Lecture 9: Introduction to Metal Optics

5 nm

What happened at the previous lectures ?

Microparticles

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

• Insulators (Rayleigh Scattering, blue sky..)

• Semiconductors (Size dependent absorption, fluorescence..)

• Metals…Resonant absorption at ωsp

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

Light interaction with small objects (d < λ)

Dielectric photonic crystal

• Molding the flow of light

What happened at the previous lectures ?

Metal Optics: An IntroductionMajority of optical components based on dielectrics

• High speed, high bandwidth (ω), but…

Some fundamental problems!

nCORE

nCLAD

Diffraction Limit

Optical mode in waveguide > λ0/2nCORE

• Does not scale well Needed for large scale integration

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

ProblemsBending losses

Solutions ?

Some:Photonic functionality based on metals?!

Metal Optics: An Introduction

Plasmon-PolaritonsWhat is a plasmon ?

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

• Metals are expected to allow for electron density waves: plasmons

Strong local field

Metal

Dielectric

z

I

E

HNote: This is a TM wave

• Sometimes called a surface plasmon-polariton (strong coupling to EM field)

Surface plasmon

Bulk plasmon• Metals allow for EM wave propagation above the plasma frequency

They become transparent!

Plasmon-Polaritons

Local Field Intensity Depends on Wavelength

Long wavelength Short wavelength

z

I

z

I

D << λo

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• ……

Local Field Intensity Depends on Wavelength

R.M. Dickson and L. A. Lyon, J. Phys. Chem. B 104, 6095-6098 (2000)

Laser excitationλ = 532 nm

8.1 µm Au rod

Light at the other end

Plasmon-Polariton Propagation in Au rod

Plasmon-Polariton Propagation in Au rod

Plasmon-Polariton Excitation using a Launch Pad

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

Plasmon-Polariton Excitation using a Launch Pad

50 nm

• Array of 50 nanometer diameter Au particles spaced by 75 nanometer

• Information transport at speeds and densities exceeding current electronics

• Enables communication between nanoscale devices

M.L. Brongersma, et al., Phys. Rev. B 62, R16356 (2000)S.A. Maier et al., Advanced materials 13, 1501 (2001)

• Guides electromagnetic energy at optical frequency below the diffraction limit

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 nmPicture taken from Nanonics

APD

APD

Computer

HV Source

Lock-In Detector

Freq. Gen.

532nm Source

PID Controller

To Piezoelectrics

Filter

Filter

Obj. Lens

Obj. LensUpper Piezo

Lower Piezo

Fiber Tip

Tuning Fork

Metal optics will make your dreams come true….Enhanced Transmission through Sub-λ Apertures

T.Thio et al., Optics Letters 26, 1972-1974 (2001).

• Ag film with a 440 nm diameter holesurrounded by circular grooves

• 3x more light than directly impingent on hole !

• Reason: Excitation of plasmon-polaritons

• Transmission enhancement of 10 xcompared to a bare hole

Enhanced Transmission through Sub-λ Apertures

Optical Properties of an Electron Gas (Metal)Dielectric constant of a free electron gas (no interband transitions)

( ) ( ) ( ){ }Re expt i t! != "E E• Consider a time varying field:

• Equation of motion electron (no damping)2

2

dm edt

= !r

E

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

• Substitution p into Eq. of motion: ( ) ( )2 2m e! ! !" =p E

( ) ( )2

2

1e

m! !

!= "p E• This can be manipulated into:

• The dielectric constant is:

• Dipole moment electron ( ) ( )t e t= !p r

2

2

2

dm edt

=p

E

( )( )

22

2 2

0 0

11 1 1

p

r

N Ne

m

!!" #

" ! " ! != + = = $ = $

p

E

!

r!

p!

Optical Properties of an Electron Gas (Metal)

Dispersion Relation for EM Waves in Electron GasDetermination of dispersion relation for bulk plasmons

( )( )

2

2

2 2

,,r

tt

c

! "=#

"

E rE r

t

• The wave equation is given by:

• Investigate solutions of the form: ( ) ( ) ( ){ }, Re , expt i i t! != " #E r E r k r

2 2 2

rc k! " =2

21

p

r

!"

!= #• Dielectric constant:

2

2 2 2 2 2

21

p

p c k!

! ! !!

" #$ = $ =% &% &

' (

!

k

p!

No allowed propagating modes (imaginary k)

ck! =

2 2 2

p c k! != +• Dispersion relation:

Note1: Solutions lie above light lineNote2: Metals: ħωp ≈ 10 eV; Semiconductors ħωp < 0.5 eV (depending on dopant conc.)

Dispersion Relation for EM Waves in Electron Gas

Dispersion Relation Surface-Plasmon PolaritonsSolve Maxwell’s equations with boundary conditions

• Maxwell’s Equations in medium i (i = metal or dielectric):

0i!" # =E 0!" =H

0µ!

"# = $!

HE

ti!"

#$ ="

EH

t

• At the boundary:

Metal

DielectriczE

H

, ,x m x dE E=

x

ym ydH H=m zm d zmE E! !=

• We are looking for solutions that look like:

( ) ( )0, ,0 expm ym xm zmH i k x k z t!= + "H

( ) ( ),0, expm xm zm xm zm

E E i k x k z t!= + "E

( ) ( )0, ,0 expd yd xd zdH i k x k z t!= + "H

( ) ( ),0, expd xd zd xd zd

E E i k x k z t!= + "E

• Mathematically: z<0

z>0

Dispersion Relation Surface-Plasmon Polaritons

i

i i!"

#$ ="

EH

t

Dispersion Relation Surface-Plasmon Polaritons

• Start with curl equation for H in medium i (as we did for EM waves in vacuum)

( ) ( )0, ,0 expi yi xi ziH i k x k z t!= + "Hwhere

( ) ( ),0, expi xi zi xi zi

E E i k x k z t!= + "E

( ) ( ), , ,0, ,0,yi yizi xi zi xi

zi yi xi yi i xi i zi

H HH H H Hik H ik H i E i E

y z z x x y!" !"

# #$ %# # # #& & & = = &' (

# # # # # #) *

zi yi i xik H E!"= #• We will use that:zm yi m xmk H E!"= #

zd yd d xdk H E!"= #

• E// across boundary is continuous: , ,x m x d

E E=

zm zdym yd

m d

k kH H

! !=

• H// across boundary is continuous: ym ydH H=

Combine with: zm zdym yd

m d

k kH H

! !=

zm zd

m d

k k

! !=

Dispersion Relation Surface-Plasmon Polaritons

Dispersion Relation Surface-Plasmon Polaritons

Relations between k vectors

• Condition for SP’s to exist: zm zd

m d

k k

! !=

• Relation for kx (Continuity E//, H//) : xm xdk k=

• For any EM wave:

zdk

zmk

Example1

m! = "

1d! =z

z

2

2 2

x zi ik k

c

!"# $

+ = % &' (

2

2

sp x i zik k kc

!"# $

= = %& '( )

• Both in the metal and dielectric:

zm zd

m d

k k

! !=

1/ 2

m d

x

m d

kc

! !"

! !

# $= % &

+' (

Dispersion relation

homework

Example Air

SiO2true at any boundary

Dispersion Relation Surface-Plasmon Polaritons

Dispersion Relation Surface-Plasmon Polaritons

1/ 2

m d

x

m d

kc

! !"

! !

# $= % &

+' (

Plot of the dispersion relation

• Last page:

• Plot dielectric constants

• Low ω: 1/ 2

limm

m d

x d

m d

kc c!

! !" "!

! !#$%

& '= () *

++ ,

• At ω = ωsp (when εm = -εd):

!

r!

p!

sp!

d!"

d! dielectric

metal

xk !"

!

k

sp!

dk

c

!"=

• Note: Solution lies below the light line

Dispersion Relation Surface-Plasmon Polaritons

Dispersion Relation Surface-Plasmon Polaritons

Dispersion relation plasma modes and SPP

• Note: Higher index medium on metal results in lower ωsp2

21

p

m d

!" "

!= # = # 2 2 2

p d! ! " !# = #

2

2

1

p

d

!!

"=

+ω = ωsp when:

Metal/air

Metal/dielectric with εd

1

p

d

!!

"=

+

,SP Air!

, dSP !"

Dispersion Relation Surface-Plasmon Polaritons

Excitation Surface-Plasmon Polaritons with Electrons

Excitation with electrons

• First experiments with high energy electrons

• Whole dispersion relation can be investigated

• Measurement: Energy lossDirection of e’s:

• Low k’s are hard!

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

kelectron >> klight

Stringent requirement on divergence e- beam

Excitation Surface-Plasmon Polaritons with Electrons

• The operating speed of data transporting and processing systems

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

Nanophotonics with Plasmonics: Alogical next step?

1825 1850 1875 1900 1925 1950 1975 2000 2025 20501

1k

1M

1G

1T

130 nm

1.8 µm

DWDMWDM

Coaxial circuits

Telephone

Transcontinental cable

Telegraph

Opera

ting s

peed (

Hz)

Time

Plasmonics?

CMOS Electronics

Photonics

Time

Communication networks

CMOS Electronics

Plasmonics

(µm-scale structures ~ 1µm)(nm-scale structures)

Nanophotonics with Plasmonics: A logical next step?

)!()bit/s(10

1/

2

2

15

max

2max

LAL

AB

L

A

RCBLCALR

<<!"#

$$#$%$

Electronics is aspect-ratio limited in speed!

As data rates AND component packing densities INCREASE,

electrical interconnects become progressively limited by RC-delay:

L

A

Why not electronics?

Why not electronics?

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:

cladding

core

a

)!(1 :size mode22/:modeguidedwell

2aë

2!2 22

<<!"#$!

"!=#+=+=

nnnaV

nnnnaVnnnnncladcorecladcore

%%&'

%&

'%%

Photonics is diffraction- limited in size!

Why not photonics?

Why not photonics?

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

!

ksp ="

c

#m#d

#m + #d

Dispersion Relation for SPPs:

λp ~ very small

Why Plasmonics?

optical ωnm-scale λ

- - -+ + + - - - + + +

E

H

Dλsp

εd > 0εm < 0

SP wavelengths can reach nanoscale at optical frequencies!SPPs are “x-ray waves” with optical frequencies

Why Plasmonics?

Why nanophotonics needs plasmons?

Courtesy of M. Brongersma