1

Plasmonics

Femius Koenderink Center for Nanophotonics AMOLF, Amsterdam

Plasmon: elementary excitation of a plasma (gas of free charges) Plasmonics: nano-scale optics done with plasmons at metal interfaces

Flavours

M-I-M Taper and wire

Hybrid V-groove Wedge/hybrid

1) Kramers-Kronig bounds on ε

2) Back to the surface plasmon

3) Pulse propagation and dispersion

Kramers-Kronig

2013 /

4

Can a material have arbitrary real and imaginary ε ? No: material response functions are constrained by causality

Frequency domain

Time domain Note how convolution turns to product

Physics: no response before cause

Causal response

Example of an allowable response function in time What does Imply for

Even and odd / real, imaginary

Three important realizations: 1) χ(t) is a real function 2) It’s even part contributes to the real part of χ(ω) 3) It’s odd part contributes to the imaginary part of χ(ω)

So what is the even / odd part ?

Define as odd part of the response function This has no physical meaning, just a math trick

So what is the even / odd part ?

The even part of χ(t) follows from the odd part

So now: Fourier transform of a product is a convolution of FTs

So what is the even / odd part ?

So what is the Fourier transform of sgn(t) ? Remember

Kramers-Kronig

Therefore the imaginary part of the frequency-dependent Permittivity completely specifies the real part Similarly

Kramers Kronig

Either you have non-dispersive vacuum ε=1, i.e., χ=0, or - A window of real χ implies a window of absorption - Real χ(ω) >0 means χ(ω) < 0 at other ω (to avoid gain) - “No dispersion” but a refractive index not 1 is impossible

Considerations hold for any physical response function

Typical solids

Absorption bands close to intrinsic resonances Real n to the red also outside absorption Most materials have ’normal dispersion’, i.e.,

goes up with energy is higher towards the blue is higher towards short

Until you go through an absorption resonance

Phase velocity

A front of constant phase follows

Phase velocity

Suppose now I have a wave packet

e.g.

Note that at x=0, this is just a pulse

Group velocity

Maybe you have dispersion

The pulse envelope has a velocity

known as the group velocity

Dispersion relation in a gas

15

Near resonance (γ ~ 0.05 ω0) - Strongly dispersive - Very low vg - Superluminal phase velocity - Negative & superluminal vg

Indices

Group index very different from phase index Negative vg Superluminal vg, vf

Region of strong absorption – Kramers-Kronig in action

Movie – dispersive propagation

Strongly dispersive, weakly absorbing ωc~ 0.85 ω0 (here loss length > 50λ) Pulse envelope tracks vg

Strongly dispersive, weakly absorbing (here loss length > 50λ) Pulse envelope tracks vg

Superluminal phase

Strongly dispersive, strongly absorbing ωc~ 1.03 ω0 vf=1.25c, vgroup=1.7c (at carrier frequency) Note how the packet - barely moves - vg >c not apparent - package break up

Loss length ~ 3 λ

Slow light, superluminal phase

Superluminal phase Slow group velocity Weak abosrption ωc~ 1.1 ω0 vf=1.15c, vgroup=c/2 (at carrier frequency) Note how the group velocity has regained meaning Loss length ~ 20 λ

Take home messages

• Phase velocity describes phase front propagation You get it from the ratio of ω/c and |k|

• For weak absorption, group velocity describes Gaussian envelope • Strong dispersion also entails strong absorption • Superluminal phase and group velocities appear • These are irrelevant to describe pulse-energy propagation

• Pulse break up (group velocity dispersion) • Strong attenuation

Note: - this has been a big controversy in Nature/Science Plasmonics is strongly dispersive and at superluminality paradox

Small scatterers and plasmonics

23

From plasmon to plasmonics

Plasmons in the bulk oscillate at ωp determined by the free electron density and effective mass

Plasmons confined to surfaces that can interact with light to form propagating “surface plasmon polaritons (SPP)”

Confinement effects result in resonant SPP modes in nanoparticles

+ + +

- - -

+ - +

k

0

2

εω

mNedrude

p =

0

2

31

εω

mNedrude

particle =

2

0

12

drude

surfaceNem

Observables

Extinction cross section [m2] Power removed from beam Incident intensity

Extinction = scattering + absorption

removed from the beam

Re-radiated into all angles

Lost as heat in the scatterer

Physical quantity - polarizability

Small object kd

Electrostatic sphere

First consider a sphere in a static field

E0 εm ε

z

r

θ

a

( )( )ar

ar>=∆Φ

Solution

20203

02

0001

42

23

2

rp

rEr

EarE

rErErE

mm

m

m

m

m

m

πεθθθ

εεεε

θ

θεε

εθ

εεεε

θ

coscos

coscos

coscoscos

+−=

+−

+−=Φ

+

−=

+−

+−=Φ

E0 εm ε

z

r

θ

a Easy to verify

30 0 with 4 2

mSI SI m

m

p E a ε εα α πε εε ε

−= = +

Inside sphere: homogeneous field Outside sphere: background field plus field of a dipole with

In the ball:

Outside:

Metal sphere

Drude model for a metal: Lorentzian `plasmon resonance’

2 23 0

2 20

1 means ( )

peasy ai i

ω ωε αω ω γ ω ω ωγ

= − = + − +

• Resonance where ε(ω0) = -2 εm • Response scales with the volume V • α exceeds V by factor 5 to 10 • Shape shift condition ε = -2 εm

30 04 with 2

measy easy

m

p E a ε επε α αε ε

−= = +

Observables

Extinction cross section [m2] Power removed from beam Incident intensity

Extinction = scattering + absorption

removed from the beam

Re-radiated into all angles

Lost as heat in the scatterer

Revisiting polarizability Classical model of harmonically bound electron describes atom, and scatterer alike as an oscillating dipole

20 0

2 20

3 ( ) ( )i t i tSIVt e e

iω ωε ω α ω

ω ω ωγ= =

− −p E E

Lorentzian resonance

Extinction: how much power is taken from the beam ? Cycle average work done by E on p

in ImdpW Edt

α∝ ⋅ ∝

Revisiting polarizability Extinction: how much power is taken from the beam (in SI units) ?

0 0

1 1Re[ ] Re[ ] Re[ ] Re[ ]T Ti t

i t i t i td eW e dt e i e dtT dt T

ωω ω ωωα= ⋅ = ⋅∫ ∫

pE E E

* * *

0

1 ( ) ( )4

Ti t i t i t i tW e e i e i e dt

Tω ω ω ωωα ωα− −= + ⋅ −∫ E E E E

Only cross terms survive cycle average * 2 2

0

1 ( | | | | ) oscill.terms ( 2 )4

T

W i i dtT

ωα ωα ω= − + + ±∫ E E

2Im | |2

W ω α= E

Revisiting polarizability Classical model of harmonically bound electron Describes atom, and scatterer alike

Scattering: how much power does p radiate ?

22

0

2

0 0

22dipole

2

0 0

2 ||4

sinsin||sin W απε

θθϕθϕπ ππ π

∝=∝⋅∝ ∫ ∫∫ ∫∫ rprdErddA

sphere

nS

Lorentzian resonance

20 0

2 20

3 ( ) ( )i t i tSIVt e e

iω ωε ω α ω

ω ω ωγ= =

− −p E E

Equate extinction to scattering (energy conservation)

Scattering

Rayleigh / Larmor

Extinction

Work done to drive p

≥

Optical theorem

2 4 k Im [ ]Easy mπ α 4 2 28 k | | [ ]3 Easy

mπ α

1. Very small particles scatter like r6/λ4 (Rayleigh) 2. For very small particles absorption wins ~ r3/λ 3. Big |α|2 implies large Im α

Equate extinction to scattering (energy conservation)

Scattering

Rayleigh / Larmor

Extinction

Work done to drive p

≥

Optical theorem

2 4 k Im [ ]Easy mπ α 4 2 28 k | | [ ]3 Easy

mπ α

Since |Im α | < |α|

33| |2 2Easy

λαπ

<

Upper bound on the strongest possible dipole scatterer

Example: simple spheres

Calculated exact cross section of Au spheres r=10 -50 nm Dashed line: σ = 4π k Im αeasy Surprises -Peak bounded by

Revisiting polarizability Classical model of harmonically bound electron describes atom, and scatterer alike as an oscillating dipole

20 0

2 20

3 ( ) ( )i t i tSIVt e e

iω ωε ω α ω

ω ω ωγ= =

− −p E E

Lorentzian resonance

Compared to electrostatics: γ must be adapted to contain radiation damping

Summary • Small particles of size < λ/10 scatter like dipoles • Generally: 3x3 polarizability tensor proportional

to V • ‘Depolarization factors’ require microscopic

model • General features of polarizability

– Optical theorem contrains α – Polarizability bounded by above by unitary limit – Radiative damping broadens resonances & raises

albedo

Photochemical imaging

39

Plasmon particles in azo-polymer Regions of high field contract Measured using atomic force microscopy Wiederrecht

Uses of particle plasmons

40

Very strong local fields |E|2: 104 x stronger than incident

Au spheres 5,8,20 nm, gaps of 1-3 nm

100 102 104

|E|2/|Ein|2

10.000 times enhanced optical field intensity

Resonance shifts

Single protein binding & unbinding

Bow ties

Measured 1000 x fold enhancement of # photons / molecule

Wenger – Inst. Fresnel, Marseille

Example – dimer in static approximation

Dimer in a static approximation

Linear problem • Symmetric, but not real matrix • 1/polarizability on the diagonal • Interaction on the off-diagonal - this will shift resonances

Hybridization (exercise)

Plasmon ruler

Idea: Alivisatos group

Yagi Uda antenna

Curto et al., Science (2010).

Directional scattering, phased arrays

Arrays of particles can result in directional scattering Arrays of particles can result in directional emission When a local source is embedded

Yagi-Uda

Optical domain: a chain of plasmon particles driven by a single molecule

Phased array

Suppose I have N dipoles, radiating into the far field as

Taylor expand for large Viewing distance

r1 r2

Far field

Overall spherical wave

If all dipoles are parallel (not meaning in phase)

Array geometry ‘Structure factor’

Single scatterer ‘Form factor’

Uses of particle plasmonics

• Enhanced fields for spectroscopy • Sensing • Enhanced fluorescence brightness • Fluorescence directionality

Slide Number 1FlavoursSlide Number 3Kramers-KronigCausal responseEven and odd / real, imaginarySo what is the even / odd part ?So what is the even / odd part ?So what is the even / odd part ?Kramers-KronigKramers KronigTypical solidsSlide Number 13Slide Number 14Dispersion relation in a gasIndicesMovie – dispersive propagationSlide Number 18Superluminal phaseSlow light, superluminal phaseTake home messagesSlide Number 22From plasmon to plasmonicsObservablesPhysical quantity - polarizabilityElectrostatic sphereSolutionMetal sphereSlide Number 29ObservablesRevisiting polarizabilityRevisiting polarizabilityRevisiting polarizabilityOptical theoremOptical theoremExample: simple spheresRevisiting polarizabilitySummaryPhotochemical imagingUses of particle plasmonsResonance shiftsBow tiesExample – dimer in static approximationSlide Number 44Plasmon rulerYagi Uda antennaDirectional scattering, phased arraysYagi-UdaPhased arrayFar fieldUses of particle plasmonics