the CNs supports strong low frequency interband plasmon modes. The weakest interaction happens when neither CN has strong low frequency plasmons. For the cases shown in Fig 12, one finds well pronounced overlapping plasmon transitions in the (12, 12) CN at ω1 = 2.18 eV and ω2 = 3.27 eV, and at ω1 = 1.63 eV and ω2 = 2.45 eV in the (17, 17) CN. At the same time, no such well defined strong low frequency excitations in the (21, 0) and (30, 0) CNs are found. Figure 12 shows that the attraction in (17, 17)@(12, 12) is much stronger than the attraction in (30, 0)@(21, 0), even though the radial sizes of the involved CNs are approximately the same. One also notes that for the case of (17, 17)@(21, 0) there is only one such low frequency excitation coming from the armchair tube and, consequently, the Casimir force has an intermediate value as compared to the above discussed two cases. We performed calculations of the Casimir force between many CN pairs and made comparisons between the relevant regions of the EELS spectra. It is found that, in general, armchair tubes always have strong, well pronounced interband plasmon excitations in the low frequency range. Zigzag and most chiral CNs have low frequency interband plasmons [37], too, but they are not as near as well pronounced as those in armchair tubes; their stronger plasmon modes are found at higher frequencies. These studies are indicative of the significance of the collective response properties of the involved CNs. Specifically, the collective low energy plasmon excitations and their relative location can result in nanotube attraction with different strengths. We further investigate ˚ ˚ The R2 = 8.22 A. this point by considering a double wall CN with radii R1 = 11.63 Aand

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Figure 13: The Casimir force per unit area as a function of the outer CN plasmon frequency, while the inner CN plasmon peak ω2 is constant. Results are shown for four values of ω2 . The dielectric functions are modeled by a generic Lorentzian as given by Eq. (65).

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dielectric function of each tube is taken to be of the generic Lorentzian form zz (R1,2, ω) = 1 −

Ω2 2 + iωΓ ω 2 − ω1,2

(65)

with the typical for nanotubes values Ω = 2.7 eV and Γ = 0.03 eV [45]. Then, the EELS spectrum has only one plasmon resonance at ω1,2 for each tube. This generic form allows us to change the relative position and strength of the plasmon peaks and uncover more characteristic features originating from the EELS spectra. In Fig. 13, the force as a function of plasmon frequency resonances of the outer CN is shown when the plasmon transition for the inner CNT is kept constant (four values are chosen for ω2 ). One sees that the local minima in F versus ω occur when ω1 and ω2 coincide. In fact, the strongest attraction happens when both CNs have the lowest plasmon excitations at the same frequency ω1 = ω2 = 0.81 eV. It is evident that the existence of relatively strong low frequency EELS spectrum and an overlap between the relevant plasmon peaks of the two structures is necessary to achieve a strong interaction. This study clearly demonstrates the crucial importance of the collective low energy surface plasmon excitations at relatively close surface-to-surface separations along with the cylindrical circular geometry of the double-wall CN system. The QED approach we used provides the unique opportunity to investigate these features together, or separately, and to uncover underlying mechanisms of the energetic stability of different double-wall

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CN combinations. An additional advantage here is that we can calculate the dielectric function explicitly for each chirality. Thus, we can determine unambiguously how the semiconducting or metallic nature of each CN contributes to their mutual interaction.

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5

Conclusion

We have shown that the strong exciton-surface-plasmon coupling effect with characteristic exciton absorption line (Rabi) splitting ∼ 0.1 eV exists in small-diameter (. 1 nm) semiconducting CNs. The splitting is almost as large as the typical exciton binding energies in such CNs (∼ 0.3 − 0.8 eV [19, 20, 24, 21]), and of the same order of magnitude as the exciton-plasmon Rabi splitting in organic semiconductors ( ∼ 180 meV [53]). It is much larger than the exciton-polariton Rabi splitting in semiconductor microcavities ( ∼ 140−400 µeV [81, 82, 83]), or the exciton-plasmon Rabi splitting in hybrid semiconductormetal nanoparticle molecules [54]. Since the formation of the strongly coupled mixed exciton-plasmon excitations is only possible if the exciton total energy is in resonance with the energy of an interband surface plasmon mode, we have analyzed possible ways to tune the exciton energy to the nearest surface plasmon resonance. Specifically, the exciton energy may be tuned to the nearest plasmon resonance in ways used for the excitons in semiconductor quantum microcavities — thermally (by elevating sample temperature) [81, 82, 83], and/or electrostatically [84, 85, 86, 87] (via the quantum confined Stark effect with an external electrostatic field applied perpendicular to the CN axis). The two possibilities influence the different degrees of freedom of the quasi-1D exciton — the (longitudinal) kinetic energy and the excitation energy, respectively. We have studied how the perpendicular electrostatic field affects the exciton excitation energy and interband plasmon resonance energy (the quantum confined Stark effect). Both of them are shown to shift to the red due to the decrease in the CN band gap as the field increases. However, the exciton red shift is much less than the plasmon one because of the decrease in the absolute value of the negative binding energy, which contributes largely to the exciton excitation energy. The exciton excitation energy and interband plasmon energy approach as the field increases, thereby bringing the total exciton energy in resonance with the plasmon mode due to the non-zero longitudinal kinetic energy term at finite temperature. The noteworthy point is that the strong exciton-surface-plasmon coupling we predict here occurs in an individual CN as opposed to various artificially fabricated hybrid plasmonic nanostructures mentioned above. We strongly believe this phenomenon, along with its tunability feature via the quantum confined Stark effect we have demonstrated, opens up new paths for the development of CN based tunable optoelectronic device applications in areas such as nanophotonics, nanoplasmonics, and cavity QED. One straightforward application like this is the CN photoluminescence control by means of the exciton-plasmon coupling tuned electrostatically via the quantum confined Stark effect. This complements the microcavity controlled CN infrared emitter application reported recently[27], offering the advantage of less stringent fabrication requirements at the same time since the planar photonic microcavity is no longer required. Electrostatically controlled coupling of two spatially separated (weakly localized) excitons to the same nanotube’s plasmon resonance would result in their entanglement [11, 12, 13], the phenomenon that paves the way for CN

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based solid-state quantum information applications. Moreover, CNs combine advantages such as electrical conductivity, chemical stability, and high surface area that make them excellent potential candidates for a variety of more practical applications, including efficient solar energy conversion [7], energy storage [14], and optical nanobiosensorics [42]. However, the photoluminescence quantum yield of individual CNs is relatively low, and this hinders their uses in the aforementioned applications. CN bundles and films are proposed to be used to surpass the poor performance of individual tubes. The theory of the excitonplasmon coupling we have developed here, being extended to include the inter-tube interaction, complements currently available ’weak-coupling’ theories of the exciton-plasmon interactions in low-dimensional nanostructures [54, 121] with the very important case of the strong coupling regime. Such an extended theory (subject of our future publication) will lay the foundation for understanding inter-tube energy transfer mechanisms that affect the efficiency of optoelectronic devices made of CN bundles and films, as well as it will shed more light on the recent photoluminescence experiments with CN bundles [43, 44] and multi-walled CNs [122], revealing their potentialities for the development of high-yield, high-performance optoelectronics applications with CNs. In addition, we have first applied the macroscopic QED approach suitable for dispersing and absorbing media to study the Casimir interaction in a double-wall carbon nanotube systems with the realistic dielectric response taken into account. We found that at distances ˚ the attraction similar to the equilibrium separations between graphitic surfaces ( ∼ 3 A), is dominated by the low energy (interband) plasmon excitations of both CNs. The key attributes of the EELS spectra are the existence of low frequency plasmons, their strong and well pronounced nature, and the overlap between the low frequency plasmon peaks belonging to the two CNTs. Thus, the chiralities of concentric graphene sheets with similar radial sizes exhibiting these features will be responsible for forming the most preferred CN pairs. As the inter-tube separation increases, the plasmon effect diminishes and the collective excitations originating from the nanotube metallic or semiconducting nature do not influence the interaction in a profound way. We expect our results to pave the way for the development of new generation of tunable optoelectronic and nano-electromechanical device applications with single-wall and multiwall carbon nanotubes. Acknowledgments I.V.B. is supported by the US National Science Foundation, Army Research Office and NASA (grants ECCS-1045661 & HRD-0833184, W911NF-10-1-0105, and NNX09AV07A). L.M.W. and A.P. are supported by the US Department of Energy contract DE-FG02-06ER46297. Helpful discussions with Mikhail Braun (St.-Peterburg U., Russia), Jonathan Finley (WSI, TU Munich, Germany), and Alexander Govorov (Ohio U., USA) are gratefully acknowledged.

Appendix A: Exciton interaction with the surface EM field We follow our recently developed QED formalism to describe vacuum-type EM effects in the presence of quasi-1D absorbing and dispersive bodies [58, 59, 60, 61, 62, 9]. The treat-

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ment begins with the most general EM interaction of the surface charge fluctuations with the quantized surface EM field of a single-walled CN. No external field is assumed to be applied. The CN is modelled by a neutral, infinitely long, infinitely thin, anisotropically conducting cylinder. Only the axial conductivity of the CN, σzz , is taken into account, whereas the azimuthal one, σϕϕ , is neglected being strongly suppressed by the transverse depolarization effect [63, 64, 65, 66, 67, 68]. Since the problem has the cylindrical symmetry, the orthonormal cylindrical basis {er , eϕ , ez } is used with the vector ez directed along the nanotube axis as shown in Fig. 1. The interaction has the following form (Gaussian system of units) ˆ int = H ˆ (1) + H ˆ (2) H int int h i X X qi qi ˆ (i) (i) (i) ˆ + ˆr(i) ˆ ˆ A(n )· p − A(n + r ) + qi ϕ(n ˆ +ˆ rn ), =− n n n mi c 2c n,i

n,i

(i)

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(66)

(i)

ˆ n are, respectively, the masses, charges, where c is the speed of light, mi , qi , ˆrn , and p coordinate operators and momenta operators of the particles (electrons and nucleus) residing at the lattice site n = Rn = {RCN , ϕn , zn } associated with a carbon atom (see Fig. 1) on the surface of the CN of radius RCN . The summation is taken over the lattice sites, and may be substituted with the integration over the CN surface using Eq. (3). The vector poˆ and the scalar potential operator ϕ tential operator A ˆ represent the nanotube’s transversely polarized and longitudinally polarized surface EM modes, respectively. They are written in the Schr¨odinger picture as follows Z ∞ c ˆ⊥ ˆ (n, ω) + h.c., (67) A(n) = dω E iω 0 Z ∞ ˆ k (n, ω) + h.c.. ˆ = dω E (68) −∇n ϕ(n) 0 (i)

(i)

ˆ ˆ + rˆn )] = 0. We use the Coulomb gauge whereby ∇n · A(n) = 0, or [ˆ pn , A(n The total electric field operator of the CN-modified EM field is given for an arbitrary r in the Schr¨odinger picture by Z ∞ Z ∞ ˆ ˆ ⊥ (r, ω) + E ˆ k (r, ω)] + h.c. ˆ (69) E(r) = dω E(r, ω) + h.c. = dω [E 0

0

with the transversely (longitudinally) polarized Fourier-domain field components defined as Z ⊥(k) ˆ 0, ω), ˆ (r, ω) = dr0 δ ⊥(k)(r − r0) · E(r (70) E where k

δαβ (r) = −∇α ∇β

1 , 4πr k

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416

I. V. Bondarev, L. M. Woods and A.Popescu

are the longitudinal and transverse dyadic δ-functions, respectively. The total field operator (69) satisfies the set of the Fourier-domain Maxwell equations ˆ ˆ ω) = ik H(r, ω), ∇ × E(r,

(72)

ˆ ˆ ω) + 4π ˆI(r, ω), ∇ × H(r, ω) = −ik E(r, c

(73)

ˆ = (ik)−1∇ × E ˆ is the magnetic field operator, k = ω/c, and where H ˆI(r, ω) =

X

ˆ δ(r − n) J(n, ω),

(74)

n

is the exterior current operator with the current density defined as follows r ~ω Reσzz (RCN , ω) ˆ ˆ f (n, ω)ez J(n, ω) = π

(75)

to ensure preservation of the fundamental QED equal-time commutation relations (see, e.g., [47]) for the EM field components in the presence of a CN. Here, σzz is the CN surface axial conductivity per unit length, and fˆ(n, ω) along with its counter-part fˆ† (n, ω) are the scalar bosonic field operators which annihilate and create, respectively, single-quantum EM field excitations of frequency ω at the lattice site n of the CN surface. They satisfy the standard bosonic commutation relations ˆ ω), fˆ†(m, ω 0)] = δnm δ(ω − ω 0 ), [f(n,

(76)

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ˆ [fˆ(n, ω), f(m, ω 0)] = [fˆ† (n, ω), fˆ†(m, ω 0)] = 0. One further obtains from Eqs. (72)–(75) that ˆ ω) = ik 4π E(r, c

X

ˆ ω), G(r, n, ω)· J(n,

(77)

n

and, according to Eqs. (69) and (70), ˆ ⊥(k) (r, ω) = ik 4π E c

X

⊥(k)

ˆ ω), G(r, n, ω)· J(n,

(78)

n

where ⊥ G and k G are the transverse part and the longitudinal part, respectively, of the total Green tensor G = ⊥ G+ k G of the classical EM field in the presence of the CN. This tensor satisfies the equation X

∇× ∇× − k2




zα

Gαz (r, n, ω) = δ(r − n)

(79)

α=r,ϕ,z

together with the radiation conditions at infinity and the boundary conditions on the CN surface. Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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All the ’discrete’ quantities in Eqs. (74)–(79) may be equivalently rewritten in continP uous variables in view of Eq. (3). Being applied to the identity 1 = m δnm , Eq. (3) yields δnm = S0 δ(Rn − Rm ). (80) This requires to redefine fˆ(n, ω) =

p

S0 fˆ(Rn , ω), fˆ† (n, ω) =

p

S0 fˆ† (Rn , ω)

(81)

in the commutation relations (76). Similarly, from Eq. (77), in view of Eqs. (3), (75) and (81), one obtains p (82) G(r, n, ω) = S0 G(r, Rn , ω), which is also valid for the transverse and longitudinal Green tensors in Eq. (78). ˆ (2) in Eq. (66) about ˆ (1) and H Next, we make the series expansions of the interactions H int int the lattice site n to the first non-vanishing terms, ˆ (1) ≈ − H int

X qi X q2 (i) i ˆ ˆ 2(n), ˆn + A(n) ·p A mi c 2mic2 n,i

(83)

n,i

ˆ (2) ≈ H int

X

(i)

qi ∇n ϕ(n) ˆ · ˆrn ,

(84)

n,i

and introduce the single-lattice-site Hamiltonian ˆ n = ε0 |0ih0| + H

X (ε0 + ~ωf )|f ihf |

(85)

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f

with the completeness relation |0ih0| +

X

ˆ |f ihf | = I.

(86)

f

Here, ε0 is the energy of the ground state |0i (no exciton excited) of the carbon atom associated with the lattice site n, ε0 + ~ωf is the energy of the excited carbon atom in the (f ) quantum state |f i with one f -internal-state exciton formed of the energy Eexc = ~ωf . In view of Eqs. (85) and (86), one has (i)

d ˆrn mi (i) ˆ mi ˆ (i) ˆ ˆ = mi = [ˆrn , Hn] = I [ˆrn , Hn] I dt i~ i~   mi X (i) (i) † ~ωf h0|ˆrn |f iBn,f − hf |ˆrn |0iBn,f ≈ i~

(i) ˆn p

(87)

f

and (i) (i) ˆrn = Iˆ ˆrn Iˆ ≈

X

 (i) (i) † h0|ˆrn |f iBn,f + hf |ˆrn |0iBn,f ,

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418

I. V. Bondarev, L. M. Woods and A.Popescu (i)

(i)

where h0|ˆrn |f i = hf |ˆrn |0i in view of the hermitian and real character of the coordinate † = |f ih0| create and annihilate, respecoperator. The operators Bn,f = |0ihf | and Bn,f tively, the f -internal-state exciton at the lattice site n, and exciton-to-exciton transitions are neglected. In addition, we also have δij δαβ =

i (i) (j) [(ˆ pn )α , (ˆrn )β ], ~

(89)

where α, β = r, ϕ, z. Substituting these into Eqs. (83) and (84) [commutator (89) goes which is to be pre-transformed as follows P into the second term of Eq. (83) ˆ α A(n) ˆ β δij δαβ /2mic2], one arrives at the following (electric dipole) apq q A(n) i j i,j,α,β proximation of Eq. (66) ˆ int = H ˆ (1) + H ˆ (2) H int int   X iωf i f ˆ † ˆ dfn · A(n) Bn,f − Bn,f + dn · A(n) =− c ~c n,f   X † + dfn · ∇n ϕ(n) ˆ Bn,f + Bn,f

(90)

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n,f

P (i) ˆ n|f i = hf |d ˆ n|0i, where d ˆn = rn is the total electric dipole moment with dfn = h0|d i qi ˆ operator of the particles residing at the lattice site n. The Hamiltonian (90) is seen to describe the vacuum-type exciton interaction with the surface EM field (created by the charge fluctuations on the nanotube surface). The last term in the square brackets does not depend on the exciton operators, and therefore results in the constant energy shift which can be safely neglected. We then arrive, after using Eqs. (67), (68), (75), and (78), at the following second quantized interaction Hamiltonian ˆ int = H

X Z n,m,f

∞ 0

(+) (−) † ˆ dω [ gf (n, m, ω)Bn,f − gf (n, m, ω)Bn,f ] f(m, ω) + h.c.,

where (±)

gf (n, m, ω) = g⊥ f (n, m, ω) ±

ω k g (n, m, ω) ωf f

(91)

(92)

with ⊥(k)

gf

(n, m, ω) = −i

X 4ωf p π~ω Re σ (R , ω) (dfn )α ⊥(k)Gαz (n, m, ω), zz CN c2 α=r,ϕ,z

and ⊥(k)

Gαz (n, m, ω) =

Z

⊥(k)

dr δαβ (n − r) Gβz (r, m, ω).

(93)

(94)

This yields Eqs. (10)–(12) after the strong transverse depolarization effect in CNs is taken into account whereby dfn ≈ (dfn )z ez . Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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Appendix B: Green tensor of the surface EM field Within the model of an infinitely thin, infinitely long, anisotropically conducting cylinder we utilize here, the classical EM field Green tensor is found by expanding the solution to the Green equation (79) in series in cylindrical coordinates, and then imposing the appropriately chosen boundary conditions on the CN surface to determine the Wronskian normalization constant (see, e.g., Ref. [118]). After the EM field is divided into the transversely and longitudinally polarized components according to Eqs. (69)–(71), the Green equation (79) takes the form X

∇× ∇× − k2


h⊥ zα

i Gαz (r, n, ω) + k Gαz (r, n, ω) = δ(r − n)

(95)

∇α⊥ Gαz (r, n, ω) = 0

(96)

α=r,ϕ,z

with the two additional constraints, X

α=r,ϕ,z

and

X

αβγ ∇β k Gγz (r, n, ω) = 0 ,

(97)

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β,γ=r,ϕ,z

where αβγ is the totally antisymmetric unit tensor of rank 3. Equations (96) and (97) originate from the divergence-less character (Coulomb gauge) of the transverse EM component and the curl-less character of the longitudinal EM component, respectively. The transverse ⊥G k αz and longitudinal Gαz Green tensor components are defined by Eq. (94) which is the corollary of Eq. (70) using the Eqs. (77) and (78). Equation (95) is further rewritten in view of Eqs. (96) and (97), to give the following two independent equations for ⊥ Gzz and k Gzz we need ∆ + k2


⊥

⊥ Gzz (r, n, ω) = −δzz (r − n) ,

k k2 kGzz (r, n, ω) = −δzz (r − n)

(98) (99)

with the transverse and longitudinal delta-functions defined by Eq. (71). We use the differential representations for the transverse ⊥ Gzz and longitudinal k Gzz Green functions of the following form [consistent with Eq. (94)] ⊥

Gzz (r, n, ω) = k



 1 ∇z ∇z + 1 g(r, n, ω), k2

Gzz (r, n, ω) = −

1 ∇z ∇z g(r, n, ω), k2

(100) (101)

where g(r, n, ω) is the scalar Green function of the Helmholtz equation (98), satisfying the radiation condition at infinity and the finiteness condition on the axis of the cylinder. Such Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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I. V. Bondarev, L. M. Woods and A.Popescu

a function is known to be given by the following series expansion √ √ ∞ S0 eik|r−Rn | S0 X ip(ϕ−ϕn ) = e g(r, n, ω) = 4π |r − Rn | (2π)2 p=−∞ Z × dh Ip (vr)Kp(vRCN ) eih(z−zn ) , r ≤ RCN ,

(102)

C

√ where Ip and Kp are the modified cylindric Bessel functions, v = v(h, ω) = h2 − k2 , and we used the property (82) to go from the discrete variable n to the corresponding continuous variable. The integration contour C goes along the real axis of the complex plane and envelopes the branch points ±k of the integrand from below and from above, respectively. For r ≥ RCN , the function g(r, n, ω) is obtained from Eq. (102) by means of a simple symbol replacement Ip ↔ Kp in the integrand. The scalar function (102) is to be imposed the boundary conditions on the CN surface. To derive them, we represent the classical electric and magnetic field components in terms of the EM field Green tensor as follows Eα (r, ω) = ik ⊥ Gαz (r, n, ω), Hα (r, ω) = −

i X αβγ ∇β Eγ (r, ω). k

(103) (104)

β,γ=r,ϕ,z

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These are valid for r 6= n under the Coulomb-gauge condition. The boundary conditions are then obtained from the standard requirements that the tangential electric field components be continuous across the surface, and the tangential magnetic field components be discontinuous by an amount proportional to the free surface current density, which we approximate here by the (strongest) axial component, σzz (RCN , ω), of the nanotube’s surface conductivity. Under this approximation, one has Ez |+ − Ez |− = Eϕ |+ − Eϕ |− = 0,

(105)

H z |+ − H z |− = 0,

(106)

4π σzz (ω)Ez |RCN , (107) c where ± stand for r = RCN ± ε with the positive infinitesimal ε. In view of Eqs. (103), (104) and (100), the boundary conditions above result in the following two boundary conditions for the function (102) Hϕ |+ − H ϕ |− =

∂g ∂g − ∂r + ∂r −

g|+ − g|− = 0,   4πi σzz (ω) ∂ 2 2 =− +k g|RCN . ω ∂z 2

(108) (109)

We see that Eq. (108) is satisfied identically. Eq. (109) yields the Wronskian of modified Bessel functions on the left, W [Ip(x), Kp(x)] = Ip (x)Kp0 (x)−Kp(x)Ip0 (x) = −1/x, which brings us to the equation −

1 RCN

=

4πi σzz (ω) 2 v Ip (vRCN )Kp(vRCN ). ω

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This is nothing but the dispersion relation which determines the radial wave numbers, h, of the CN surface EM modes with given p and ω. Since we are interested here in the EM field Green tensor on the CN surface [see Eq. (93)], not in particular surface EM modes, we substitute Ip (vRCN )Kp(vRCN ) from Eq. (110) into Eq. (102) with r = RCN . This allows us to obtain the scalar Green function of interest with the boundary conditions (108) and (109) taken into account. We have √ Z eih(z−zn ) iω S0 δ(ϕ − ϕn ) dh , g(R, n, ω) = − 8π 2σzz (ω)RCN C k2 − h2

(111)

where R = {RCN , ϕ, z} is an arbitrary point of the cylindrical surface. Using further the residue theorem to calculate the contour integral, we arrive at the final expression of the form √ c S0 δ(ϕ − ϕn ) iω|z−zn |/c e , (112) g(R, n, ω) = − 8πσzz (ω)RCN which yields ⊥

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k

Gzz (R, n, ω) ≡ 0,

(113)

Gzz (R, n, ω) = g(R, n, ω),

(114)

in view of Eqs. (100) and (101). The fact that the transverse Green function (113) identically equals zero on the CN surface is related to the absence of the skin layer in the model of the infinitely thin cylinder (see, e.g., Ref. [118]). In this model, the transverse Green function is only non-zero in the near-surface area where the exciton wave function goes to zero. Thus, only longitudinally polarized EM modes with the Green function (114) contribute to the exciton surface EM field interaction on the nanotube surface.

Appendix C: Diagonalization of the Hamiltonian (1)–(13) We start with the transformation of the total Hamiltonian (1)–(13) to the k-representation using Eqs. (5) and (9). The unperturbed part presents no difficulties. Special care should (±) be given to the interaction matrix element g f (n, m, ω) in Eq. (13). In view of Eqs. (114), (112) and (3), one has explicitly (±)

gf (k, k0, ω) =

1 X (±) 0 gf (n, m, ω) e−ik·n+ik ·m N n,m

p iω π~ω Re σzz (ω) dfz p R2CN S0 =± 2πc σzz (ω)RCN N N S02 Z 2π Z ∞ 0 0 × dϕn dϕmδ(ϕn − ϕm ) e−ikϕ ϕn +ikϕ ϕm dzn dzm eiω|zn −zm |/c−ikz zn +ikz zm , 0

−∞

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I. V. Bondarev, L. M. Woods and A.Popescu

where we have also taken into account the fact that the dipole matrix element (dfn)z = ˆ n)z |f i is actually the same for all the lattice sites on the CN surface in view of their h0|(d P ˆ n)z |f i. equivalence. As a consequence, (dfn)z = dfz /N with dfz = n h0|(d The integral over ϕ in Eq. (115) is taken in a standard way to yield Z

2π

0

dϕn dϕmδ(ϕn − ϕm ) e−ikϕ ϕn +ikϕ ϕm = 2πδkϕ kϕ0 .

(116)

0

The integration over z is performed by first writing the integral in the form Z

∞

dzn dzm ... = lim

Z

L→∞

−∞

L/2

dzn

Z

−L/2

L/2

dzm ... −L/2

(L being the CN length), then dividing it into two parts by means of the equation eiω|zn −zm |/c = θ(zn − zm ) eiω(zn −zm )/c + θ(zm − zn ) e−iω(zn −zm )/c, and finally by taking simple exponential integrals with allowance made for the formula 2 sin[L(kz − kz0 )/2] . L→∞ L(kz − kz0 )

δkz kz0 = lim

After some simple algebra we obtain the result  Z ∞ iω|zn −zm |/c−ikz zn +ikz0 zm 2 dzn dzm e = lim L 1 − L→∞

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−∞

2iω/c 2 L [kz − (ω/c)2]



δkz kz0 .

In view of Eqs. (116) and (117), the function (115) takes the form p   f iω d πS0 ~ω Re σzz (ω) 2iω/c z (±) 0 lim 1 − δkk0 . gf (k, k , ω) = ± (2π)2c σzz (ω)RCN L→∞ L [kz2 − (ω/c)2]

(117)

(118)

We have taken into account here that δkϕ kϕ0 δkz kz0 = δkk0 , as well as the fact that (RCN L/N S0 )2 = 1/(2π)2. This can be further simplified by noticing that only absolute value squared of the interaction matrix element matters in calculations of observables. We then have 2 α 2iω/c =1+ α ≈ 1+ 1 − 2 2 2 2 L [kz − (ω/c) ] u u + α2 with u = (ckz /ω)2 − 1, and α = (2c/Lω)2 being the small parameter which tends to zero as L → ∞. Using further the formula (see, e.g., Ref. [74]) δ(u) =

α 1 lim 2 , π α→0 u + α2

and the basic properties of the δ-function, we arrive at 2 2iω/c = 1 + πc|kz | [ δ(ω + ckz ) + δ(ω − ckz )] lim 1 − L→∞ L [kz2 − (ω/c)2] 2

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Exciton-Plasmon Interactions in Individual Carbon Nanotubes

423

p Re σ (ω) 2 1 zz . = Re σzz (ω) σzz (ω)

(120)

We also have

Equation (118), in view of Eqs. (119) and (120), is rewritten effectively as follows (±)

gf (k, k0, ω) = ±iDf (ω) δkk0

(121)

with Df (ω) =

fp

πS0 ~ω Re[1/σzz (ω)] (2π)2c RCN

ω dz

r

1+

πc|kz | [δ(ω + ckz ) + δ(ω − ckz )] . (122) 2

In terms of the simplified interaction matrix element (121), the k-representation of the Hamiltonian (1)–(13) takes the following (symmetrized) form X ˆ =1 ˆ k, H H 2

(123)

k

where ˆk = H

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+ +

Z

  † † Ef (k) Bk,f Bk,f + B−k,f B−k,f

(124)

f

h i ˆ ω) + fˆ† (−k, ω)f(−k, ˆ dω ~ω fˆ† (k, ω)f(k, ω)

∞

0 ∞

XZ f

X

 h i † dω iDf (ω) Bk,f + B−k,f fˆ(k, ω) − fˆ† (−k, ω) + h.c.

0

with Df (ω) given by Eq. (122). To diagonalize this Hamiltonian, we follow Bogoliubov’s canonical transformation technique (see, e.g., Ref. [74]). The canonical transformation of the exciton and photon operators is of the form Bk,f =

X h

i uµ (k, ωf )ξˆµ (k) + vµ (k, ωf )ξˆµ† (−k) ,

(125)

i u∗µ (k, ω)ξˆµ(k) + vµ∗ (k, ω)ξˆµ† (−k) ,

(126)

µ=1,2

fˆ(k, ω) =

X h

µ=1,2 † where the new operators, ξˆµ (k) and ξˆµ (k) = [ξˆµ(k)]†, annihilate and create, respectively, the coupled exciton-photon excitations of branch µ on the nanotube surface. They satisfy the bosonic commutation relations of the form

h

i ξˆµ (k), ξˆµ† 0 (k0) = δµµ0 δkk0 ,

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(127)

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which, along with the reversibility requirement of Eqs. (125) and (126), impose the following constraints on the transformation functions uµ and vµ X  u∗µ (k, ωf )uµ0 (k, ωf ) − vµ (k, ωf )vµ∗0 (k, ωf ) f ∞

Z

+

0

  dω uµ (k, ω)u∗µ0 (k, ω) − vµ∗ (k, ω)vµ0 (k, ω) = δµµ0 ,

X  u∗µ (k, ωf )uµ (k, ωf 0 ) − vµ∗ (k, ωf )vµ (k, ωf 0 ) = δf f 0 , µ

X  u∗µ (k, ω)uµ(k, ω 0) − vµ∗ (k, ω)vµ(k, ω 0) = δ(ω − ω 0 ). µ

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Here, the first equation guarantees the fulfilment of the commutation relations (127), whereas the second and the third ensure that Eqs. (125) and (126) are inverted to yield ξˆµ (k) as given by Eq. (15). Other possible combinations of the transformation functions are identically equal to zero. The proper transformation functions that diagonalize the Hamiltonian (124) to bring it to the form (14), are determined by the identity h i ˆk . (128) ~ωµ (k) ξˆµ(k) = ξˆµ (k), H Putting Eqs. (15) and (124) into Eq. (128) and using the bosonic commutation relations for the exciton and photon operators on the right, one obtains ( k-argument is omitted for brevity) Z ∞   ∗ (~ωµ − Ef ) uµ (ωf ) = −i dω Df (ω) uµ (ω) − vµ∗ (ω) , Z 0∞   (~ωµ + Ef ) vµ (ωf ) = i dω Df (ω) uµ (ω) − vµ∗ (ω) , 0 X   Df (ω) u∗µ (ωf ) + vµ (ωf ) , ~ (ωµ − ω) uµ (ω) = i f

~ (ωµ +

ω) vµ∗ (ω)

X   =i Df (ω) u∗µ (ωf ) + vµ (ωf ) . f

These simultaneous equations define the complex transformation functions uµ and vµ uniquely. They also define the dispersion relation (the energies ~ωµ , µ = 1, 2) of the coupled exciton-photon (or exciton-plasmon, to be exact) excitations on the nanotube surface. Substituting uµ and vµ∗ from the third and forth equations into the first one, one has   Z ∞ 4Ef ω|Df (ω)|2 u∗ (ωf ) = 0, dω ~ωµ − Ef − ~ωµ + Ef 0 ~(ωµ2 − ω 2) µ whereby, since the functions u∗µ are non-zero, the dispersion relation we are interested in becomes Z ∞ ω|Df (ω)|2 2 2 = 0. (129) (~ωµ) − Ef − 4Ef dω ~(ωµ2 − ω 2) 0

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Exciton-Plasmon Interactions in Individual Carbon Nanotubes

425

The energy E0 of the ground state of the coupled exciton-plasmon excitations is found by plugging Eq. (15) into Eq. (14) and comparing the result with Eqs. (123) and (124). This yields   Z ∞ X X E0 = − ~ωµ (k) |vµ (k, ωf )|2 + dω |vµ (k, ω)|2. k, µ=1,2

f

0

Using further Df (ω) as explicitly given by Eq. (122), the dispersion relation (129) is rewritten as follows Z ∞  Ef S0 |dfz |2 ω 4Re[1/σzz (ω)] π(c|kz |)5Re[1/σzz (c|kz |)] 2 2 (~ωµ ) − Ef = dω + . ωµ2 − ω 2 ωµ2 − (c|kz |)2 4π 3c2R2CN 0

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∗ (−ω), which originates Here we have taken into account the general property σzz (ω) = σzz from the time-reversal symmetry requirement, in the second term on the right hand side. This term comes from the two delta functions in |Df (ω)|2, and describes the contribution of the spatial dispersion (wave-vector dependence) to the formation of the exciton-plasmons. We neglect this term in what follows because the spatial dispersion is neglected in the nanotube’s axial surface conductivity in our model, and, secondly, because it is seen to be very small for not too large excitonic wave vectors. Thus, converting to the dimensionless variables (17), we arrive at the dispersion relation (16) with the exciton spontaneous decay (recombination) rate and the plasmon DOS given by Eqs. (18) and (19), respectively. Lastly, bearing in mind that the delta functions in |Df (ω)|2 are responsible for the spatial dispersion which we neglect in our model, and therefore dropping them out from the squared interaction matrix element (121), we arrive at the property (45).

Appendix D: Effective longitudinal potential in the presence of the perpendicular electrostatic field Here we analyze the set of equations (27)–(29), and show that the attractive cusp-type cutoff potential (32) with the field dependent cutoff parameter (33) is a uniformly valid approximation for the effective electron-hole Coulomb interaction potential (30) in the exciton binding energy equation (29). We rewrite Eqs. (27) and (28) in the form of a single equation as follows   2 d 2 + q + p cos ϕ ψ(ϕ) = 0 . (130) dϕ2 p Here, ϕ = ϕe,h , ψ = ψe,h , q = RCN 2me,h εe,h /~, and p = ±2eme,h R3CN F/~2 with the (+)-sign to be taken for the electron and the (–)-sign to be taken for the hole. We are interested in the solutions to Eq. (130) which satisfy the 2π-periodicity condition ψ(ϕ) = ψ(ϕ + 2π). The change of variable ϕ = 2t transfers this equation to the well known Mathieu’s equation (see, e.g., Refs. [119, 114]), reducing the solution’s period by the factor of two. The exact solutions of interest are, therefore, given by the odd Mathieu

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I. V. Bondarev, L. M. Woods and A.Popescu

functions se2m+2 (t = ϕ/2) with the eigen values b2m+2 , where m is a nonnegative integer (notations of Ref. [119]). These are the solutions to the Sturm-Liouville problem with boundary conditions on functions, not on their derivatives. It is easier to estimate the z-dependence of the potential (30) if the functions ψe,h (ϕe,h ) are known explicitly. So, we do solve Eq. (130) using the second order perturbation theory in the external field (the term p cos ϕ). The second order field corrections are also of practical importance in the most of experimental applications. The unperturbed problem yields the two linearly independent normalized eigen functions and the eigen values as follows q exp(±ijϕ) RCN (0) (0) √ , q=j= 2me,h εe,h (131) ψj (ϕ) = ~ 2π (0)

with j being a nonnegative integer. The energies εe,h (j) are doubly degenerate with the

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(0)

exception of εe,h (0) = 0, which we will discard since it results in the zero unperturbed band gap according to Eq. (8). The perturbation p cos ϕ does not lift the degeneracy of the unperturbed states. Therefore, we use the standard nondegenerate perturbation theory with the basis wave functions set above (plus sign selected for definiteness) to calculate the energies and the wave functions to the second order in perturbation. The standard procedure (see, e.g., Ref. [95]) yields !   2 2 6 me,h e RCN 2 (0) ϑ(j − 2) 1 + F ψj e,h (ϕe,h ) ψj e,h (ϕe,h ) = 1 − 2~4 [(j − 1)2 − j 2]2 [(j + 1)2 − j 2]2   (0) (0) ϑ(j − 2)ψj−1 e,h (ϕe,h ) ψj+1 e,h (ϕe,h ) me,h eR3CN  + F (132) ± (j − 1)2 − j 2 (j + 1)2 − j 2 ~2   (0)  m2 e2 R6 ϑ(j − 2)ϑ(j − 3)ψ (0) (ϕe,h ) ψ (ϕ ) e,h j−2 e,h j+2 e,h e,h CN 2 + F . +  [(j − 1)2 − j 2][(j − 2)2 − j 2] [(j + 1)2 − j 2][(j + 2)2 − j 2] ~4 Here, j is a positive integer, and the theta-functions ensure that j = 1 is the ground state of the system. The corresponding energies are as follows εe,h =

~2 j 2 me,h e2 R4CN wj 2 − F 2~2 2me,h R2CN

(133)

with wj given by Eq. (34), thus, according to Eq. (8), resulting in the nanotube’s band gap as given by Eq. (37). From Eq. (132), in view of Eq. (131), we have the following to the second order in the field  eR3CN wj 1 2 2 cos ϕ − m cos ϕ ) F (134) 1 − 2 (m |ψe (ϕe )| |ψh(ϕh )| ≈ h h e e 4π 2 ~2 #
e2 R6CN vj 2 e2 R6CN wj2 2 2 2 F − 4µMex cos ϕe cos ϕh F , +2 mh cos 2ϕh + me cos 2ϕe ~4 ~4

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427

where   ϑ(j − 2) ϑ(j − 3) 1 1 + + . vj = 2 2 2 2 2 2 2 2 (j − 1) − j (j − 2) − j (j + 1) − j [(j + 1) − j ][(j + 2)2 − j 2] Plugging Eqs. (134) and (31) into Eq. (30) and noticing that the integrals involving linear combinations of the cosine-functions are strongly suppressed due to the integration over the cosine period, and are therefore negligible compared to the one involving the quadratic cosine-combination, we obtain Veff (z) = −

Z

e2 4π 2

Z dϕe

2π 0

2π

dϕh 0

1 − 2 cos ϕe cos ϕh ∆j (F ) {z 2 + 4R2CN sin2 [(ϕe − ϕh )/2]}1/2

(135)

with ∆j (F ) given by Eq. (34). The next step is to perform the double integration in Eq. (135). We have to evaluate the two double integrals. They are I1 =

Z

Z

2π 0

and I2 =

2π

dϕe

Z

0

Z dϕe

2π 0

{z 2

+

4R2CN

dϕh sin2[(ϕe − ϕh )/2]}1/2

(136)

2π

dϕh cos ϕe cos ϕh . 2 2 2 1/2 0 {z + 4RCN sin [(ϕe − ϕh )/2]}

(137)

We first notice that both I1 and I2 can be equivalently rewritten as follows Z 2π Z 2π Z ϕe dϕe dϕh ... = 2 dϕe dϕh ...

Z

2π

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0

0

0

(138)

0

due to the symmetry of the integrands with respect to the (ϕe = ϕh )-line. Using this property, we substitute ϕh with the new variable t = sin[(ϕe − ϕh )/2] in Eqs. (136) and (137). This, after simplifications, yields Z 2π Z I1 = 4 dϕe 0

and

sin(ϕe /2) 0

[(1 −

Z 2π Z 2 I2 = 4 dϕe cos ϕe 0

t2 )(z 2

dt + 4R2CN t2 )]1/2

(139)

sin(ϕe /2) 0

dt (1 − 2t2 ) . [(1 − t2 )(z 2 + 4R2CN t2 )]1/2

(140)

Here, the inner integrals are reduced to the incomplete elliptical integrals of the first and second kinds (see, e.g., Ref. [114]). We continue the evaluation of Eqs. (139) and (140) by expanding the denominators of the integrands in series at large and small |z| as compared to the CN diameter 2RCN . One has " #       1 2RCN t 2 3 2RCN t 4 5 2RCN t 6 1 1 1− ≈ + − + ... |z| 2 |z| 8 |z| 16 |z| (z 2 + 4R2CN t2 )1/2

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I. V. Bondarev, L. M. Woods and A.Popescu

for |z|/2RCN  1, and Z

sin(ϕe /2) 0

dt f (t) 1 = 2RCN [(1 − t2 )(z 2 + 4R2CN t2 )]1/2

lim (|z|/2RCN )→0

Z

sin(ϕe /2)

f (t) dt √ 2 |z|/2RCN t 1 − t

for |z|/2RCN  1 [f (t) is a polynomial function]. Using these in Eqs. (139) and (140), we arrive at "    2#   |z| |z| 4R 4π ln  CN − 1  , 2R 1  4 2RCN  RCN |z| CN I1 ≈ "  2  4 #  2  |z|  4π 1− 1 2RCN + 9 2RCN  1 ,  |z| 4 64 2RCN |z| |z| and

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"     2 #  |z| |z| 4π 1 ln 4RCN − 1 + 3   , 2R 1  8 2RCN  RCN 2 |z| CN I2 ≈  2 "  2#  2  |z| 2R 2R  3 π CN CN  1− 4 , 1  4|z| 2RCN |z| |z| Plugging these I1 and I2 into Eq. (135) and retaining only leading expansion terms yields     e2 [1−∆j (F )] |z| 4R  CN  ln , 2R 1 − πRCN |z| CN (141) Veff(z) ≈  2  |z| e  − 1 , |z| 2RCN We see from Eq. (141) that, to the leading order in the series expansion parameter, the perpendicular electrostatic field does not affect the longitudinal electron-hole Coulomb potential at large distances |z|  2RCN , as one would expect. At short distances |z|  2RCN the situation is different, however. The potential decreases logarithmically with the field dependent amplitude as |z| goes down. The amplitude of the potential decreases quadratically as the field increases [see Eq. (34)], thereby slowing down the potential fall-off with decreasing |z|, or, in other words, making the potential shallower as the field increases. Such a behavior can be uniformly approximated for all |z| by an appropriately chosen attractive cusp-type cutoff potential with the field dependent cutoff parameter. Indeed, consider the dimensionless function f (y) = −2RCN  Veff /e2 of the dimensionless variable y = |z|/2RCN . Then, according to Eq. (141), one has     Φ1(y) = 2 [1 − ∆j (F )] ln 2 , 0 < y  1 π y f (y) =  Φ (y) = 1 , y  1 2 y Now introduce the function Φ(y) =

1 y + y0

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(142)

Exciton-Plasmon Interactions in Individual Carbon Nanotubes

429

Figure 14: The dimensionless function (142) with the zero-field cutoff parameter (143). See text for details. with the cutoff parameter y0 selected in such a way as to satisfy the condition Φ(1) = [Φ1(1) + Φ2(1)]/2. This yields

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y0 =

π − 2 ln 2 [1 − ∆j (F )] . π + 2 ln 2 [1 − ∆j (F )]

(143)

Figure 14 shows the zero-field behavior of the Φ(y) function as compared to the corresponding Φ1 (y) and Φ2 (y) functions. We see that Φ(y) gradually approaches Φ2 (y) = 1/y for increasing y > 1. For decreasing y < 1, on the other hand, Φ(y) is very close to the logarithmic behavior as given by Φ1 (y), with the exception that there is no divergence at y ∼ 0 due to the presence of the cutoff. The cutoff parameter (143) is field dependent, decreasing as the field grows, which is consistent with the behavior of the original potential (141). Multiplying Eq. (142) by the dimensional factor −e2 /2RCN  and putting y = |z|/2RCN , we obtain the attractive longitudinal cusp-type cutoff potential (32) we build our analysis on in this paper.

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[88] T.Ando, J. Phys. Soc. Jpn. 75, 024707 (2006). [89] T.Ogawa and T.Takagahara, Phys. Rev. B 44, 8138 (1991). [90] A.Jorio, C.Fantini, M.A.Pimenta, R.B.Capaz, Ge.G.Samsonidze, G.Dresselhaus, M.S.Dresselhaus, J.Jiang, N. Kobayashi, A.Gr¨uneis, and R.Saito, Phys. Rev. B 71, 075401 (2005). [91] A.Suna, Phys. Rev. 135, A111 (1964). [92] Y.Miyauchi, R.Saito, K.Sato, Y.Ohno, R.Iwasaki, T.Mizutani, J.Jiang, and S.Maruyama, Chem. Phys. Lett. 442, 394 (2007). [93] I.V.Bondarev, Y.Nagai, M.Kakimoto, T.Hyodo, Phys. Rev. B 72, 012303 (2005). [94] R.Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, New York, 1995). [95] L.D.Landau and E.M.Lifshits, Quantum Mechanics: Non-Relativistic Theory (Pergamon, New York, 1977). [96] L.P.Gor’kov and L.P.Pitaevski, Dokl. Akad. Nauk SSSR 151, 822 (1963) [English transl.: Soviet Phys.—Dokl. 8, 788 (1964)]. [97] C.Herring and M.Flicker, Phys. Rev. 134, A362 (1964); C.Herring, Rev. Mod. Phys. 34, 631 (1962).

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[98] F.Wang, et. al., Phys. Rev. B 70, 241403(R) (2004). [99] D.Abramavicius, et. al., Phys. Rev. B 79, 195445 (2009). [100] A.Srivastava and J.Kono, Phys. Rev. B 79, 205407 (2009). [101] M.J.Sparnaay, Physica 24, 751 (1958). [102] S.K.Lamoreaux, Phys. Rev. Lett. 78, 5 (1997). [103] U.Mohideen and A.Roy, Phys. Rev. Lett. 81, 4549 (1998); F.Chen and U.Mohideen, Rev. Sci. Instrum. 72, 3100 (2001). [104] J.N.Munday, F.Capasso, and V.A.Parsegian, Nature 457, 170 (2009). [105] M.Bordag, U.Mohideen, and V.M.Mostepanenko, Phys. Rep. 353, 1 (2001). [106] V.A.Parsegian, van der Waals forces (Cambridge University Press, CityplaceCambridge, 2005). [107] R.F.Rajter, R.Podgornik, V.A.Parsegian, R.H.French, and W.Y.Ching, Phys. Rev. B 76, 045417 (2007). [108] A.Popescu, L.M.Woods, and I.V.Bondarev, Phys. Rev. B 77, 115443 (2008). Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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[109] M.Bordag, B.Geyer, G.L.Klimchitskaya, and V.M.Mostepanenko, Phys. Rev. B 74, 205401 (2006). [110] C.T.Tai, Dyadic Green Functions in Electromagnetic Theory , 2nd Ed. (IEEE Press, Piscatawnay, NY, 1994). [111] L.W.Li, M.S.Leong, T.S.Yeo, and P.S.Kooi, J. Electr. Waves Appl. 14, 961 (2000). [112] I.Cavero-Pelaez and K.A.Milton, Annals of Phys. 320, 108 (2005). [113] K.A.Milton, L.L.DeRaad, Jr., and J.Schwinger, Ann. Phys. 115, 388 (1978). [114] Handbook of Mathematical Functions , edited by M.Abramovitz and I.A.Stegun (Dover, New York, 1972). [115] N.G. van Kampen, B.R.A.Nijboer, and K.Schram, Phys. Lett. A 26, 307 (1968). [116] F.Intravaia and A.Lambrecht, Phys. Rev. Lett. 94, 110404 (2005). [117] J.F.Dobson, A.White, and A.Rubio, Phys. Rev. Lett. 96, 073201 (2006). [118] J.D.Jackson, Classical Electrodynamics (Wiley, New York, 1975). [119] Higher Transcendental Functions , edited by H.Bateman and A.Erd´elyi (Mc GrawHill, New York, 1955). Vol. 3. [120] S.Liu, J.Li, Q.Shen, Y.Cao, X.Guo, G.Zhang, C.Feng, J.Zhang, Z.Liu, M.L.Steigerwald, D.Xu, and C.Nuckolls, Angew. Chem. 48, 4759 (2009).

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[121] P.L.Hern´andes-Martinez and A.O.Govorov, Phys. Rev. B 78, 035314 (2008). [122] H.Hirori, K.Matsuda, and Y.Kanemitsu, Phys. Rev. B 78, 113409 (2008).

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In: Plasmons: Theory and Applications Editor: Kristina N. Helsey, pp. 437-466

ISBN: 978-1-61761-306-7 c 2011 Nova Science Publishers, Inc.

Chapter 17

S URFACE P HENOMENA C AUSED BY O PTICAL S PIN -O RBIT I NTERACTION: A P EDAGOGICAL T HEORY USING Q UANTUM M ECHANICAL C ONCEPTS I. Banno∗ Interdisciplinary Graduate School of Medicine and Engineering, University of Yamanashi 4-3-11 Takeda, Kofu, Yamanashi 400-8511, Japan

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Abstract This chapter gives a pedagogical treatment of optical system with plane interfaces (multilayers) , emphasizing analogy with one-dimensional quantum system. Familiar optical phenomena such as refraction, total reflection, Brewster’s total transmission, surface plasmon polariton, and optical tunneling effect, etc. are described in a manner compatible with quantum mechanics. To make clear the analogy between optical and quantum systems, dual vector potential is used as the minimum degrees of freedom of electromagnetic field in optical regime (the regime with negligible magnetic response of the matter). The source of dual vector potential is separated into surface and volume magnetic currents; the surface source is responsible for the singularity expressed by Maxwell’s boundary conditions. Furthermore, it is shown that the surface source originates in the optical spin-orbit interaction and that the coexistence of spin-orbit interaction (surface source) and volume source is essential for surface effects such as Brewster’s total transmission and surface plasmon polariton.

∗ E-mail

address: [email protected]

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PACS Keywords: boundary effects, plasmons on interfaces, bound state, spin-orbit effect, optical tests of quantum theory. Keywords: Maxwell’s boundary conditions, surface plasmon polariton, Brewster angle, Rytov-Vladimirsky effect, theoretical optics

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1.

Introduction

Suppose a vacuum-dielectric flat interface exposed to a plane wave incident light with ppolarization as described in Fig.1. This simple system represents the spin-orbit interaction of electromagnetic(EM) field in the interface region. The wave vector of incident light changes at the interface region, V01 and generates two outgoing branches of the transmission and reflection. With this deformation of the orbit, the polarization vector also changes because the electric field is transversal both in the regions of vacuum and dielectric, V0 and V1 . The polarization vector is the spin degrees of freedom (the inner degrees of freedom) of the EM field and is affected by the orbit (wave vector or momentum) of the EM field. This fact means that there is spin-orbit interaction in the interface region, V01 . The spin-orbit interaction in optics is often referred as Rytov-Vladimirsky effect because it was found by Rytov [1] and was subsequently discussed by Vladimirsky [2] in the context of geometrical phase; lately it was understood as an example of Berry phase [3]. In ordinary theory of classical optics, the spin-orbit interaction is considered through Maxwell’s boundary conditions (MBCs), that is, the relationships between the EM field in the two sides, V0 and V1 . In such a boundary value problem, the interface region, V01 is treated as a black box and the physics in it is not inquired. However, the spin-orbit interaction in the interface region plays a central role in surface optical effects such as Brewster’s total transmission and surface plasmon polariton. There are two purposes of this chapter: The first is to understand various optical phenomena using well-established quantum-mechanical concepts. The second is to show that optical spin-orbit interaction leads to the singularity of EM field (expressed by MBCs), and that the coexistence of the spin-orbit interaction (surface source) and inhomogeneity of dielectric function (volume source) is essential for Brewster’s total transmission and surface plasmon polariton. In the following, the EM field in optical regime is expressed in terms of the dual vector potential, which was introduced in Ref. [4] as the minimum degrees of freedom of EM field. A physicist knows by his experience that the description in terms of the minimum degrees of freedom might bring a clear physical picture. However, even in such a simple system of Fig.1, ordinary theoretical optics does not use EM potential (the minimum degrees of freedom) but use electric and magnetic fields (redundant degrees of freedom). This irrationality might come from delta-function-related singularity appearing in the wave equation of EM potential; see Sec5.3.. Such a singularity should be a difficulty to use EM potential in theoretical optics before the delta function had been invented by Dirac in the context of quantum mechanics. Perhaps, this is one of the historical cause that the electric and magnetic fields with MBCs have been widely used and the physics in the interface region has been still

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Surface Phenomena Caused by Optical Spin-Orbit Interaction

V0 ε 0

(0)

k

^e (0)

^e (r)

439

k(r)

V01 V1 ε 1

^e (t)

k(t) Figure 1. A system of vacuum-dielectric interface exposed to a plane wave incident light with p-polarization. Among two types of transversal linearly polarization modes, ppolarized field (or transversal magnetic field) has perpendicular polarization component to the plane interface, while s-polarized field (or transversal electric field) has only the parallel component.

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hindered. The outline of this chapter is as follows: In Sec.2., dual vector potential is introduced as the minimum degrees of freedom of EM field in the optical regime and its wave equation are derived. In Sec.3., the transversal dual vector potential in the system with plane interfaces is decomposed into two fields, each of which satisfies one-dimensional wave equation and, in the steep interface limit, MBCs and corresponding transfer matrix are derived. Sec.4. describes various optical phenomena in a unified and compatible manner with one-dimensional quantum mechanics. In Sec.5., the surface source (source localized in interface region) of dual vector potential is identified as spin-orbit interaction. In particular, Sec.5.3. makes clear the singularity of the spin-orbit interaction (surface source originating MBCs) and shows that the coexistence of surface and volume sources are essential for surface effects such as Brewster’s total transmission and surface plasmon polariton. The conclusion is given in Sec.6.. Four appendices are attached for calculation details.

2.

Classical Optics in terms of Dual Vector Potential

Following Ref. [4], I would like to reconsider the classical optics in terms of the dual vector potential, which is the minimum degrees of freedom of EM field in optical regime (the regime with negligible magnetic response of the matter). A description in terms of the dual vector potential brings a clear physical picture and one may interpret the surface effects of EM field, referring to quantum-mechanical concepts such as the spin-orbit interaction, the total transmission, the bound state, and so on.

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440

2.1.

I. Banno

Definition of Dual Vector Potential

Let us introduce the dual vector potential. Assuming that there is only induced electric charge density and current density i.e., no external electric charge density and no external electric current density, Maxwell’s equations are: ∇ · B(r,t) = 0 ,

∇ × E(r,t) + ∂t B(r,t) = 0 ,

∇ · D(r,t) = 0 ,

∇ × H(r,t) − ∂t D(r,t) = 0 ,

(1)

and the equations for matter are: B(r,t) = µ0 H(r,t) + M(r,t) , D(r,t) = ε0 E(r,t) + P(r,t) ,

(2)

where the notations are as follow: E, B, H, and D are the electric field, magnetic flux density, magnetic field, and electric flux density (or displacement vector field), respectively. P and M are the polarization and magnetization of the matter, respectively. ε0 and µ0 permittivity and permeability of vacuum, 1 respectively. Now note that there is dual symmetry in (1) and (2). Actually (1) and (2) are invariant under the dual transformation ( the mutual exchange between the electric quantities and the magnetic ones): E ⇐⇒ −H, D ⇐⇒ B,

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ε0 ⇐⇒ −µ0 , P ⇐⇒ M.

(3)

On the other hand, it is well known that the minimum degrees of freedom of EM field is neither E nor B but the EM potential A and φ. The ordinary EM potential is defined as B = ∇ × A,

E = −∂t A − ∇φ.

(4)

As a guide to introduce dual EM potential, let us consider optics with a hypothetical matter that has magnetic response but no electric response, i.e. M 6= 0 and P = 0. In such a system, the following wave equation for A in the radiation gauge is derived from (1) and (2). ∇ × ∇ × A(r,t) + ε0 µ0 ∂t2 A(r,t) = ∇ × M(r,t) ,

(5)

∇ · A(r,t) = 0 ,

(6)

φ(r,t) = 0 .

(7)

In (5)-(7), the source of the field A is ∇ × M = µ0 ×(magnetizing electric current density); this transversal vector source yields the transversal field ( A in the radiation gauge). Note that the condition for the radiation gauge is satisfied in the whole space including the interface region. The matter of interest in optics is just dual to that discussed in the above case, that is, the matter has electric response and a negligible magnetic response ( P 6= 0 and M = 0). 1 The

permeability, µ0 is 1/ε0 c2 (c is light velocity) and is not an elementary physical constant. But it is used for visibility of dual symmetry. Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

Surface Phenomena Caused by Optical Spin-Orbit Interaction

441

Actually, this is the situation of optical regime. In connection with the duality, it is natural to introduce dual EM potential defined as D = −∇ × C,

H = −∂t C − ∇χ.

(8)

Applying the dual transformation to (5)-(7), i.e. (A, φ, M) =⇒ (−C, −χ, P), one obtains the wave equation followed by the dual EM potential, ∇ × ∇ × C(r,t) + ε0 µ0 ∂t2 C(r,t) = −∇ × P(r,t) ,

(9)

∇ · C(r,t) = 0 ,

(10)

χ(r,t) = 0 .

(11)

The minimum degrees of freedom of the EM field in the optical regime is the dual EM potential in the radiation gauge and the wave equation (9) for the transversal field C is effective all over the regions including the interface region. This fact enable to set our hand to the interface region and make clear the physics of the surface phenomena in optics. The source of the dual vector potential is ∇ × P = −ε0 × (polarizing magnetic current density); this source is dual to ∇ × M = µ0 ×(magnetizing electric current density), see (3). Equations (9)-(11) are equivalent to Maxwell’s equations under M = 0. In fact, one can know D, E, B and H from C using the following relationships:

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2.2.

D(r,t) = −∇ × C(r,t) ,

(12)

E(r,t) = (−∇ × C(r,t) − P(r,t))/ε0 ,

(13)

B(r,t) = µ0 H(r,t) = −µ0 ∂t C(r,t) .

(14)

Surface and Volume Source of Dual Vector Potential

Suppose that the incident field oscillates with a constant angular frequency ω [or a constant wavenumber k0 = ω/c = ω(ε0 µ0 )1/2 ] and that P is defined through a local and linear dielectric function ε(r) as   ε0 ∇ × C(r,t) , (15) P(r,t) = (ε(r) − ε0 )E(r,t) = − 1 − ε(r) where ε(r) is assumed to be a smooth function for a time. Omitting the common time dependence in C(r,t) = C(r) exp(−iωt), (9) and (10) together with (15) lead to the following Helmholtz equation and the condition for the radiation gauge: ∇ × ∇ × C(r) − k02 C(r) = −Vˆs [C](r) − Vˆ v[C](r) for r ∈ V0 ∪V01 ∪V1 , ∇ · C(r) = 0 , ∇ε(r) × ∇ × C(r) , Vˆs [C](r) ≡ − ε(r) ε(r) − 1)k02 C(r) . Vˆv [C](r) ≡ −( ε0

(16) (17) (18) (19)

The source is divided into two parts. One is (surface magnetic current density) ×(−ε0 ), namely Vˆs[C], which is responsible for the MBCs as described in the following sections. Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

442

I. Banno

The other is (volume magnetic current density)×(−ε0 ), namely Vˆv[C]. In the following, −Vˆs [C] (−Vˆv[C]) is simply referred to as “surface (volume) magnetic current density”, ignoring the constant factor ε0 . The dual vector potential, of which sources are surface and volume magnetic current was firstly introduced in the context of near-field optics. The surface source of the dual vector potential predominates over the volume source in a small-scaled material and leads to a simple physical picture [4, 6]. In the regions V0 and V1 , ordinary EM potential can maintain radiation gauge ( ∇ · A = 0 and φ = 0), therefore, might describe optics in the same manner as the dual vector potential does, if MBCs are used for the interface region V01 as a black box. For example, ordinary vector potential can be used to construct transfer matrix, which is derived in the next section using the dual vector potential. However, ordinary EM potential cannot maintain the radiation gauge in V01 and the discussion in Sec.5. is not available.

3.

Optical System with Plane Interfaces

In this section, we treat systems composed of plane interfaces (or multilayers) and a monochromatic light. The wave equation of the dual vector potential is reduced to onedimensional Helmholtz equation and the transfer matrix is constructed.

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3.1.

Wave equations for p- and s-polarization Modes

Let us set x3 -direction as the direction normal to the interfaces and the scattering plane as x3 -x1 plane, and assume that the incident light comes from the side of matter with the well-defined wave vector k1 [Fig.2(a)]. In systems of our interest, the momentum component parallel to the interface ( ~kk eˆ 1 ) is conserved because of the translational symmetry along any directions parallel to the interface. Using this fact, one may decompose C(r) into two transversal modes as follows:
C(r) = eˆ 2 p(x3 ) exp(ikkx1 ) + ∇ × eˆ 2 s(x3 ) exp(ikkx1 ) .

(20)

One can check that both terms in (20) are transversal, i.e., ∇ · C(r) = 0 for any p(x3 ) and s(x3 ). The fields p(x3 ) and s(x3 ) represent amplitude of two independent transversal modes (p- and s-polarized modes, respectively), which are essentially the surface-parallel ( x2 -) component of magnetic and electric fields, respectively. Actually, one may derive next relationships, substituting (20) into (12)-(14): D1 (r) = ∂3 p(x3 )eikk x1 , E2 (r) = D3 (r) =

(21)

k02 s(x3 )eikk x1 /ε0 , −ikk p(x3 )eikk x1 ,

H1 (r) = B1 (r)/µ0 = −ick0 ∂3 s(x3 )eikk x1 , ikk x1

H2 (r) = B2 (r)/µ0 = ick0 p(x3 )e

,

ikk x1

H3 (r) = B3 (r)/µ0 = −ck0 kk s(x3 )e

.

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(22)

Surface Phenomena Caused by Optical Spin-Orbit Interaction

(a)

(b)

k0

k1

1exp(+k13x3) 1

r θr

θt

θ in

^e

1

^e

2

k// k13 V1 ε 1

r exp(-k13x3) ^e

t exp(+k03x3)

t 0

0 exp(-k03x3)

3

(c)

α’ exp(+k13x3) α exp(+k03x3)

k03 V01

443

V0 ε 0

α’

α

β’

β

β’ exp(-k13x3)

β exp(-k03x3)

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Figure 2. (a) The system with a plane interface; θin , θr , and θt are the incidence, reflection, and transmission (refraction) angles, respectively. The wave numbers in the matter and vacuum are k1 and k0 and their projection to the x3 -direction are k13 and k03 , respectively. The surface-parallel (x1 -) components of all the branches have a common value, kk ; this is the momentum conservation originating in the translational invariance along the surface parallel directions. The polarization vectors of p- and s-polarized modes are indicated by white arrows (kˆ × eˆ 2 ) and circles with dot ( eˆ 2 ), respectively, where kˆ is the unit vector of the wave number vector in each branch. (b) The reduced description for the system of (a); the propagation in each branch of p- or s-polarized mode is projected to the x3 -direction. (c) The same as (b) but for general case; there are two incidences. Substituting (20) into the wave equation (16) and taking the inner product with eˆ 2 and eˆ 1 (or eˆ 3 ), one can obtain one-dimensional Helmholtz equations for p(x3 ) and s(x3 ), respectively: 2 (−∂23 − k03 )p(x3 ) = −Vs [p](x3 ) −Vv[p](x3 ),

(23)

2 )s(x3) (−∂23 − k03 2 where k03

(24)

= ≡

−Vv [s](x3), k02 − kk2 ,

∂3 ε(x3 ) ∂3 ζ(x3 ), ε(x3 )   ε(x3 ) − 1 k02 ζ(x3 ) Vv[ζ](x3 ) ≡ − ε0 Vs[ζ](x3 ) ≡

(25) (ζ = p or s).

(26)

The Helmholtz equations (23) and (24) are just like time-independent Schr¨odinger equation in one-dimensional system and hint that the optical phenomena might be explained with the help of quantum-mechanical concepts.

3.2.

Maxwell’s Boundary Conditions across a Steep Interface

To check the effect of surface source, let us consider an interface region with small width in the neighbourhood of x3 = 0 ([−η/2, +η/2] 3 x3 ) and assume that the η is sufficiently small than the light wavelength ( k03 η, k0 η  1). The interface region is so thin that one cannot Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

444

I. Banno

feel wavelength (or wave number) in it; this means that we may ignore the wavenumberdependent term (retardation effects) in the wave equation. See Ref. [4] for a simple explanation for the retardation effect. Then, (23) and (24) are reduced to the next equations: −∂23 p(x3 ) = −

∂3 ε(x3 ) ∂3 p(x3 ), ε(x3 )

(27)

−∂23 s(x3 ) = 0

(28)

One can easily integrate these equations and obtain the solution in the interface region as ∂3 p(x3 ) = (const.) × ε(x3 ), p(x3 ) = (const.) × ∂3 s(x3 ) = (const.0 ),

Z x3

dx03 ε(x03 ) + p(−η/2),

(29)

s(x3 ) = (const.0 ) × (x3 + η/2) + s(−η/2).

(30)

−η/2

The boundary condition at the steep interface 2 is obtained by setting x3 = +η/2 in (29) and (30) and taking the limit of η/2 → +0:  x3 =+0 1 = 0, ∂ p(x ) = 0, (31) [p(x3 )]xx33 =+0 3 3 =−0 ε(x3 ) x3 =−0

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[s(x3 )]xx33 =+0 =−0 = 0,

[∂3 s(x3 )]xx33 =+0 =−0 = 0,

(32)

From (21) and (22), one may be confirmed that (31) and (32) represents MBCs:   x3 =+0 = 0 or [B ] = 0 , [E1 ]xx33 =+0 [D3 ]xx33 =+0 2 x3 =−0 =−0 =−0 = 0,   [E2 ]xx33 =+0 or [B3 ]xx33 =+0 [B1 ]xx33 =+0 =−0 = 0 =−0 = 0 , =−0 = 0.

(33) (34)

After all, Vs in (23) leads to the discontinuity of ∂3 p(x3 ) and continuity of p(x3 ) across x3 = 0, while absence of Vs in (24) leads to continuity of s(x3 ) and its derivative across x3 = 0. The boundary condition for s-polarization mode (32) is the same as for de Broglie field in non-relativistic regime.

3.3.

Transfer Matrices for s- and p-polarization Modes

A steep interface may be describe by the next limiting form of dielectric function,  for x3 > 0,  ε0 (ε + ε1 )/2 for x3 = 0, ε(x3 ) = ε0 + (ε1 − ε0 )θ(−x3 ) =  0 for x3 < 0. ε1

(35)

Substituting (35) for (25) and (26) leads to the next boundary value problems : 2 )p(x3 ) = −Vv[p](x3 ) for x3 6= 0 with (31), (−∂23 − k03 2 2 (−∂3 − k03 )s(x3 ) = −Vv[s](x3 ) for x3 6= 0 with (32),   ε1 − 1 k02 ζ(x3 ) (ζ = p or s). where Vv[ζ](x3 ) = −θ(−x3 ) ε0 2 Remark

(36) (37) (38)

that, even in the limit of steep interface, the interface region is a volume (line section in the present one-dimensional case) expressed by ∂V ⊗ (+0), where +0 means infinitesimal width including the steep interface. Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

Surface Phenomena Caused by Optical Spin-Orbit Interaction

445

The boundary condition (31) means a certain singularity of the field p(x3 ) across x3 = 0, while the boundary condition (32) means regularity of the field s(x3 ) across x3 = 0. The wave equation for s-polarization mode, (24) or (37) has no singularity at the interface and is essentially equivalent to time-independent Schr¨odinger equation. The above boundary value problems can be solved using the transfer matrix method. In the region V0 (x3 > 0), the general solutions of (36) and (37) may be expressed in terms of superposition of two independent solutions: p(x3 > 0) = α p exp(+ik03 x3 ) + β p exp(−ik03 x3 ),

(39)

s(x3 > 0) = αs exp(+ik03 x3 ) + βs exp(−ik03 x3 ),

(40)

2 k03

=

k02 − kk2 ,

(41)

where αs’ and βs’ are the amplitudes to be determined; see Fig.2(c). In the same way, the general solutions in V1 (x3 < 0) are: p(x3 < 0) = α0p exp(+ik13 x3 ) + β0p exp(−ik13 x3 ),

(42)

α0s exp(+ik13 x3 ) + β0s exp(−ik13 x3 ), ε1 k12 − kk2 , k12 ≡ k02 . ε0

(43)

s(x3 < 0) =

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2 k13

≡

(44)

In (39) and (42) concerning p(x3 ), there are four unknown quantities [Fig.2(c)] and the boundary conditions (31) provide two relationships between these unknown quantities. Therefore, the relationship between the field in V0 and that in V1 is represented by a matrix, namely, transfer matrix, T0←1,p . Considering (40) and (43) in the same way, the boundary conditions (32) concerning s(x3 ) leads to transfer matrix, T0←1,s. As the result, one may obtain  0    αp αp T0←1,p 0 = , (45) βp βp       ε0 k13 ε0 k13 1 1 1 + 1 − 2 ε1 k03  2  ε1 k03   , (46) where T0←1,p ≡   ε0 k13 ε0 k13 1 1 2 1 − ε1 k03 2 1 + ε1 k03  0    αs αs T0←1,s 0 = , (47) βs βs       k13 k13 1 1 1 + 1 − 2 k03  2  k03   . (48) where T0←1,s ≡   k13 k13 1 1 1 − 1 + 2 k03 2 k03

4.

Optical Phenomena Referring to Quantum Mechanical Concepts

In this section, quantum-mechanical concepts are used to describe Snell’s law, Fresnel’s formulas, and various optical phenomena in Fig.3. It is assumed that the incident light comes from material region. To know the case that the incident light comes from the vacuum region, what one should do is only to exchange indices for the vacuum and matter, i.e., ε0 ↔ ε1 , k0 ↔ k1 , and k03 ↔ k13 .

Plasmons: Theory and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook Central,

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