Revisiting Fresnel & refractive ine What is the refractive ine of a ielectric Metals an plasmons
Squeezing plasmons in a nanowire Moe with 150 nm SPP l < 1 mm At l 1.550 mm
Snell s law Generic solution steps: Step 1: Whenever translation invariance: Use to fin allowe refracte wave vectors conservation together with
Sketch of k conservation k conservation: The only way for the phase fronts to match everywhere, any time on the interface
Amplitue s-polarization Remember Now eliminate t to obtain reflection coefficient r (equal m)
Amplitue s-polarization The parallel components Remember of E an H are continuous across the interface at any location on the interface, & any time Now eliminate t to obtain reflection coefficient r (equal m)
What you see from this problem Scattering: incient fiel (plane wave) is split by object Reflections: are specular whenever translation invariance rules Refraction: Snell s law is just wave vector conservations Total internal reflection: if wave vector is too long to be conserve across the interface Bounary conitions etermine everything to o with amplitue
What s oes nature give us? Why?
Optical materials Optics eal with plane waves of spee with Metals: reflective Insulators: transparent
Refractive ine Insulators 4 3 B Si GaAs TiO (pigment) Density raises Semiconuctors help 1 0-1 Water Silicon nitrie Si 3 N 4 glass SiO Metamaterial (Nature (008)) 0.4 0.7 1.0 1.3 1.6 1.9 Wavelength (micron) All s between 1 an 4 Spoof (later class)
Dielectrics Dielectric materials: All charges are attache to specific atoms or molecules Response to an electric fiel : Microscopic isplacement of charges Macroscopic material properties: electric susceptibility, ielectric constant (or relative ielectric permittivity)
Atomic polarization Equation of motion of electron: : amping coefficient for given material : restoring-force constant resonance frequency Assume is varying harmonically, an also
Typical solis multiple resonances for electrons per molecule: Where is the oscillator strength or (quantum mechanically) the transition probability is a comple number:
Optical materials Optics eal with plane waves of spee with c/ n n m Metals: reflective Insulators: transparent
Problem with trapping light Scalar wave equation for light Schröinger equation V < U Confine states U < V Light states are always above the potential maimum in ielectrics Trap requires either interference or negative (metals plasmonics)
What o we know about metals? Metals contain free electrons in an ionic backbone: plasma Static electric fiels o not penetrate into metals. Fiels are shiele by surface charges Questions: - What is the permittivity of a metal at high frequencies? - How quickly can charges move to shiel fiels?
Surface charge Plasma frequency free charge + oscillation Suppose we isplace the whole electron gas by a istance n electrons at ensity N n/z ions - Surface charge Collective plasma oscillation: frequency
Quick estimate Silver: 1 electron per atom Mass ensity: 10.5 g/cm 3 Atomic weight: 108 g/mole N=5.8 10 8 e- per m 3 e=1.6 10-19 C m=9 10-31 kg 0 =8.8 10-1 F/m p Ne m 0 p = 1.37 10 16 s -1 l p = 140 nm Actual value renormalize by m *, effective mass of conuction electrons
Drue moel Drue moel: conuction electrons with amping: equation of motion ee 1 0 it e Free electrons: no restoring force t τ t m For the conuctivity s we fin the Drue moel : σ0 j Ne σ E σ = t 1 iωτ σ 0 = Ne m Drue moel: - The DC conuctivity is set by the ensity of electrons - The AC conuctivity rops for frequencies approaching the electron relaation rate τ
Converting conuctivity to Compare Mawell in two forms (without/with currents): D E H 0 H t t iσ iσ 0 / 0 p 1 1 1 ω 0 ω 1 iωτ ω iω j ω 0 E t with collision rate g1/t an plasma frequency p Ne m 0 Special frequency scale epening on metal
Measure ata an moel for Ag 50 0-50 -100 Measure ata: ' " Drue moel: ' " Moifie Drue moel: ' -150 00 400 600 800 1000 100 1400 1600 1800 " Wavelength (nm) ' Drue moel: ' 1 ' p, p, " " p g Moifie Drue moel: p Contribution of boun electrons Ag: 3. 4 g
Take home message: 1) A ielectric has > 1, real Boun charges ) A perfect metal has = -, real perfect screening by free charges 3) A real metal has Re < 0 up to the UV/visible 4) Im signifies loss. For metals this is Ohmic resistance
From plasmon to plasmonics + + + - - - Plasmons in the bulk oscillate at p etermine by the free electron ensity an effective mass rue p Ne m 0 k + - + Plasmons confine to surfaces that can interact with light to form propagating surface plasmon polaritons (SPP) rue surface 1 Ne m 0 Confinement effects result in resonant SPP moes in nanoparticles rue particle 1 3 Ne m 0
Photonics with guie moes Confining light in 1 or D to guie it along a plane or line
k z -ais 4 k -ais n slab /c n cla /c Guie moe if: - In the high ine meium k z >0 - In the low ine meium k z <0 -eponential tail
Dielectric slab z Assume eponential outsie, oscillatory in the slab Symmetric moes Asymmetric moes
Dielectric slab 10 5 B /k z tan k z 0 Symmetric moes -5 /c) beyon is k z imaginary -10 0 4 k z A symmetric ielectric D slab has a symmetric guie moe for any thickness
Dielectric slab 10 B Micro- photonics 0 Symmetric moes Low-loss transport, but: -10 0 4 Vertical confinement is limite to l/ Wavelength is boune by the ielectrics k z Hence, tight bens are leaky & chips are big A symmetric ielectric D slab has a symmetric guie moe for any thickness 5-5 /k z tan k z Dielectric waveguies are wiely use in /c) beyon is k z imaginary
Squeezing plasmons in a nanowire Moe with 150 nm SPP l < 1 mm At l 1.550 mm
Surface plasmon polariton Polariton: light couple to a material resonance Plasmon polariton: EM wave couple to plasma oscillations t z k k i z e E t z E 0,,, t z k k i m m z e E t z E 0, ),, ( For propagating boun waves: - k is real - k z is imaginary z
Dispersion relation Derivation of surface plasmon ispersion relation: k() Wave equation: 1 m, E E m,, m 0 0, m H H, m, m We assume a transverse magnetic moe H is perpenicular to k an parallel to the surface 0 H, m, z, t H y,, m e 0 c m, z t i k k zt
Dispersion relation Wave equation gives z,, m y,, m, m y,, m ( k k ) H H c Bounary conitions H H y, y, m Parallel H continuous E, m k 0 k H 1 1 0 H 0 k m, z y y m,, m z 0 k H y Parallel E continuous across interface
Circles kz, ( k ) c Dielectric: equation of a circle Also: imaginary k z for very large k Metal: equation of a circle with an imaginary raius simply means: always eponentially confine in some irection
Dispersion relation Wave equation gives Bounary conition yiels kz, ( k ) k k z, z, m c kz, m m ( k ) c m How to fin a solution? Reshuffle an fin k k k z, z, m m m c c k k
Dispersion relation -irection: z-irection: m k k ' ik " c m kz, k ' z, ik " z, c m 1/ 1/ Boun SP moe: Requirement 1: k z imaginary: m + < 0, Requirement : k real: m < 0 Conclusion: m < -
1/ " ' m m c ik k k -irection: c k Note: in regular ielectric: Dispersion relation Conclusion for the wave vector along the plane For a lossless metal ( m real, negative), this implies a guie, propagating moe as long as 0 m <
Measure ata an moel for Ag 50 0-50 -100 Measure ata: ' " Drue moel: ' " Moifie Drue moel: ' -150 00 400 600 800 1000 100 1400 1600 1800 " Wavelength (nm) ' Drue moel: ' 1 ' p, p, " " p g Moifie Drue moel: p Contribution of boun electrons Ag: 3. 4 g
Surface plasmon ispersion relation: ck Weakly boun Almost free photons p 1 p k Raiative moes Strongly ' m > 0) boun Very plasmonic Quasi-boun moes < ' m < 0) m c m real k real k z 1/ z Dielectric: Metal: m = m ' + m " Boun moes (' m < ) real k imaginary k z Re k
Surface plasmon ispersion relation: p ck Raiative moes k ' m > 0) m c m real k real k z 1/ p 1 z Dielectric: Metal: m = m ' + m " Boun moes (' m < ) real k imaginary k z Re k
Further stuff to remember 1. Moe is truly transverse magnetic. Moe has strong longituinal E-fiel 3. Moe has E-fiel normal to substrate Note that k k z, / m 1/ E k m, z m, 0 Moe sticks out a istance ~ l/5 or so from substrate In the metal, the penetration epth is just about the skin epth k 1 zm, kh H y y 1 ~ m
Surface plasmon ispersion relation: ck k m c m large k small wavelength 1/ 3.4 ev (360 nm) Ag/SiO Ar laser: l vac = 488 nm l iel = 387 nm l SP = 100 nm X-ray wavelengths at optical frequencies Re k
Even better at this Insulator-metal-insulator Metal-insulator-metal
Converse geometry: MIM Symmetric moe only Very ispersive Relatively low loss
Surface plasmon photonics Eperimental evience for surface plasmons What can we o with surface plasmons?
Energy lost, DE 1. Driving with electrons Surface plasmon polariton: light + charge oscillations 75 kev EELS on 16 nm thin aluminium Drive by e - Bulk plasmon +-+-+-+-+- Surface plasmons Dk DE Pettit, Vincent & Silco, PRB 1975 Phonons Ballistic Deflection angle ~ Dk
. Coupling to light Problem: wave vector of plasmon is too long for free photons +-+-+-+-+- Light can only couple in if How can we offer bigger wave vectors?
. Coupling to light Matching the wave vector of plasmons Prism coupling Attenuate total reflection Minimum traces (,k SPP )
Prism coupling Ecite air-sie plasmon from the glass sie
SPR bio-sensing 10+ companies: Biacore Biosensing instr. Gol with protein-selective funtionalization Senses D of ielectric sie
Plasmons guie in a strip Light converte into SPPs guie along Au strips on glass (40 nm high, l=800 nm) Boun & confine in irection normal to the strip Weeber et al. / Dijon
Squeezing plasmons in a nanowire Moe with 150 nm SPP l < 1 mm At l 1.550 mm
Metal-Insulator-Metal Measure l SPP ~ l/1
Hybri guiing geometries Zhang group Berkeley Low-ine gap of < 5 nm high Moe area 1% of cyliner area Worl s most compact laser