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Surface Plasmon Polaritons (SPPs) - Introduction and basic properties - Overview - Light-matter interaction - SPP dispersion and properties Standard textbook: - Heinz Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings Springer Tracts in Modern Physics, Vol. 111, Springer Berlin 1988 Overview articles on Plasmonics: - A. Zayats, I. Smolyaninov, Journal of Optics A: Pure and Applied Optics 5, S16 (2003) - A. Zayats, et. al., Physics Reports 408, 131-414 (2005) - W.L.Barnes et. al., Nature 424, 825 (2003) OverviewOverview Replace ‘slow’ electronic devices with ‘fast’ photonic ones Photonic crystals SPs is the way to concentrate and channel light using subwavelength structures SPs structures at the interface between a metal and a dielectric material Transverse magnetic in character Æ electric field normal to the surface Perpendicular direction Æ evanescent field Momentum mismatch in the SP dispersion curve OverviewOverview surface plasmon polariton optics (SPP) Band-gap effects, SPP waveguiding along straight and bent line, Gold stripes OverviewOverview enhanced optical transmission ElementaryElementary excitations excitations and and polaritonspolaritons Elementary excitations: • Phonons (lattice vibrations) • Plasmons (collective electron oscillations) • Excitions (bound state between an excited electron and a hole) Polaritons: Commonly called coupled state between an elementary excitation and a photon = light-matter interaction Plasmon polariton: coupled state between a plasmon and a photon. Phonon polariton: coupled state between a phonon and a photon. PlasmonsPlasmons A plasmon is the quantum of the collective excitation of free electrons in solids. Electron plasma effects are most pronounced in free-electron-like metals. The dielectric constant of such materials can be expressed as 2 ⎛⎞ω p εωm ()=− 1⎜⎟ ⎝⎠ω ω < ωp Æ εm < 0 Æ wavevector of light in the medium is imaginary Æ no propagating electromagnetic modes ω > ωp Æ εm→1 Æ altered by intraband transitions in noble metals A combined excitation consisting of a surface plasmon and a photon is called a surface plasmon polariton (SSP). Æ different nature, such as phonon–polariton, exciton–polariton, etc. PlasmonsPlasmons 23 −3 • free electrons in metal are treated as an electron liquid of high density n ≈10 cm • longitudinal density fluctuations (plasma oscillations) at eigenfrequency ω p 4πne2 • quanta of volume plasmons have energy , in the order 10eV hω p = h m0 Volume plasmon polaritons propagate through the volume for frequencies ω > ω p Surface plasmon polaritons Maxell´s theory shows that EM surface waves can propagate also along a metallic surface with a broad spectrum of eigen frequencies from ω = 0 up to ω = ω p 2 Particle (localized) plasmon polaritons PlasmonPlasmon resonance resonance positions positions in in vacuumvacuum + + + + Bulk ω p metal ε = 0 ---- Metal ++ -- ++ -- ++ -- surface ε = −1 ω p 2 drude model + Metal sphere + + ε = −2 ω p 3 localized SPPs - - drude - model SurfaceSurface Plasmon Plasmon Photonics Photonics Optical technology using Also called: - propagating surface plasmon polaritons • Plasmonics - localized plasmon polaritons • Plasmon photonics • Plasmon optics Topics include: Localized resonances/ - nanoscopic particles local field enhancement - near-field tips Propagation and guiding - photonic devices - near-field probes Enhanced transmission - aperture probes - filters Negative index of refraction - perfect lens and metamaterials SERS/TERS - surface/tip enhanced Raman scattering Molecules and - enhanced fluoresence quantum dots NanophotonicsNanophotonics using using plasmonic plasmonic circuits circuits Nanoscale plasmon waveguides • Proposal by Takahara et. al. 1997 hω Metal nanowire Diameter << λ • Proposal by Quinten et. al. 1998 hω Chain of metal nanoparticles Diameter and spacing << λ Atwater et.al., MRS Bulletin 30, No. 5 (2005) First experimental observation by Maier et. al. 2003 Subwavelength-scaleSubwavelength-scale plasmon plasmon waveguides waveguides Krenn, Aussenegg, Physik Journal 1 (2002) Nr. 3 SomeSome applications applications of of plasmonplasmon resonant resonant nanoparticles nanoparticles • SNOM probes T. Kalkbrenner et.al., J. Microsc. 202, 72 (2001) • Sensors • Surface enhanced Raman scattering (SERS) • Nanoscopic waveguides for light S.A. Maier et.al., Nature Materials 2, 229 (2003) Light-matter interactions in solids EM waves in matter Lorenz oscillator Isolators, Phonon polaritons Metals, Plasmon polaritons Literature: - C.F.Bohren, D.R.Huffman, Absorption and scattering of light by small particles - K.Kopitzki, Einführung in die Festkörperphysik - C.Kittel, Einführung in die Festkörperphysik EM-wavesEM-waves in in mattermatter (linear(linear media)media) -- definitions definitions Polarization Complex refractive index P = ε 0 χE with suszeptibility χ N = n + iκ = ε Complex dielectric function Electric displacement ε = ε′ + iε′′ = 1+ χ D = ε 0E + P = ε 0 ()1+ χ E = ε 0εE Relationship between N and ε ε′ = n2 +κ 2 ε′′ = 2nκ EM-wavesEM-waves in in mattermatter (linear(linear media)media) -- dispersion dispersion 2π E = E ei()kr −ωt wavevector k = 0 λ i()kr −ωt frequency ω = 2πf B = B0e Dispersion in transparent media without absorption: c c k and ε are real ω = k = k ε n ε >0 Dispersion generally: ω 2 ()k⋅ k = ε ε(ω) and thus k = k´ + ik´´ are complex numbers! c 2 i(kr−ωt )()i k′r−ωt −k′′r E = E0e = E0e e propagating wave exponential decay of amplitude HarmonicHarmonic oscillator oscillator (Lorentz) (Lorentz) modelmodel - iωt m&x& + bx& + Kx = eE = eE0e ω0 + e m iΘ eE x = 2 2 E = Ae ω0 −ω − iγω m ω0 p = ex = ε 0αE P = Np = ε 0 χE 2 ω p ε =1+ χ =1+ 2 2 ω0 −ω − iγω 2 ω0 = Km γ = bm 2 2 ωp = Ne / mε0 plasma frequency One-oscillatorOne-oscillator (Lorentz) (Lorentz) modelmodel N = n + iκ = ε (n −1)2 +κ 2 R = ()n +1 2 +κ 2 from Bohren/Huffman WeakWeak and and strongstrong molecular molecular vibrations vibrations weak oscillator: ε‘> 1 strong oscillator: ε‘< 0 2 1 25 25 eps' eps'' eps' eps'' 1,6 0,8 20 20 1,2 0,6 15 15 0,8 0,4 10 10 0,4 0,2 5 5 0 0 0 0 -0,4 -0,2 -5 -5 -0,8 -0,4 -10 -10 860 880 900 920 940 860 880 900 920 940 wavenumber / cm -1 wavenumber / cm -1 caused by: molecular vibrations crystal lattice vibrations examples: PMMA, PS, proteins SiC, Xonotlit, Calcite, Si3N4 OpticalOptical properties properties of of polarpolar crystalscrystals note: lattice has transversal T and longitudinal L γ = 0 oscillations but only transversal phonons can be excited by light A n = 0, κ = ε N = n + iκ = ε B n = ε , κ = 0 ω 2 A k = i ε ω c (n −1)2 +κ 2 ()k ⋅k = 2 ε R = c ω ()n +1 2 +κ 2 B k = ε c BBA i()kr−ωt A: total reflection E = E0e B: transmission and reflection SiCSiC - - single single oscillator oscillator model model γ > 0 GeneralGeneral dispersiondispersion for for nonconductor nonconductor ω = 0 resonances 0 restoring forces permanent dipoles ω > 0 e.g. water 0 phonons molecular vibrations DispersionDispersion inin polarpolar crystals crystals - - phonon phonon polaritons polaritons γ = 0 Dispersion relation for γ = 0 ω 2 ω 2 ω 2 −ω 2 between TO and LO k 2 = ε = ε ()ω L there is no solution for 2 2 s 2 2 real values ω and k c c ωT −ω frequency gap: E ∝ e−iωte−kr no propagation reflection strong coupling of mechanical and electromagnetic waves, polariton like photon like MetalsMetals --Drude Drude modelmodel (e/m)E m&x&+ bx& + Kx = eE → x = 2 2 ω0 − ω − iγω 1 γω A = ; θ = tan −1 2 2 2 2 ω2 − ω2 (ω0 − ω ) + (γω) 0 2 ωεp 0 E px=→==eNe P x 22 ωω0 −−iγω Ne2 K ω 2 ω 22==; ω p p 0 εω()=+ 1 22 mmε 0 ωω0 −−iγω PE()ω = εχω ()() ω 0 →=+ε ()ωχω 1 () DEEP()ωεεωωε==+00 ()() () ω () ω MetalsMetals -- drude drude model model - Intraband transitions ω ω - longitudinal plasma oscillations γ = 0 k = i ε k = ε - ω0 = 0 i.e. no restoring force c c - ω = plasma frequency i(kr−ωt ) P E = E0e 2 n = 0, κ = ε n = ε , κ = 0 ω p ε =1+ 2 2 ω0 −ω − iγω ω0 = 0 ω 2 ε =1− p ω 2 − iγω total reflection transmission ω 2 ω 2 ′ p p ε =1− 2 2 ≈1− 2 ω + γ ω ω >> γ = 1/τ (1/ collision time) collisions usually by electron-phonon scattering ω 2γ ω 2γ ε′′ = p ≈ p 2 2 3 generally: ω()ω + γ ω γ > 0 leads to damping of transmitted wave n > 0, κ > 0 AluminiumAluminium ωL = ωP ωT = 0 MetalsMetals --Drude Drude modelmodel εm (ωp)= 0 Æ longitudinal field Æ bulk plasma oscillation caused by Coulomb forces. if γ≠0, εm (ω)= 0 Æ ω = ωp+iγ/2 Æ plasmon is the quantum of plasma oscillation with energy ħ ωp and lifetime τ=2/γ. plasmon is not an electron but a collection of electrons. The damping constant γ is related to the average collision time 1/ τ Æ interactions with the lattice vibrations: electron-phonon scattering. non conductor = metals at high frequencies: intraband transitions acts mainly at low frequencies Æ Drude model as well. at frequencies >> ωp metals are transparent: ultraviolet transparency. MetalsMetals --Drude Drude modelmodel The Drude model needs to consider the effect of bound electrons Æ lower lying shells Equation of motion has to include also the “restoring force” ω 2 p mx++ bxαεω x = eE →Interband () =+ 1 22 ωω0 −−iγω α ω2 = 0 m FreeFree andand boundbound electronselectrons inin metalsmetals Bound electrons contribute like a Lorenz oscillator ε metal = ε drude + ε bound ε bound where 2 ω p,d ε drude =1− 2 ω − iγ dω ε drude 2 ω p, j ε bound = ∑ 2 2 j ω0 −ω − iγ jω MetalsMetals -- plasmon plasmon polaritons polaritons Plasmon polariton dispersion (γ = 0) ω 2 2 2 2 ω = ω p + c k ω = ck ω p E ∝ e−iωte−kr no propagation 0 k reflection DielectricDielectric function function of of metalsmetals andand polarpolar crystalscrystals Metal Polar crystal collective free electron oscillations (plasmons) strong lattice vibrations (phonons) no restoring force Reststrahlenband plasma frequency transversal optical longitudinal optical (longitudinal oscillation) phonon frequency, TO phonon frequency, LO Surface Plasmon Polaritons (SPPs) - Introduction and basic properties - Overview - Light-matter interaction - SPP dispersion and properties Standard textbook: - Heinz Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings Springer Tracts in Modern Physics, Vol.
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  • Double Negative Dispersion Relations from Coated Plasmonic Rods∗

    Double Negative Dispersion Relations from Coated Plasmonic Rods∗

    MULTISCALE MODEL. SIMUL. c 2013 Society for Industrial and Applied Mathematics Vol. 11, No. 1, pp. 192–212 DOUBLE NEGATIVE DISPERSION RELATIONS FROM COATED PLASMONIC RODS∗ YUE CHEN† AND ROBERT LIPTON‡ Abstract. A metamaterial with frequency dependent double negative effective properties is constructed from a subwavelength periodic array of coated rods. Explicit power series are developed for the dispersion relation and associated Bloch wave solutions. The expansion parameter is the ratio of the length scale of the periodic lattice to the wavelength. Direct numerical simulations for finite size period cells show that the leading order term in the power series for the dispersion relation is a good predictor of the dispersive behavior of the metamaterial. Key words. metamaterials, dispersion relations, Bloch waves, simulations AMS subject classifications. 35Q60, 68U20, 78A48, 78M40 DOI. 10.1137/120864702 1. Introduction. Metamaterials are artificial materials designed to have elec- tromagnetic properties not generally found in nature. One contemporary area of research explores novel subwavelength constructions that deliver metamaterials with both a negative bulk dielectric constant and bulk magnetic permeability across cer- tain frequency intervals. These double negative materials are promising materials for the creation of negative index superlenses that overcome the small diffraction limit and have great potential in applications such as biomedical imaging, optical lithography, and data storage. The early work of Veselago [39] identified novel ef- fects associated with hypothetical materials for which both the dielectric constant and magnetic permeability are simultaneously negative. Such double negative media support electromagnetic wave propagation in which the phase velocity is antiparallel to the direction of energy flow and other unusual electromagnetic effects, such as the reversal of the Doppler effect and Cerenkov radiation.