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Electromagnetic modes in cylindrical structures
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by Jakub Pritz.
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2008.
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Thesis (M.S.)University of South Florida, 2008.
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Text (Electronic thesis) in PDF format.
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ABSTRACT: Nanostructures have received much attention from the physical and engineering communities in the past few years. The understanding of the behavior of nanostructures in various conditions is warranted since the applications of such materials in optics, electronics, and mechanics is ever expanding. This thesis investigates a specific type of structure, a concentric cylindrical. More specifically, the dispersion relation of radiating and nonradiating plasmon polaritons (quasiparticles resulting from interactions of photons and surface electrons) is studied under varying conditions. We intend to show the influence of changing the thickness of the layers, the number of layers, the curvature of each layer, and the type of material the layers has on the dispersion relation.By first solving Maxwell's equations in cylindrical coordinates and applying boundary conditions, we developed a matrix equation through which we were able to obtain the dispersion relation for an N layered cylindrical system characterized by a specified dielectric function placed into a background. For the nonradiative modes we used the bisection method to obtain the dispersion relation; however, since radiative modes encompass virtual modes, which contain real and imaginary components, a Newton method was used to gather that data. The dielectric functions for silver and carbon dielectric functions were used to describe the material layers within the radiative and nonradiative regimes. The results show that curvature changes influence the surface plasmon polariton dispersion by either red shifting or blue shifting the energetics. Lifetimes and damping are seen to be influenced by the curvature as well.The addition of more layers to the system results in an increase in the complexity of the dispersion energetics. The results obtained would help provide better scanning tips within the optical microscopy field. Also, these results can have direct application to the field of photonics. Finally, these results also help provide the foundations to understanding the fundamentals of longranged forces in cylindrical layered structures.
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Mode of access: World Wide Web.
System requirements: World Wide Web browser and PDF reader.
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Advisor: Lilia Woods, Ph.D.
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Optical properties
Dispersion
Concentric layers
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Electromagnetic Modes in Cylindrical Structures by Jakub Pritz A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Physics College of Arts and Sciences University of South Florida Major Professor: Lilia Woods, Ph.D. Sarath Witanachchi, Ph.D. Ivan Oleynik, Ph.D. Date of Approval: November 13, 2008 Keywords: Plasmon, Nanophotonics, Optical Prop erties, Dispersion, Concentric Layers Copyright 2008, Jakub Pritz
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i Table of Contents List of Tables ii List of Figures iii Abstract iv I. Introduction 1 II. Electromagnetic Modes 5 III. The Model 11 IV. The NonRadiative Regime 19 V. The Radiative Regime 28 VI. Conclusions 37 References 40 Bibliography 44
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ii List of Tables Table 1 Characteristics of Different Plasm on Polariton Modes 7 Table 2 Constants used in the Silver Dielectric Function 19 Table 3 Constants used in the Carbon Nanot ube Dielectric Function 34
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iii List of Figures Figure 1. Illustration of Different M ode Types 8 Figure 2. Multiwalled Concentric Cylindr ical Layers 11 Figure 3. NonRadiative Dispersion Curves and Number of Layers 21 Figure 4. NonRadiative Dispersion Curves and Plasmon Frequency 23 Figure 5. NonRadiative Dispersion Curv es and Thickness 24 Figure 6. NonRadiative Dispersion Curv es and Changing Radii 25 Figure 7. NonRadiative Dispersion Curves and Changing one Radius 25 Figure 8. Radiative Ag Dispersion Curv es One Layers 29 Figure 9. Radiative Ag Dispersion Cu rves Two Layers 31 Figure 10. Radiative Ag Dispersion Curves Thr ee Layers 33 Figure 11. Radiation CN Dispersion Curv es for Various Ns 35
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iv Electromagnetic Modes in Cylindrical Structures Jakub Pritz ABSTRACT Nanostructures have received much atte ntion from the physical and engineering communities in the past few years. The unde rstanding of the behavior of nanostructures in various conditions is warrant ed since the applications of such materials in optics, electronics, and mechanics is ev er expanding. This thesis i nvestigates a specific type of structure, a concentric cylindrical. More sp ecifically, the dispersi on relation of radiating and nonradiating plasmon polaritons (quasiparticles resulting from interactions of photons and surface electrons) is studied under varying conditions. We intend to show the influence of changing the thickness of th e layers, the number of layers, the curvature of each layer, and the type of material the layers has on the dispersion relation. By first solving MaxwellÂ’s equations in cylindrical coordinates and applying boundary conditions, we developed a matrix equation through which we were able to obtain the dispersion relation for an N layered cylindrical sy stem characterized by a specified dielectric function placed into a background. For the nonradiative modes we used the bisection method to obtain the di spersion relation; howev er, since radiative modes encompass virtual modes, which c ontain real and imaginary components, a Newton method was used to gather that data The dielectric func tions for silver and carbon dielectric functions were used to descri be the material layers within the radiative and nonradiative regimes.
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v The results show that curvature change s influence the surface plasmon polariton dispersion by either red shifting or blue shif ting the energetics. Lifetimes and damping are seen to be influenced by the curvature as well. The addition of more layers to the system results in an increase in the co mplexity of the dispersion energetics. The results obtained would help provide be tter scanning tips within the optical microscopy field. Also, these results can have direct application to the field of photonics. Finally, these results also help provide th e foundations to understanding the fundamentals of longranged forces in cylindrical layered structures.
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1 I. Introduction Electromagnetic excitations have long been of interest to phys icists. Beginning in the 1940Â’s, Fano used a specific type of el ectromagnetic excitation, known as plasmon polaritons (PP), to explain the WoodÂ’s anomal ies seen from metal di ffraction gratings at optical frequencies (1). Ngai later extende d this work (2) which confirmed previous experiments performed by Spicer (3) and La nder (4). Meanwhile, Fuchs and Kliewer added to the theoretical understanding by st udying these excitations for ionic slab geometries (5, 6). Beck then used PPs to deve lop a dispersion relation for Al and Mg (7). Later, theoretical work took into account diffe rent geometries such as thin metallic films (8) and multilayers (9, 10). Today, PPs are still used for theoretical research when investigating plasmon hybridizat ion in spherical nanoparticles (11) and when considering surface defects (12). Experimentally, PPs were first seen in an electron energy loss experiment conducted by Powell, et al (13). Since then, PPs have been used in various experimental studies including: surface plasmon micr oscopy (14), biosensing (15), and electrochemistry (16). These excitations ha ve also been observed optically (17, 18) and thus have found use in characterizing and meas uring optical constants for metals (19). The miniaturization of technology to the na noscale regime is a fairly recent trend that has been gaining much interest. Materi als can be made such that importance on their spatial dimensionality becomes a factor for material properties. For example, in two dimensions superlattices and thin films can be created using different techniques (20, 21). While constructing superlattices nonlinear dielectric propert ies can be achieved making
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2 application in voltage tunable devices avai lable (22). One dime nsional nanotubes have been shown to have high tensile strength (23) Nanofibers, filaments, and wires (24) as well as quasione dimensional microtubules (2 5) and nanorods (26) have also been fabricated. Quasizero dimensi onal structures have been synt hesized with examples being nanospheres (27) and nanodots (28). Zero dimens ional structures are aptly suited in for use as fluorescence (29, 30) devices. Research into the optical properties of these nanostructures has motivated the development of more efficient and novel devi ces. In particular, there has been much work done on one and quasione dimensional na nostructures and their optical properties. The study of infrared gaps for carbon nanotubes ( 31) could lead to more efficient thermal imaging devices. Knowledge of the optical re sponse for nanocylinder arrays aids in the development of improved biological and ch emical sensors (32). Electromagnetic excitations have also been investigated but ma inly in simple cylindrical geometries such as: cylinders made from one material (33 Â– 36) and different dielectric layers (37 39). However, not much theoretical investigatio n has gone into varyi ng the geometry of cylindrical structures. With manufacturing techniques that can control the growth of single walled nanotubes (40), multiwalled nanot ubes (41), vary the dielectric materials within cylindrical layers (42), and apply thin film coatings on nanotubes (43), the need to develop an understanding of th e relationship between varyi ng the cylindrical geometry and PP dispersion is validated. Knowledge from these studies about elec tromagnetic modes has already led to development of useful devices. Optical swit ches (44), waveguides (45), and transmission of light through metallic layers (46) all rely on these modes. In fact, a term has been
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3 given to the area of research for devices buil t based on plasmon properties: plasmonics. PP modes also serve as the basi s for photonic circuits, the desi gn of electronic circuits in the nanoscale regime (47). As well as their applications, these electromagnetic excitations help to describe some fundamental forces of nature. The va n der Waals and Casimir forces can both be determined from the electromagnetic modes of the materials present (48, 49). More specifically, the interaction en ergy between objects can be de fined as the change in the zero point energy of the longitudinal electr omagnetic modes of the system. These longitudinal modes are characterized by surface ex citations (48). PPs also come into play when calculating reflectance, transmission, and absorption coefficients since these coefficients are effected by in teraction of surfac e topology with elect romagnetic waves. Therefore, a deep theoretical understand ing of PP modes is warranted given the importance it has presented itself to the scientific field. As stated before, fabrication of specifi c structures in th e one and quasione dimensional regime is now possible; however investigations of electromagnetic modes with varying cylindrical geometry have been limited. Electromagnetic modes also provide a fundamental role in the van der W aals and Casimir forces. Therefore, the goal of this paper is to provide a model for su rface plasmon polariton (SPP) dispersion under a variety of cylindrical geometrical conditions such as varying number of layers (from single layered to three layers), varying the thickness of each layer, changing the inner and outer radii, and/or changing the dielectric material. There are two regimes for which these conditions were applied: the radiativ e and nonradiative regimes of the dispersion relation. The nonradiative regime are for those excitations whose momentum, k is
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4 greater than an excita tion in vacuum that has the same frequency, ; or simply, the nonradiative regime is to the right of the vacuum light line while the radiative regime is to the left (48). With this work, one will be able to better understand the roles PP modes play in energy distribution for cylindrical structures and how this varies with geometry; thus, construction of devices with this type of symmetry can benefit from these results.
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5 II. Electromagnetic Modes Electromagnetic fields immerse th e world around us. They propagate via oscillations in the electric and magnetic fi elds as defined by MaxwellÂ’s equations: 0 ) ( ) ( r B r E t r E r B t r B r E ) ( ) ( ) ( ) ( E and B are the electric and magnetic fields, respectively. Both can depend on the position, r and frequency, of the excitation; is the permeability, is the permittivity, and is the free electron density. Qu anta of these electromagnetic oscillations are known as photons ; these are the particles that carry the electromagnetic interactions between objects. Photons travel indefinitely until they interact with a new medium, defined by a change in either , or Surfaces provide a natural place for these parameters to change, thus photons ma y interact with elements within this new medium. Light can transmit th rough without much change or even reflect off; however, due to electromagnetic interactio ns some of the photons inter act along a thin layer of the surface. A portion of the energy of these photons is now contained along the surface; therefore, interfaces between objects of different dielectr ic property are capable of supporting energy from these photonic interactions. Any sort of matter can support various type s of excitations and particle modes. For example, the unit cell is the base stru cture for any crystalline system and its oscillation in time defines the two types of phonons, optical and acoustic. An acoustic phonon is a quantized vibration of the unit cell, while optical phonons have the elements 1) 2) 3) 4)
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6 in the unit cell vibrating individually (50). Acoustic phonons always exist in medium, provided that the temperature does not equal zero. However, optical phonons require a finite amount of energy to be excited usually provided by electromagnetic radiation (48). Another type of collective excitation, plasmons, strongly depend on the property of the medium. They are defined as quantized oscillations of the electron density within the material. An example of a quasiparticle supported by conductive materials is an exciton. They are defined by a one to one coupling between an electron and a hole through a Coulombic interaction. A polariton results when a photon interacts w ith any charged excitation or particle in a quantized manner. These types of interactions occur when a photon changes a medium, i.e. passing through a surface or boundary. Continuity requires that the electric and magnetic fields on both sides of the in terface be equal; this is expressed by the following boundary conditions: 0 ) ( ) (1 2 1 2 n B B n D D K H H n E E n ) ( 0 ) (1 2 1 2 Where ,2 1 2 1 2 1E D ,2 1 2 1 2 1B H n is the unit vector perp endicular to the interface, is the surface charge density, and K is surface current. Interactions with different types of charged excitations in mediums are given specific names. For example, a polariton interacting with an optical phono n is called a phononpolar iton. Coupling an exciton with a polariton produces an excitonpolariton. Finally, when a quantized interaction occurs between a photon and a plasmon the name plasmonpolariton (PP) is given (48). 5) 6) 7) 8)
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7 The focus of this work is to investig ate PP interactions in a multilayered concentric cylindrical structure There are two general regimes where this occurs in matter. For PPs, interactions within the bulk of an object result in bulk modes, while interactions occurring on th e surface of an object result in what are called SPP. Bulk modes, such as bulk plasmons and phonons, arise when there is a nonzero longitudinal electric field, the electric field parallel to the direction of motion, and the longitudinal current is taken to be zero. In a zero transverse electric field, where the field vector points perpendicular to the motion of the wave, coupled with a zero transverse current leads to a dispersion of light within the medium. The frequency of the interaction along with its associated wave number defines the dispersion of an interaction. Bulk modes arise from only taking into considera tion MaxwellÂ’s equations within the medium; therefore, boundary conditions are not taken into consideration. When boundary conditions are taken into consideration surface modes arise. Table 1. Listing of the different m ode characteristics. (48) These boundary conditions, Eqs. 58, and the dielectric function defi ne all the properties of surface modes. As shown in Table 1, the dielectric function helps to characterize these modes. The real part of the dielectric func tion is associated with energy stored in the medium; the imaginary portion of the dielectri c function deals with dissipation. When there is no damping, Â” 0, in the system, Fano modes are the only means of energy Dielectric Function = Â’ + i Â” Mode Real Part Imaginary Part Fano Â’ < 1 Â” <<  Â’ Evanescent 1 < Â’< 0 Â”  Â’ Brewster Â’ > 0 Â” <<  Â’ Zenneck Â’ << 1 Â” >>  Â’
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8 propagation. Once damping is introduced, Brewster, Zenneck, and Evanescent modes replace Fano modes as means of energy propagation within the medium. The four modes, Fano, Brewster, Evanes cent, and Zenneck, are characterized by the dispersion relation, ) ( k (48). Brewster modes are bound to the surface of the medium. Evanescent modes decay rapidly. Zenneck modes are present in both the bulk and surface and so are rarely present in very thin materials. Fano modes are considered the true surface mode due to having a phase velo city less then the speed of light (48). Figure 1 The solid and dashed curves are the dispersions of the modes without and with damping, respectively, when one demands that frequency is real valued. If one instead demands that k/ T,1 is real valued the real part of the complex frequency follows the circles, even in the case of damping. (48) Figure 1) shows the areas where the diffe rent surface modes occupy. The solid and dashed curves are the dispersions of these modes without and with damping, respectively. The circles are the real part of the complex frequency when k or is taken to be real valued. The circles follow the same pattern even in the presence of damping. The = ck line is the light line.
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9 Because there are different characterizations for these modes, different applications can take advantage of the diff erent PP modes. For example, ellipsometry (technique for finding dielectric properties of thin films) relies on Brewster and Fano modes (51) since these modes are strictly bound to the surface. Evanescent modes are useful for constructing bandpass filters (52) ; meanwhile, Brewster and Zenneck modes can assist light transmission in metallic f ilms (53). Other optical microscopy techniques such as nearfield microscopy can rely on all the types of modes (54, 55). Since these modes are essentially ener getic modes at th e boundaries between objects, information about these modes will tell us much about the interactions between objects. For example, the stability of a st ructure can be gleaned from knowing how/what modes are propagating. Frictional forces be tween objects dictate ware and sheer stresses that occur at the interfaces of different materials. These fr ictional forces originate from electromagnetic interaction between the materi als. Adhesion is another force resulting from electromagnetic bonding of two objects (56) Specifically, these forces result from van der Waal interactions and as will be shown, the van der Waal force results directly from the electromagnetic modes. Vacuum fluctuations of the electromagne tic field in the vicinity of object boundaries give rise to surface ar ea dependent forces. To st udy these forces, one needs to account for the interaction ener gy of boundaries and see how it changes as a function of object separation. This is exactly the same as taking the summed zeropoint energy of electromagnetic modes and seeing how that changes as a function of object separation (56, 57). For example, in the van der Waal force one studies the interaction between
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10 unpolarized atoms or molecules; while the Ca simir force results from interactions of macroscopic objects (56). To derive these modes, one must first start by expressing the electromagnetic radiation in mathematical terms. MaxwellÂ’s equations, Eqs. 1 Â– 4, provide this. Solving these partial differential equations for the E and B field will give the correct form one needs to have an expression for the electrom agnetic radiation. The PP modes are then solved after the boundary conditions, Eqs. 58, have been applied to the electromagnetic expressions derived from Eqs. 1 Â– 4. The procedure described above is a genera l way of obtaining the PP modes. This manuscript focuses on cylindrical geometries; therefore, it is more natural to express the solutions to MaxwellÂ’s equations and the applied boundary conditions within a cylindrical coordinate format. The next sect ion will do this and describe the model used to obtain the dispersion relations for this cylindrical system. Geometry will play a key part in our model and influen ces of number of layers, curvat ure, and layer thickness will be described. Our model will take into co nsideration different dielectric functions as well.
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11 III. The Model Figure 2 Multilayer concentric cylindrical layer system. There are N layers with radius Rj, thickness i, and dielectric function i( The environment medium has a dielectric constant 0 The z axis is perpendicular to the page. Since the goal of this manuscript is to develop a model for PP dispersion for a cylindrical geometry it is only natural to use cy lindrical coordinates. Figure 2) displays the parameters we used to desc ribe the structure under investigat ion. It is a cross section of a multiwalled cylinder displayed azimuthally. RN represents the inner radius layer number N (counting from the inside to out). N represents the thickness for each layer, while N is the dielectric function for layer N 0 ) ( ) ( ) ( ) (2 2 2 r E r r c r E t ie r E r E ) ( ) ( ikz t ie e E r E ) ( ) ( 2 2 2 2) ( ) ( ) ( z r E r E r E 9) 10) 12) 11)
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12 As stated before, the solutions to Maxw ells equations will be solved for in cylindrical coordinates (r, z, ) The wave equation, Eq. 9, is derived from Maxwells equations, Eqs. 1 4. Next, we assume an electric field form that can be separated into spatial and time dependent values, Eq. 10, and has the form shown in Eq. 11. The Laplacian part of Eq. 9 is sp lit into its perpendicular and z parts as defined in Eq. 12. 0),()(22 tiikz jeeE Here j 2 = .22 2k cjj Using Eqs. 9 13, we find the solutions for the) ,(E and ),(B fields in each region j Each field has three components: , and z Results for ),(E and ) ,(B are: )( 0)}()({tkziin jjn j n n jjn j n j zeegF fCE ))()(())()(({' 0 2 jjn j njjn j n j jjn j n n jjn j n jj jgG fD i gF fC kn E )}()(())()(({' 0 2 jjn j njjn j n j jjn j n n jjn j n jj jgF fC ik gG fD n E )( 0)}()({tkziin jjn j n n jjn j n j zeegG fDB ))}()(( ))()(({' 0 2 jjn j njjn j n j jj jjn j n n jjn j n jj jgF fC i gG fD kn B )}()(())()(({' 0 2 jjn j njjn j n j jjn j n n jjn j n jj jj jgG fD ik gF fC n B Eqs. 14 19 show these results for each coordinate z, and for the electric and magnetic field (note the symbols for vect or representation have been dropped). j is the 14) 15) 16) 17) 18) 19) 13)
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13 radial vector for layer j (Figure 2)), j is the dielectric function, k is the wave number, and is the frequency. In this work we consider only nonmagnetic materials thus j = 1 everywhere. ,j nC ,j nD ,j nF andj nG are unknown coefficients wh ich will be solved for later. The terms f and g are substitutes for Bessel and Hankel functions with argument jj. Depending on the regime being considered, f and g represent different functions. More specifically, when in the radiative regime, ) ( ) (j j n j j nJ f and ) ( ) (1 j j n j j nH g ; for the nonradiative regime, ) ( ) (j j n j j nI f and ). ( ) (j j n j j nK g Consequently, ) (' j j ng g and likewise ) (' j j nf f Finally, n represents the order of the Be ssel and/or Hankel function used. As described earlier, the next step is to apply boundary conditions to solve for the coefficients: ,j nC ,j nD ,j nF and.j nG The boundary conditions for cylindrical coordinates are: ) ( ) ( ) ( ) ( ) ( ) (1 1 1 1 j j z j j z j j j j j j j j j jE E E E E E ) ( 1 ) ( 1 ) ( ) ( ) ( ) (1 1 1 1 j j z j j j z j j j j j j j j jB B B B B B Note, since there are only four coefficients, one needs to on ly use four of the boundary conditions to write equations for the coeffi cients. The result of applying the boundary conditions and solving for th e coefficients leads to: 20) 21) 22) 23) 25) 24)
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14 )} ) ( ) ( ) ( ) ( )( )) ( ) ( ) ( ) ( ( )) ( ) ( ) ( ) ( {( 11 1 2 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 j n j n j n j nG D F Cj j n j j n j j n j j n j j j j j j j j j n j j n j j n j j n j j j j j j n j j n j j n j j n j j j j j j nR g R g R g R f ( R k n i R g R g R g R g R g R f R g R f C )} G ) ( ) ( D ) ( ) ( )( F )) ( ) ( ) ( ) ( ( C )) ( ) ( ) ( ) ( {( 1j n 1 j n 1 2 1 1 2 1 1 j n 1 1 1 1 j n 1 1 1 1 1 1 j j n j j n j j n j j n j j j j j j j j j n j j n j j n j j n j j j j j j n j j n j j n j j n j j j j j j nR f R g R f R f ( R k n i R f R g R f R g R f R f R f R f F } G )) ( ) ( ) ( ) ( ( D )) ( ) ( ) ( ) ( ( ) F ) ( ) ( C ) ( ) ( )( 1 1 { 1j n 1 1 1 j n 1 1 1 j n 1 j n 1 2 1 2 1 1 1 j j n j j n j j n j j n j j j j n j j n j j n j j n j j j j n j j n j j n j j n j j j j j j nR g R g R g R g R g R f R g R f R g R g R g R f ( R ink D } G )) ( ) ( ) ( ) ( ( D )) ( ) ( ) ( ) ( ( ) F ) ( ) ( C ) ( ) ( )( 1 1 { 1j n 1 1 1 j n 1 1 1 j n 1 j n 1 2 1 2 1 1 1 j j n j j n j j n j j n j j j j n j j n j j n j j n j j j j n j j n j j n j j n j j j j j j nR f R g R f R g R f R f R f R f R f R g R f R f ( R ink G 26) 27) 28) 29)
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15 Where j+1=)()()()(1 1 1 1 jjnjjnjjnjjnRgRfRgRf These equations were solved for the outer layer coefficients in te rms of the inner layer coefficients. Similar results would be obtained in solving for the inside layer coefficients in terms of the outer layer coefficients; however, every j + 1 j and every j j + 1 . The coefficients can be arranged into the following matrix form: j n j n j n j n jj Gn jj Gn jj Fn jj Fn jj Dn jj Dn jj Cn jj Cn j n j n j n j nG F D C G F D Cj1,j Gn, j1,j Gn, j1,j Fn, j1,j Fn, j1,j Dn, j1,j Dn, j1,j Cn, j1,j Cn, ,1 ,1 ,1 ,1 ,1 ,1 ,1 ,1 1 1 1 1 Where the following substitutions have been made: ] [ 11 1 1 1 1 1 jjnjjnjjnjjn jj jj j j Cngf gf jjnjjn j j j j jj j j j Cngf in 1 2 1 1 2 1 1 1 1 ,1 jjnjjnjjnjjn jj jj j j Cngg gg 1 1 1 1 1 1 ,1 jjnjjn j j j j jj j j j Cngg in 1 2 1 1 2 1 1 1 1 ,1 jjnjjn jj j j j j Dngf in 1 2 1 2 1 1 1 ,11 1 jjnjjnjjnjjn j j j j Dngf gf 1 1 1 1 1 ,1 jjnjjn jj j j j j Dngg in 1 2 1 2 1 1 1 ,11 1 31) 36) 37) 34) 32) 35) 33) 30)
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16 jjnjjnjjnjjn j j j j Dngg gg 1 1 1 1 1 ,1 jjnjjnjjnjjn jj jj j j Fnff ff 1 1 1 1 1 1 ,1 jjnjjn j j j j jj j j j Fnff in 1 2 1 1 2 1 1 1 1 ,1 jjnjjnjjnjjn jj jj j j Fnfg fg 1 1 1 1 1 1 ,1 jjnjjn j j j j jj j j j Fnfg in 1 2 1 1 2 1 1 1 1 ,1 jjnjjn jj j j j j Gnff in 1 2 1 2 1 1 1 ,11 1 jjnjjnjjnjjn j j j j Gnff ff 1 1 1 1 1 ,1 jjnjjn jj j j j j Gnfg in 1 2 1 2 1 1 1 ,11 1 jjnjjnjjnjj n n j j j j Gnfg fg 1 ' 1 1 1 1 ,1 One can now write a matrix e quation that couples the first layer with the second, the second with the third, third w ith the fourth, etc. Therefore, one can couple the inner layer with the outermost layer and because of the si mple switching of indices one has a relation between the outermost layer and the inne rmost. For purposes of solving for the dispersion relation, it will be c onvenient to write the innermos t layer with itself going up through the layers and back down. The following shows this process to layer N and back. 46) 39) 43) 38) 40) 41) 42) 44) 45)
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17 1 1 1 1 1,2 Gn, 1,2 2,1 Gn, 1,2 1,2 Fn, 1,2 2,1 Fn, 1,2 1,2 Dn, 1,2 2,1 Dn, 1,2 1,2 Cn, 1,2 2,1 Cn, 1 ,2 j1,j Gn, 3,2 Gn, 2,3 2,3 j1,j Fn, 3,2 Fn, 2,3 2,3 j1,j Dn, 3,2 Dn, 2,3 2,3 j1,j Cn, 3,2 Cn, 2,3 2,3 2,12 Gn, 2,12 Gn, 2,12 2,12 2,12 Fn, 2,12 Fn, 2,12 2,12 2,12 Dn, 2,12 Dn, 2,12 2,12 2,12 Cn, 2,12 Cn, 2,12 2,12 12,2 Gn, 12,2 12,2 Gn, 12,2 12,2 Fn, 12,2 12,2 Fn, 12,2 12, 2 Dn, 12,2 12,2 Dn, 12,2 12,2 Cn, 12,2 12,2 Cn, 12,2 2,3 Gn, 2,3 Gn, 3,2 3,2 2,3 Fn, 2,3 Fn, 3,2 3,2 2,3 Dn, 2,3 Dn, 3,2 3,2 2,3 Cn, 2,3 Cn, 3,2 3,2 2,1 1,2 Gn, 2,1 2,1 2,1 1,2 Fn, 2,1 2,1 2,1 1,2 Dn, 2,1 2,1 2,1 1,2 Cn, 2,1 2,1 1 1 1 1 ... ... n n n n Gn Gn Fn Fn Dn Dn Cn Cn GnGn FnFn DnDn CnCn NNNNNN Gn NN Gn NNNNNN Fn NN Fn NNNNNN Dn NN Dn NNNNNN Cn NN Cn NNNN Gn NNNN Gn NNNN F n NN NN Fn NNNN Dn NN NN Dn NNNN Cn NNNN Cn GnGn FnFn DnDn CnCn Gn GnGn Fn FnFn Dn DnDn Cn CnCn n n n nG F D C G F D C For the nonradiative regime: ,2,1 ,, GFn ,2,1 ,, GFn ,1,2 GF,n, ,2,1 ,, GFn ,12,2 ,,,, NN GFDCn,12,2 ,,,, NN GFDCn ,2,12 ,, NN DCn ,2,12 ,, NN DCn ,2,12 DC,n, NN ,2,12 DC,n, NN ,1,2 ,,,, GFDCn and 1,2 G F, D, C,n, are all zero within Eq. 47. The zeroes are required from the contin uity of the modified Bessel functions at 0 and The limiting conditions are different for the radiative regime since the Bessel and Hankel functions used are substituted in for the f s and gs. Here ,2,1 ,, GFn ,2,1 ,, GFn ,1,2 GF,n, ,2,1 ,, GFn ,1,2 ,,,, GFDCn and 1,2 G F, D, C,n, are zero within Eq. 47. Performing the matrix multiplication simplifies to: 1 1 1 1 n n nn nn n nD C D C 1 11 0n n nn n nD C 147) 48) 49)
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18 The solution to Eq. 49 is either the trivial solution wh ere the coefficients are zero or the determinant of the matrix containing th e layer relation information being zero. The latter gives the dispersion re lation for the entire system. Note, that the equation above contains all the information about each layerÂ’ s dielectric properties, the thickness of each layer, the curvature for each layer, and the number of layers present.
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19 IV. The NonRadiative Regime This chapter discusses results obtained in performing calculations for the nonradiative regime for the plasmon polariton disper sion (58). This is the regime to the right of the vacuum light line; altern atively, this is the area where j 2 < 0 In the definitions for the coefficients in Eqs. 31 46, both k and are real, and j 2 = 2 2 2k cj j In this case, f=I( jj) and g=K( jj) in Eqs. 31 Â– 46. A number of parameters have been genera lized for this system For example, the system was imbedded in vacuum, thus 0( )=1 the system is nonmagnetic so is taken to be unity and the layer thickness, is uniform for all layers. Previous studies (59, 60) have achieved good results for dielectric response propertie s of silver nanostructures when the form of the silver dielectric function for the layer material was taken to be: i iL L p 2 2 2 2) ( ) ( Here, is dielectric constant of the material, p the plasmon frequency, and is the electronic lifetime. The last term in Eq. 50 takes into account interband transitions and is fitted in order to obtain agreement with experimental data: the frequency, L, the strength, and the spectral width of the Lorentz oscillator, p (THz) L (THz) (THz) bulk (THz) vF (m/s) A 3.91 13420 6870 12340 .76 33.3 1.4E6 3 Table 2. Values of the optimized parameters used in the dielectric function to fit experimental data for silver (59, 60). 50)
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20 There are a couple of limiting cases for th e dielectric function as described in Eq. 50. When all the damping parameters are taken to be zero, i.e. = 0 and L = 0 the dielectric function models a perfect conduc tor. When L = 0 but 0 interband transitions are not taken into account and the material. F bulkAv Eq. 51 shows the damping parameter used; is the layer thickness, vf is the Fermi velocity, A is a constant taken to fit experimental data (61, 62). The additional damping term, bulk, accounts for bulk damping; it accounts da mping from interactions of electronelectron and electronphonon. These additional scattering terms result from adjusting our system to fall within the nanoscale range wher e the size of our system is smaller then the electronic mean free path. Table 2 give s all the numerical values entered in ( ) for silver (59, 61). The damping parameters and make the dielectric function complex, thus the SPP frequency also becomes complex resulti ng in a time decay of the electric and magnetic fields. The real part of represents the energy of the SPP modes and the imaginary part of which has a negative sign, re presents the damping of the electromagnetic fields. Fi gure 3 shows the energy of th e SPP excitations supported by the concentric cylindrical laye rs. The damping of the SPP mode s is found to be at least three orders of magnitude smaller than the actual energy and its absolute value increasing with the increase of k Here we only focus on the real part of (k) 51)
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21 2 4 6 8 10 ceV 2 4 6 8 10 V e n0 n1 n2 2 4 6 8 10 ceV 2 4 6 8 10 V e n0 n1 n2 2 4 6 8 10 ceV 2 4 6 8 10 V e n0 n1 Figure 3 Energy dispersion of SPP modes as a function of wave vector k : a) one cylindrical layer ( N = 1) Â– R = 50 nm; b) two cylindrical layers ( N=2) R1= 10 nm, R2= 50 nm; and c) three cylindrical layers ( N = 3) R1 = 10 nm, R2 = 50 nm, R3 = 90 nmThe layer thickness is the same in each case nm The plasmon energy is taken to be the same in each case p = 14 eV The straight line represents the light lineck c) a) b)
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22 The relation between number of layers and its dispersion curves is illustrated in Figure 3. The thickness was chosen to be the same for each layer and for each graph. In the N = 1 case, Figure 3a), each order of Bessel function is shown to have two modes. As the number of layers increases ( N = 2 for Figure 3b) and N = 3 for Figure 3c)) the number of dispersion curves is shown to incr ease as well. This is attributed to the increase in the number of surfaces present, which in turn are able to support a larger number of modes. Similar results were obt ained in previous studies conducted for thin slabs (63), cylinders with a dielectric core (37), and coated spherical shells (64). Two sets of modes are clearly shown in Figure 3 for the one layered case. All the dispersions go to zero as the wave vector goes to zero as well. For one layer, each line is well defined; however, as the number of layers increase, the dispersions start to overlap, cross, or split suggesting strong interactions with the interfaces. A strong nonlinear behavior is seen at small wave vector values; nevertheless, th is behavior is not seen at larger wave vector values as the behavior becomes dispersionless. As N increases, this nonlinear regime extends over a larger k region. Different kinds of materials may influen ce the dispersion modes for a concentric cylindrical structure. To investigate this a simple Drude formula for the dielectric function without damping and interband transitions was taken. By varying the characteristic plasmon frequency, p, dependence on material can be investigated within the dispersion modes. 2 2) ( p 52)
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23 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 ceV 2 4 6 8 10 12 14 16 18 20 22 24 26 V e p9eVp16eVp24eVp35eV Figure 4 Energy dispersion fo r the lowest energy ( n = 0) SPP eigenmodes of a system with N = 2 layers for different values of the plasma frequency. The radii are R1 = 10 nm and R2 = 50 nm and the thickness for each layer is nm The light line ck is also given. Figure 4) shows the influence of varyi ng the plasmon frequency for the dispersion modes. Each layer was to take n to be the same material, same thickness, same curvature, and surrounded by free space. As p increases, it takes longer for the dispersion to plateau. This is attributed to the material acting more like a perfect conductor since the increase in p increases the density of electrons. Also, for larger values of p, the dispersion tends to stay somewhat parallel to the light line for longer periods in the wave vector. Ergo, for materials with larger plas mon frequency values, the excitations will experience retardation later when compared to materials with a smaller plasmon frequency. Previous studies have shown curvatur e to be an important influence on the dispersion modes in a given system (37, 65 67). For example, the dispersion energetics for PPs change when one varies the size of a cylinder with a dielectric core (37) and the size for spherical systems (67). Decreasing the radius for each ring in a cylindrical array
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24 2 4 6 8 10 ceV 2 4 6 8 10 y g r e n EV e __ ___ ___ ___ ___ ___.1nm.5nm1nm5nm8nm10nm Figure 5 Energy dispersion for the lowest n = 0 eigenmode for different thicknesses. The system consists of two concentric layers with radii R1 = 10 nm and R2 = 50 nm The plasmon energy is p = 14 eV The light line is ck imbedded in a metallic film has been shown to enhance optical transmission. Figures 5) and 6) will show that dispersion in the PP modes occurs for the system of concentric cylindrical layers. This is ascribed to curv ature being a prominent feature when varying thickness and radii for a layered system. Keeping the radii the same in a two laye red system, Figure 5) shows the result of changing the thickness of each layer by the same amount. Only the lowest order, n = 0 is shown but similar results are obtained for higher order branches ( n = 1, 2, Â… ). When the thickness is of magnitude smaller, the pl ateau in the dispersion occurs at a lower value for the wave vector. For thicknesses above half that of the sma ller radii, the plateau takes longer to get to and th e nonlinearity occu rs until a lager wa ve vector value. Generally, as the thickness increases the m odes for the PP excitations are pushed up into higher energetic values. Similar dependence on thickness have been shown for cylinders with a dielectric core (37). Unlike in that study, here we have ex amined a broader range and show greater variations in the dispersi on as the variation take s place over a larger thickness range.
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25 2 4 6 8 10 ceV 2 4 6 8 10 y g r e n EV e ___ ___ ___ ___ R110nmR240nm R130nmR260nm R150nmR280nm R170nmR2100nm Figure 6 Energy dispersion for the lowest SPP eigenmodes for the n = 0 (solid line) and n = 1 (dashed line) branches. The thickness of each layer is nm and the plasmon energy is p = 14 eV The light line is ck 2 4 6 8 10 c e V 2 4 6 8 10 y g r e n EV e a___ ___ ___ ___R110nmR2100nm R130nmR2100nm R150nmR2100nm R170nmR2100nm 2 4 6 8 10 c e V 2 4 6 8 10 y g r e n EV e b___ ___ ___ ___R130nmR240nm R130nmR260nm R130nmR280nm R130nmR2100nm Figure 7. Energy dispersion for the lowest n = 0 SPP eigenmodes: a) the inner radius is varied; b) the outer radius is varied. The thickness of each layer is nm and the plasmon energy is p = 14 e V. The light line is ck Another way to influence the curvature of our system is to change the inner radii for the layers. This was investigated for a two layered system and the thickness of each layer was kept constant. Figur e 6) demonstrates the effect of changing the inner radii of each layer for order n = 0 and n = 1 Both inner radii are taken to increase simultaneously. As the radii increases in th e system, a la the curvature of the system decreases, the dispersion modes of the system have less energy then before. This occurs for both orders of the modes as they are seen to be very close in energy even overlapping
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26 and crossing at certain point s. Comparing this result with that from changing the thickness, one sees the opposite effect occu rring. Therefore, tw o contending effects occur as one influences the curvature properties of a concentric multilayered cylindrical system. As a final analysis, we investigated th e effects of only changing one of the inner radii as the other was fixed. Thickness was al so kept constant duri ng the investigation for the lowest order of the two layered system. Figure 7a) shows the effects of changing the first inner radius, R1, while varying the second inner radius, R2. Figure 7b) displays shows the reverse, the first inner radius is fi xed while the second inner radius is allowed to vary. By varying the radii this way, it provides a model of spaci ng two layers either closer or farther apart. When the two layers are closer to e ach other, the di spersion mode is decreased and a large area of nonlinearity appears. Neverthe less, at large wave vector values each dispersion approaches the same va lue. Figure 7) also shows different effects occur if the curvature of the system is change differently. In particular, when the first inner radius is decreased and the second kept fixed, (k) shifts up in energy. The reverse happens when R1 is fixed and R2 is allowed to decrease; (k) decreases in energy. To sum up, several factors such as thickness of each layer, the radius of each layer, the overall radius of the system, a nd the proximity of the layers to each other influence the dispersion modes of a concentric cylindrical stru cture. As described earlier, varying these parameters leads to competing effects in the dispersion. Increasing the thickness of each layer (which brings th e layers closer together) increases (k) ; however, if one varies the inner radii, R1 and R2, in such a way as to bring the layers closer, (k) decreases. If the difference between the inner radii is kept constant ( R2 Â– R1 = constant ),
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27 but the overall curvature of the system decreases, (k) is suppressed. Similar results are obtained when one fixes R2 but allows R1 to increase; however, the opposite occurs to (k) when R1 is kept constant but R2 is allowed to increase. Knowing that geometry greatly influe nces the dispersion modes for PPs in multilayered concentric cylinders and that by varying itÂ’s geometry one can take advantage of higher or lower PP modes shoul d be a benefit to optical microscopic techniques. Scanning nearfie ld optical microscopy uses tips which work for a specific mode (54, 68). To scan other modes, the si ze tip is varied or a coating applied. The results from this investigation show that one layered cylinders only have two modes available; however, if one would were to use a multilayered tip an increase in the range of modes would be accessible. Also, if the tip were needed to access a specific frequency, all that would be needed to be done is use different initial polarizations ( n = 0, 1, Â… ) since the mode branches tend to be close or even overlap.
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28 V. The Radiative Regime The radiative regime deals with dissipa tive, radiating PP modes and a number of changes are needed within the model in order to accurately and aptly describe it. The radiative regime is the area to the left of the light line, = ck where j 2 = 2 2 2 c kj j The wave vector, k is still real; however, becomes complex, = Â’ + i Â” In this regime, f=J( jj) in Eqs. 31 46 and g=H1( jj ) ; here H1 is the first Hankel function H1=J( jj)+iN( jj). The complex transcendental equation (Eq. 49) is solved again; therefore, no analytical solution is possible. Nevertheless, numerical eval uations are possible and this section discusses the results obtained (69) for carbon and silver nanocylinders. For silver, variations in layer thickness, number of layers, and change of the inner radii were considered when performing the calculations For carbon nanotubes, number of layers and changing of the inner radii were considered. First, we calculate the dispersion relations for a single silver layer. We examine the role of the size of the radial dimensi on of the layer in terms of its thickness and diameter. The dispersion relations for the real and imaginary parts are shown in Figure 8), where only the lowest level modes for the first three n are given. The light lines for this structure are determined by the vacuum:kc (middle straight line), and by the silver material: kcAg ) ( (6). The energy for each mode increases as the thickness (Figure 8a) or the radius R (Figure 8b) of the layer increases, showing that as
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29 the curvature is decreased the modes approach the ones for a planar layer. Similar trend for Re of the radiative modes was found for a full cylinder imbedded in vacuum (36). For larger k the modes approach the silver light line indicating that they become more photonlike. The imaginary part of the solutions fo r the modes are shown in Figures 8c) and 8d). Our results display a trend towards developing a deeper and relatively well pronounced minimum for n = 0 as the layer becomes thicker, thus showing the increase of damping for these modes. The imaginary part  Im for n = 0 mimics a similar behavior found for the lifetimes of these mode s in a planar layer (6) when its thickness is increased. However, we find the opposite trend for  Im for n = 0 when the radius is increased, namely,  Im decreases as R increases showing a longer lived mode. 0 2 4 6 8 10 12 0123 0123 3.0 2.5 2.0 1.5 1.0 0.5 0.0 n = 0 = 1 nm n = 1 = 5 nm _ n = 2 = 10 nm . n = 0 R = 10 nm __ n = 1 R = 40 nm _ R = 60 nm . (a) (b) (c) (d) Im (eV) ck (eV) ck (eV) IRe ( eV ) Figure 8 a) Real, and c) imaginary parts of the dispersion for N=1 Ag layer for three values of the layer thickness and R = 10nm b) real, and d) imaginary parts of the dispersion for N = 1 Ag layer for several values of the radius and nm.
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30 It is interesting to see that for n = 1 there is a minimum in Im for a thicker layer while no such minimum is found for a thinner one Â– Figure 8c). Also, no such feature is obtained for the n=2 modes or for the n=1 for the specific in Figure 8d) when R is varied. In fact, for R=60 nm Im is very small, thus the mode becomes well defined and long lived. In previous work s (6, 36), the minimum in Im has been associated in some cases with the existence of a Brewster angle for the particular mode s. The Brewster angle is defined as the angle of incidence for which there is no reflected power of the electromagnetic waves in the structure (70). For cylindrical system s the angle is given as ) ( / tan2 2 2 2 2 k c k(36). If the positions of maxima of Im are in the close vicinity of the 2tan curve in the vs. k plane, then the mode is considered to be a Brewster mode. For n=0 ( = 10 nm ) and n=1 ( = 10 nm ) from Figure 8c), the modes minima are very close to the 2tan curve. The other minima for n=0 from Figure 8c) are found to be far away from 2tan curve. The n=0 ( = 10 nm) mode in Figure 8d) is the only mode seen to have a minimum close to the 2tan line. Our results show that both, the thickness and radius of the cylindrical layer, influence the radiative modes. A Brewster mode may or ma y not be found for a certain combination of R and This is unlike the results for the radiative modes of a full cylinder, where Brewster modes exist for all n (36), and unlike the cas es of planar layers for which the Brewster modes are dete rmined by the initial polarization (6)
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31 In Figure 9 we show the results for the first and second lowest laying n=0, 1, 2 modes of a system composed of two Ag laye rs for three values of the thickness. The energetics here appears more complicated as compared to the one layered system. This is attributed to the presence of more inte rfaces facilitating the scattering of the electromagnetic waves. An interesti ng behavior is found for the lowest n=0, 2 modes for all These start with relatively dispersionless Re at ~ 0.4 eV just below the light line for Ag, then cross the vacuum light line, c ontinuing into the regi me of surface plasmon polaritons. The n=0 modes are very close in energy for the different while the ones for n=2 show some differences in the ck = 0.51.3 eV region. These particular modes have made the crossover from wave guided to nonradiative localized at the surfaces of the cylindrical structure. The crossover happens at a particular value of the wave vector where the argument of the Bessel and Hankel functions, R, changes sign. Due to this argument sign change, a transformati on to the modified Bessel functionsnIand 0.0 0.4 0.8 1.2 1.6 000.20.40.60.81.01.21.41.61.8 2.0 1.5 1.0 0.5 n = 0 = 1 nm n = 0 = 5 nm n = 1 = 8 nm . n = 2 n = 2 Re (eV) Im (eV) ck (eV) (a) (b) Figure 9 a) Real part and b) imagin ary part of the dispersion for N=2 Ag layers for three values of the thickness. The radii arenmR 101andnmR 202
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32nKoccurs. Thus the dispersion relation equati on now describes the nonradiative regime, however is still complex, therefore solutions for Im are still possible. The lowest n=1 excitations do not exhibit this behavior, they are entirely radiative. Such guided to surface electromagnetic excitations have also been found for planar multilayered materials (71), where their existence depe nds on the combination of the dielectric properties of the different materials involved. Our results show that the curvature and the number of layers in the system are also important. Figure 9b) shows that  Im for all n shows no minimum in its behavior, except the guided n=0 exhibiting a shallow minimum at ck ~ 1.7 eV No Brewster mode is found here. Radiative modes with larger n are more heavily damped due to the larger values of Im The lifetime for the crossover modes for n=0, 2 for the different does not show much deviation. Im decreases to very small values after ck ~ 1.5 eV indicating their nonradiativ e nature. The fact that 0Im for these modes is explained by the existence of small losses in the Ag material due toD andD
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33 Next, we consider a system composed of three Ag cylindrical layers. The role of thickness and overall radius size is investigated. In Figure s 10a) and 10c), we show the dispersion relations of the two lowest laying n=0, 1, 2 excitations for three values of the thickness. One sees again that the crossove r wave guided to surface PP modes exist for n=0, 2, but not for n=1 as in the case for N=2 layers. For these,Re does not show much difference for the different Also, the thickness does not se em to influence significantly the low k regime forRe for the radiative n=0, 1 modes, where the energy is almost the same for all The overall size of the system affects the low k regime, where for the larger radii (Fig. 10b) the modes are higher in energy as compared to the ones for the smaller radii. At larger k they all become almost parallel to the Ag light line withRe being slightly higher. The results from Figure 10c) show that for some n and Im can have local minima at a specific wave vector k. For example, for the radiative n=0, a local minimum 0 1 2 3 4 0.00.71.42.12.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.00.71.42.12.8 n = 0 = 1 nm __ n = 0 = 5 nm n = 1 = 8 nm . n = 2 n = 2 n = 0 n = 0 n = 1 R1 = 10 nm R1 = 60 nm R2 = 20 nm R2 = 70 nm R3 = 30 nm R3 = 80 nm (a) (b) (c) (d) ck (eV) ck (eV) Re (eV) Im (eV) Figure 10 a) Real part, and c) imagin ary parts for the dispersion of N=3 Ag layers for several values of the layer thickness, R1 = 10 nm R2 = 20 nm R3 = 30 nm ; b) real, and d) imaginary parts of the dispersion for N=3 Ag layers for several values of the radii and =5 nm
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34 is found at ck ~ 0.65 eV = 1 nm and ck ~ 0.3 eV = 8 nm. On the other hand, for the radiative n=2 only a global mini mum is found at ck ~ 0.25 eV. One also sees that as the overall size of the system is varied from having larger curvature to having smaller curvature, the dispersion forRe does not change much. However, the modes are more heavily damped for the structure with larger curvature (smaller layer radii). Further, we consider concentric cylindr ical layers made out of carbon in an attempt to model the radiative PP excita tion spectrum in carbon nanotubes. Several studies related to mechanical and optical pr operties have shown that in many instances carbon nanotubes behave similarly as continuous cylindrical structur es (72 Â– 75). Thus it becomes eminent that such continuous models can be a relatively fa st tool for useful analysis of their properties. Here we assume that the separation between the layers in the carbon concentric cylindrical stru cture is of the same order as the equilibrium distance in multiwalled carbon nanotubes ~ 3.5 (72). Calculations for the dispersion spectra for other thicknesses ranging from = 1 nm to = 2 nm were done and the results showed to be very similar, therefore we present only the radiative PP for nm 5 1 For the dielectric function we also em ploy an effective dielectric function expression similar to the Eq. (50) with a combination of a Drude term and localized Lorentzian absorption peaks: j j j pj D Di i ) ( ) ( ) (2 2 2 2 c p (eV) pj (eV) j (eV) j (eV) (eV) 8.5 0.64 0.45 0.02 0.042 0.25 Table 3. Values of the parameters used in the dielectric function for carbon nanotubes (31). 53)
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35 The values for the frequency i ndependent dielectric constant plasma frequencyD carrier relaxation rateD Lorentzian central frequencypj strength of each Lorentzian oscillator pj, and spectral width of each Lorentzian oscillator j, are listed in Table 3, where the optical spectrum of a film of carbon nanotube was measured and modeled successfully using the effective expression from Eq. 53). Figure 11 displays the results for the lo west lying modes for a system with N=1, 2, 3 carbon layers. For one layer, crossover guided modes are found only for n=0, 2 However, for N=2, 3 such modes are found for all n. Some features such as local minima and maxima are found inIm for N=2 and N=3 layers. These are attributed to the presence of more scattering interfaces and their closen ess. Due to the smaller damping constant for the carbon system (58, 75), the modes tend to be less da mped as compared to those for silver layers with comparable sizes. These results also show that materials characterized with different dielectric f unctions can have different effect on the dispersion of the electromagnetic modes of a system with compatible structure characteristics. For example, for Ag N=1Im exhibits minima for different n, but for C N=1 ,Im does not with the excep tion of the second n=0 mode as seen in Figure 11b). 0 2 4 6 0.00510152025 10 08 06 0.4 02 n = 0 R = 10 nm n = 1 n = 2 0 1 2 3 0.00.51.01.52.02.5 0.75 0.50 0.25 R1 = 10 nm R2 = 14 nm 0 1 2 3 000.51.0152.02.5 15 10 05 R1 = 10 nm R2 = 14 nm R3 = 18 nm ck ck ck Re Im a) b) c) d) e) f) Figure 11 Real part of the dispersion for a) N = 1 ; c) N = 2 ; e) N =3 layers. Imaginary part of the dispersion for b) N = 1 ; d) N = 2 ; f) N = 3 layers. The thickness in each case is = 1.5 nm
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36 Also, for N=3 all studied modes of the Ag syst em show significant damping over a shorter wave vector range as compared to the ones for a C syst em. The broad minima found in Figures 11d) and 11f) for some n arenÂ’t associated with the2tan curve.
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37 VI. Conclusions In conclusion, I have presented a simple model to calculate the energy dispersion relation of the radiative and nonradiative plasmon polariton modes of a multilayered infinitely long cylindrical system. The final expression for the dispersion is obtained by solving the MaxwellÂ’s equations with appropr iate boundary conditions and it is written in a matrix form. It takes into account the finite speed of light, the di electric properties of the materials, the number of la yers, and their curvature. The general expressions presented here can be used for other studies as well. For example, all or some layers can be made to have different thickne ss and/or different dielectric properties (different p). The system can also be submerged in to an environment with 0( ) 1 Thus multilayered cylindrical structures provide many ways of tailoring the optical properties of nanostr uctures demonstrating their versatility in optical applications. The results from the nonradiative regime show that the surface plasmon polariton spectrum becomes more complex as the number of layers increases. In the multilayered cylindrical system, the energy subbranches for each branch n are a result from strong interaction between electromagnetic oscillati ons from all interfaces Many of the modes from different n can cross each other, split into subbran ches at a particular wave vector or overlap. Such complexity, particularly th e closeness of the branches, might make it difficult to distinguish them experimentally. On the other hand, if a specific mode range is needed, the excitation can be done with different initial polarizations. The numerical
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38 evaluations for the N = 2 case indicate that varying th e dielectric properties, the thickness, and/or the size of one or both layers can ch ange the SPP modes energy significantly. We demonstrate that the SPP ener gy can be redshifted or blueshifted in different ways simply by modifying the size and curvature of the systems. For the radiative regime, specific calculations were done for a system composed of silver or carbon layers. The dielectric function for each material is taken from available experimental measurements, a nd it includes a Drudelike contribution and interband optical transitions. We find that some modes are more heavily damped than other. Also, for some systems crossover mode s from the radiative region to the guided region can be found. Our results show that th e energy spectra are complex resulting from the interplay between the numbe r of layers, their proximity and the dielectric function parameters, but due to the rather technically involved expression for the energy spectrum implicit conclusions are not possible. Howeve r, our method can be easily applied to any system consisting of cylindrical layers submer ged into a medium with different structural and dielectric characteristics. Thus many ways of tailoring th e plasmon polariton properties of such nanostructures can be achieved for appropriate experimental applications. These kinds of cylindrical structures may find applicati ons in scanning nearfield optical microscopy where they would be used as the scanning tips. There are many advantages as indicated by our results for doing this: mo re modes are available for excitations, thus greater range of operation; a range of excitations can be excited by different initial polarizations; blue shifting and red shifting if the dispersion spectra can be achieved in many more ways as compared to using just coated cylindrical tips, thus
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39 there are many more possibilities for engine ering the system for the desired range of operation.
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