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Home , Dispersion relation

Lecture 9: Introduction to Metal

5 nm What happened at the previous lectures ?

What happened at the previous lectures ? Light interaction with small objects (d < λ)

• Insulators (Rayleigh Scattering, blue sky..)

• (Size dependent absorption, fluorescence..)

• Metals…Resonant absorption at ωsp

Microparticles

• Particles with d ≈ λ (λ-independent scattering, white clouds)

• Particles with d >> λ (Intuitive ray-picture useful)

Dielectric photonic crystal

• Molding the flow of light Metal Optics: An Introduction

Metal Optics: An Introduction Majority of optical components based on dielectrics • High speed, high bandwidth (ω), but… • Does not scale well Needed for large scale integration Problems

Bending losses Diffraction Limit

nCLAD nCORE

Optical mode in > λ0/2nCORE

Solutions ?

Some fundamental problems!

Some: Photonic functionality based on metals?!

J. D. Joannopoulos, et al, Nature, vol.386, p.143-9 (1997) Plasmon-

Plasmon-Polaritons What is a plasmon ?

• Compare electron gas in a metal and real gas of molecules

• Metals are expected to allow for electron density : plasmons Bulk plasmon • Metals allow for EM propagation above the plasma They become transparent! Surface plasmon z E Strong local field Dielectric

Metal I H Note: This is a TM wave

• Sometimes called a surface plasmon- (strong coupling to EM field) Local Field Intensity Depends on

Local Field Intensity Depends on Wavelength

z z

D << λo

I I

Long wavelength Short wavelength

Characteristics plasmon-polariton • Strong localization of the EM field • High local field intensities easy to obtain

Applications: • Guiding of light below the diffraction limit (near-field optics) • Non-linear optics • Sensitive optical studies of surfaces and interfaces • Bio-sensors • Study film growth • …… Plasmon-Polariton Propagation in Au rod

Plasmon-Polariton Propagation in Au rod

Laser excitation λ = 532 nm

8.1 µm Au rod

Light at the other end

R.M. Dickson and L. A. Lyon, J. Phys. Chem. B 104, 6095-6098 (2000) Plasmon-Polariton Excitation using a Launch Pad

Plasmon-Polariton Excitation using a Launch Pad

J.R. Krenn et al., Europhys.Lett. 60, 663-669 (2002) 50 nm

• Array of 50 nanometer diameter Au particles spaced by 75 nanometer • Guides electromagnetic energy at optical frequency below the diffraction limit • Enables communication between nanoscale devices • Information transport at speeds and densities exceeding current electronics

M.L. Brongersma, et al., Phys. Rev. B 62, R16356 (2000) S.A. Maier et al., Advanced materials 13, 1501 (2001) Purdue Near-Field Optical Microscope

• Nanonics MultiView 2000 • NSOM / AFM • Tuning Fork Feedback Control – Normal or Shear Force • Aperture tips down to 50 nm • AFM tips down to 30 nm • Radiation Source – 532 nm Picture taken from Nanonics APD

To Piezoelectrics HV Source Filter Freq. Gen. Obj. Lens

Upper Piezo Fiber Tip 532nm Source

Lock-In Detector Lower Piezo Tuning Fork PID Controller Obj. Lens

Filter Computer

APD Enhanced Transmission through Sub-λ Apertures

EnhancMede tTarl aonpstimcsis wsilol nm tahkreo uyoguhr S dureba-λm As pceormtuer etrsue….

• Ag film with a 440 nm diameter hole surrounded by circular grooves • Transmission enhancement of 10 x compared to a bare hole • 3x more light than directly impingent on hole !

• Reason: Excitation of plasmon-polaritons

T.Thio et al., Optics Letters 26, 1972-1974 (2001). Optical Properties of an Electron Gas (Metal)

Optical Properties of an Electron Gas (Metal) Dielectric constant of a free electron gas (no interband transitions) • Consider a time varying field: E(t)= Re{E(!)exp("i!t)} d 2r • Equation of motion electron (no damping) m 2 = !eE 2 dt d p 2 m 2 = e E • Dipole moment electron p(t)= !er(t) dt

• Harmonic time dependence p(t)= Re{p(!)exp("i!t)}

• Substitution p into Eq. of motion: "m! 2p(!)= e2E(!) e2 1 • This can be manipulated into: p(!)= " 2 E(!) m ! Np ! Ne2 1 ! 2 • The dielectric constant is: ( ) p " r =1+ # = =1$ 2 =1$ 2 " 0E(!) " 0m ! !

! r

! p ! Relation for EM Waves in Electron Gas

Dispersion Relation for EM Waves in Electron Gas Determination of dispersion relation for bulk plasmons "2 E r,t • The is given by: ! r ( ) 2 2 2 = # E (r,t) c "t • Investigate solutions of the form: E (r,t)= Re{E (r,!)exp(ik "r # i!t)}

2 2 2 ! " r = c k 2 2 2 " ! p # 2 2 2 2 ! p ! %1$ 2 & = ! $! p = c k • Dielectric constant: " r =1# 2 % ! & ! ' ( ! = ! 2 + c2k 2 • Dispersion relation: ! p ! = ck

! p

No allowed propagating modes (imaginary k) k Note1: Solutions lie above light line

Note2: Metals: ħωp ≈ 10 eV; Semiconductors ħωp < 0.5 eV (depending on dopant conc.) Dispersion Relation Surface-Plasmon Polaritons

Dispersion Relation Surface-Plasmon Polaritons Solve Maxwell’s equations with boundary conditions • We are looking for solutions that look like: E z Dielectric

Metal x H

H = 0, H ,0 expi k x + k z "!t • Mathematically: z<0 d ( yd ) ( xd zd ) Ed = (Exd ,0, Ezd )expi(kxd x + kzd z "!t)

z>0 Hm = (0, H ym ,0)expi(kxm x + kzm z "!t)

Em = (Exm ,0, Ezm )expi(kxm x + kzm z "!t) • Maxwell’s Equations in medium i (i = metal or dielectric): !H "E " #!i E = 0 ! " H = 0 "# E = $µ #$ H = !i 0 !t "t • At the boundary:

Ex,m = Ex,d ! m Ezm = ! d Ezm H ym = H yd Dispersion Relation Surface-Plasmon Polaritons

Dispersion Relation Surface-Plasmon Polaritons

• Start with curl equation for H in medium i (as we did for EM waves in vacuum) "E #$ H = ! i i i "t where Hi = (0, H yi ,0)expi(kxi x + kzi z "!t)

Ei = (Exi ,0, Ezi )expi(kxi x + kzi z "!t)

$ #H zi #H yi #H xi #H zi #H yi #H xi % ' & , & , & ( = (ikzi H yi ,0,ikxi H yi )= (&i!"i Exi ,0,i!"i Ezi ) ) #y #z #z #x #x #y *

• We will use that: kzi H yi = #!"i Exi kzm H yi = #!" m Exm k k kzd H yd = #!" d Exd zm zd H ym = H yd ! m ! d

• E// across boundary is continuous: Ex,m = Ex,d

• H across boundary is continuous: H = H // ym yd k k k k zm = zd Combine with: zm zd H ym = H yd ! m ! d ! m ! d Dispersion Relation Surface-Plasmon Polaritons

Dispersion Relation Surface-Plasmon Polaritons

Relations between k vectors k z zd ! =1 k k d • Condition for SP’s to exist: zm = zd Example ! m ! d k zm ! m = "1

z Air • Relation for kx (Continuity E//, H//) : kxm = kxd Example SiO true at any boundary 2

2 2 2 # ! $ • For any EM wave: kx + kzi = "i % & ' c (

2 ! Dispersion relation • Both in the metal and dielectric: k k # $ k 2 sp = x = "i & ' % zi 1/ 2 ( c ) " # ! m! d $ kx = % & k k c ' ! m + ! d ( zm = zd ! m ! d homework Dispersion Relation Surface-Plasmon Polaritons

Dispersion Relation Surface-Plasmon Polaritons

Plot of the dispersion relation 1/ 2 # $ • Last page: " ! m! d kx = % & c ' ! m + ! d ( ! r • Plot dielectric constants dielectric ! d metal

! p ! "! d !sp

1/ 2 ! " & ! m! d ' " k • Low ω: k = ( ! = " d x c lim ) * c d c !m #$% + ! m + ! d , ! !sp • At = (when = - ): ω ωsp εm εd kx ! "

• Note: Solution lies below the light line k Dispersion Relation Surface-Plasmon Polaritons

Dispersion Relation Surface-Plasmon Polaritons

Dispersion relation plasma modes and SPP

Metal/air !SP,Air

Metal/dielectric with εd "SP,!d

• Note: Higher index medium on metal results in lower ωsp 2 2 ! p ! ! p = when: 2 2 2 2 p ω ωsp " m =1# 2 = #" d ! #! p = #" d! ! = ! = ! 1+ " d 1+ " d Excitation Surface-Plasmon Polaritons with Electrons

Excitation Surface-Plasmon Polaritons with Electrons Excitation with electrons

• First experiments with high energy electrons

• Measurement: Energy loss Direction of e’s:

• Whole dispersion relation can be investigated

• Low k’s are hard!

Example: 50 keV has a λ = 0.005 nm << λlight

kelectron >> klight

Stringent requirement on divergence e- beam Nanophotonics with Plasmonics: A logical next step?

Nanophotonics with Plasmonics: A logical next step?

• The operating speed of data transporting and processing systems

Photonics (µm-scale structures ~ 1µm) 1T Plasmonics?(nm-scale structures) ) DWDM z

H WDM (

d CMOS Electronics 1e G e 130 nm p s

Coaxial circuits g 1.8 µm n i

1t M a r e

p Transcontinental cable Communication networks O 1k Telephone CMOS Electronics Plasmonics Telegraph 1 1825 1850 1875 1900 1925 1950 1975 2000 2025 2050 Time Time  The ever-increasing need for faster information processing and transport is undeniable

Electronic components are running out of steam due to issues with RC-delay times Why not electronics?

Why not electronics?

As data rates AND component packing densities INCREASE, electrical interconnects become progressively limited by RC-delay:

A L

1 A R $ L / A % C $ L # B $ $ max RC L2 A # B " 1015 ! (bit/s)(A << L2 !) max L2

Electronics is aspect-ratio limited in speed! Why not photonics?

Why not photonics?

The bit rate in optical communications is fundamentally limited

only by the carrier frequency: Bmax < f ~ 100 Tbit/s (!), but light propagation is subjected to diffraction:

core

a cladding

2' 2! n = n +%n = n +%n # V = a n2 ! n2 " a 2n%n core clad & core clad ë well ! guided mode : V $ ' # a " & / 2 2n%n ! mode size : %n<<1(!)

Photonics is diffraction- limited in size! Why Plasmonics?

Why Plasmonics?

" #m#d D ksp = λ c #m + #d sp E εd > 0 Dispersion Relation for SPPs: - - -+ + + - - - + + + εm < 0 H ! optical ω nm-scale λ

! m (") # $! d (")

λp ~ very small

SP can reach nanoscale at optical ! SPPs are “x-ray waves” with optical frequencies Why nanophotonics needs plasmons?

Courtesy of M. Brongersma