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Bề mặt plasmon
Excitation Lecture 10: Surface Plasmon Excitation nm Summary The dispersion relation for surface plasmons • Useful for describing plasmon excitation & propagation This lecture: ! !p !sp k sp Coupling light to surface plasmon-polaritons • Using high energy electrons (EELS) • Kretschmann geometry k //, SiO2 ! = " d sin # = k sp c • Grating coupling • Coupling using subwavelength features • A diversity of guiding geometries E kSiO2 H Θ k sp Dispersion Relation Surface-Plasmon Polaritons Plot of the dispersion relation 1/ "# ! ! $ • Last page: k x = % m d & c ' !m + !d ( !r • Plot of the dielectric constants: !d • Note: k x ! " when εm = −εd "! d Define: ω = ωsp when εm = −εd 1/ & ! ! ' " • Low ω: k x = lim ) m d * c ! m #$% + ! m + ! d , Dielectric Metal !p !sp kx = " ( !d c ! !sp ! ! "d c Light line in dielectric • Solutions lie below the light line! (guided modes) kx Dispersion Relation Surface-Plasmon Polaritons Dispersion relation bulk and surface plasmons !SP , Air "SP ,! Metal/air Metal/dielectric with εd d • Note: Higher index medium on metal results in lower ωsp !p ω = ωsp when:" m = # = #" d ! 2 p ! # ! = #" d ! 2 !p !2 = 1+ "d != !p 1+ "d Excitation Surface-Plasmon Polaritons (SPPs) with Light Problem SPP modes lie below the light line • No coupling of SPP modes to far field and vice versa (reciprocity theorem) • Need a “trick” to excite modes below the light line Trick 1: Excitation from a high index medium • Excitation SPP at a metal/air interface from a high index medium n = nh ! !sp ! =c k !e ! c = k nh kh > ksp k Air k sp kh k • SPP at metal/air interface can be excited from a high index medium! • How does this work in practice ? Excitation Surface-Plasmon Polaritons with Light Kretschmann geometry (Trick 1) k sp • Makes use of SiO2 prism k //,SiO2 • Create evanescent wave by TIR θ • Strong coupling when k//,SiO2 to ksp From dispersion relation E kSiO2 H • Reflected wave reduced in intensity ! !sp ! =c k ! c = k n !e Note: we are matching energy and momentum k Air k sp k SiO2 k Surface-Plasmon is Excited at the Metal/Air Interface Kretschmann geometry k sp , Air Θ E ! • Makes use of SiO2 prism k sp , SiO2 • Enables excitation surface plasmons at the Air/Metal interface k SiO2 H • Surface plasmons at the metal/glass interface can not be excited! Light line air Light line glass !sp , AIR !sp , SiO2 Surface plasmon Metal/Air interface Surface plasmon Metal/Glass interface !e k //, SiO2 = " d k Air k SiO2 k sp , Air k sp , SiO2 k ! sin # = k sp c Quantitative Description of the Coupling to SPP’s Calculation of reflection coefficient • Assume plane polarized light • Find case of no reflection d E • Solve Maxwell’s equations for • Solution (e.g transfer matrix theory! ) R= Plane polarized light p are ik where r the amplitude reflection coefficients p r p E E D H = p 01 p 12 p p 01 12 r + r exp (2ik z1d ) + r r exp (2ik z1d ) "k k # "k k # rikp = % zi $ zk & % zi + zk & ' !i !k ( ' !i !k ( Also known as Fresnel coefficients (p 95 optics, by Hecht) Notes: Light intensity reflected from the back surface depends on the film thickness There exists a film thickness for perfect coupling (destructive interference between two refl beams) When light coupled in perfectly, all the EM energy dissipated in the film) Dependence on Film Thickness Critical angle Θ E R H laser detector θ Raether, “Surface plasmons” • Width resonance related to damping of the SPP • Light escapes prism below critical angle for total internal reflection • Technique can be used to determine the thickness of metallic thin films Quantitative Description of the Coupling to SPP’s Intuitive picture: A resonating system ' • When ! m >> !r …well below ωsp: "! d • and '' ' ! m