Lectures 9: Surface Plasmon Polaritons

Surface Plasmon Polaritons (SPPs) Introduction and basic properties

Standard textbook:- Heinz Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings

Springer Tracts in Modern Physics, Vol. 111, Springer Berlin 1988

Overview articles on Plasmonics:- A. Zayats, I. Smolyaninov, Journal of Optics A: Pure and Applied Optics 5, S16 (2003)- A. Zayats, et. al., Physics Reports 408, 131-414 (2005) - W.L.Barnes et. al., Nature 424, 825 (2003)

- Overview

- SPP dispersion

- SPP excitation

- active switching

Elementary excitations and polaritonsElementary excitations and polaritons

FromBoardman

Elementary excitations:

• Phonons• Plasmons• Excitons (bound electron-hole pair)

Polaritons:

Coupled state between an elementaryexcitation and a photon.

Plasmon polariton: coupled statebetween a plasmon and a photon.Phonon polariton: coupled statebetween a phonon and a photon.

As (visible) photons have wavelengthsmuch larger than lattice constants theyinteract only with long wavelengthpolar elementary excitations. Polar = excitations have longwavelength electromagnetic fields.

• free electrons in metal: electron liquid of high density

• longitudinal density fluctuations (plasma oscillations) at eigenfrequency

• quanta of volume plasmons have energy , ~ 10eV

propagate through the volume for frequencies

00

2

εω

mne

p hh =

Volume plasmon polaritons

323cm10 −≈n

Surface plasmon polaritons

PlasmonsPlasmons

Maxell´s theory shows that EM surface waves can propagate also along a metallic surface with a broad spectrum of eigen frequencies

from ω = 0 up to 2pωω =

Particle (localized) plasmon polaritons(later)

pωω >

pω

2pω

pω

3pω

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

+ + + +

+++

---

Bulkmetal

Metalsurface

Metal spherelocalized SPPs

Plasmon resonance positions in vacuumPlasmon resonance positions in vacuum

0=ε

1−=ε

2−=εdrudemodel

- - - -

drudemodel

2

2

1ωω

ε pm −=

Drude model

Volume plasmon polaritons in metalsVolume plasmon polaritons in metals

Volume plasmon polariton dispersion

2222 kcp += ωω

0

ck=ω

0 ε 0

ω ω

ωp

Dielectric function

2

2

1ωω

ε pm −=

Drude model(see Nanooptic I, Chap. 5)

Volume plasmon

Surface plasmon polaritons (SPPs)Surface plasmon polaritons (SPPs)

Solution of the Maxwell equations for interface metal/dielectric

Excitaton of a coupled state between photons and plasma oscillationsat the interface between a metal and a dielectric

- radiative surface plasmons are coupled with propagating EM waves- nonradiative surface plasmons do not couple with propagating EM waves- for perfectly flat surfaces SPPs are always nonradiative!

In contrast to conventional waveguides: field on both sides are evanecent

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

longitudinal surface wave

dielectric

metal

εd ω( )

εm ω( )zx

Ez

z zkzeE Im−∝

SPP - propagationSPP - propagation

dielectric waveguidingvs.

plasmon waveguiding

( )tzkxki zxe ω−±±= 0SP EESP

xkλ

π2=′

xxx kikk ′′+′=

Derivation of SPP dispersion – boundary conditionsDerivation of SPP dispersion – boundary conditions

( ) ( )tzkxkizmxmm

zmxmeEE ω−−= ,0,E

( ) ( )tzkxkiymm

zmxmeH ω−−= 0,,0H

( ) ( )tzkxkiydd

zdxdeH ω−+= 0,,0H

( ) ( )tzkxkizdxdd

zdxdeEE ω−+= ,0,E

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

longitudinal surface wave

dielectric

metal

εd ω( )

εm ω( )zx

xdxm EE =

ydym HH =

Boundarybonditions (z=0)

zddzmm EE εε =

xdxm kk =

+ + - -

0=yE

xE

zE

0== zx HH

zx

⊗yH

0>z

0<z

Derivation of SPP dispersionDerivation of SPP dispersion

EHtc ∂

∂=

1rot ε

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛∂=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛×

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∂∂∂

z

x

ty

z

y

x

E

E

cH 01

0

0ε

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∂

∂−

z

x

yx

yz

E

E

cH

H00 ωε

Maxwell eq.:

z-component:

Diel.:

Metal:zmmymxm E

cHk ωε−=+

zddydxd Ec

Hk ωε−=+

x-component:

Diel.:

Metal: ymzmymz HkH +=∂−

ydzdydz HkH −=∂−

xmmymzm Ec

Hk ωε−=+

xddydzd Ec

Hk ωε+=+

( )tzkxki zdxde ω−+

( )tzkxki zmxme ω−−

Diel.:

Metal:

xmmymzm Ec

Hk ωε−=

xddydzd Ec

Hk ωε=

ydym HH =xdxm EE =

0=+m

zm

d

zd kkεε

Boundaryconditions:

Derivation of SPP dispersionDerivation of SPP dispersion

I:

II:

I / II:xd

xm

d

m

yd

ym

zd

zm

EE

HH

kk

εε

−=

d

m

zd

zm

kk

εε

−=

x-component:

Diel.

Metal

dm

dmx c

kεε

εεω+

⎟⎠⎞

⎜⎝⎛=

22

( )dm

dzd c

kεε

εω+

⎟⎠⎞

⎜⎝⎛=

222

( ) xdmm k real and 0Re →>< εεε

kzd and kzm are imaginary

xmxd kk =

Dispersion relation of SPPsDispersion relation of SPPs

at interface metal/dielectric:

2

222

ckk dzdx

ωε=+

2

222

ckk mzmx

ωε=+

dielectric:

metal:

0=+m

zm

d

zd kkεε

2222 kkkk zyx =++

generally:

( )dm

mzm c

kεε

εω+

⎟⎠⎞

⎜⎝⎛=

222

02

2

cωε

photonin air

kx

ω

dpSP ε

ωω+

=1

1

xck=ω

dεε −→′

surface plasmon polariton

2222xp kc+= ωω

ωp

plasmonpolariton

xdm

dm ckεεεεω +

=

Dispersion relation of SPPsDispersion relation of SPPs

surface plasmonsnon-propagatingcollective oscillationsof electron plasmanear the surface

volume plasmon

Dielectric function for silverDielectric function for silver

D. GüntzerZulassungsarbeit

www.phog.physik.uni-muenchen.de

Dispersion relation for SPPs on silver (with damping) Dispersion relation for SPPs on silver (with damping)

D. Güntzer, Zulassungsarbeit, www.phog.physik.uni-muenchen.de

real part

xk ′

imaginary part

xk ′′

Drude model

1=dε

SPP propagation lengthSPP propagation length

pωω

mε ′

mε ′′

)( vacxL λ

γωωω

εip

m +−= 2

2

1

2.0=γ

0.4 0.6 0.8 1

-10-7.5

-5-2.5

2.55

7.510

0.2 0.4 0.6 0.8 1

12

51020

50100200 2

pSP

ωω =

( ) xkxkixik xxx eeex ′′−′== 00 EEE

propagating term exponential decayin x-direction

1+=′′+′=

m

mxxx c

kikkε

εωmetal/airinterface

xx k

L′′

=21 propagation

length

intensity !

Example silver: m 22 :nm 5.514 μλ == xLm 500 :nm 1060 μλ == xL pωω

SPP field perpendicular to surfaceSPP field perpendicular to surface

( ) zkzez Im0

−= EEz

z kL

Im1

=

z-decay length(skin depth):

Examples:

silver:

gold:

nm 24 and nm 390 :nm 600 ,, === mzdz LLλ

nm 31 and nm 280 :nm 600 ,, === mzdz LLλ

Ez

zdielectric

metal

εd ω( )

εm ω( )zx

xk

SPPs have transversal and longitudinal el. fieldsSPPs have transversal and longitudinal el. fields

xz

xz E

kkiE =

At large values,

the el. field in air/diel. has a strongtransvers Ez component compared to thelongitudinal component Ex

mε ′

In the metal Ez is small against Ex

At large kx, i.e. close to ε = - εd, both components become equal

xz iEE ±= (air: +i, metal: -i)

m

d

x

zm iEE

εε

−−=

pω

ωmε ′

xk′

SPω

1−

zk ′

The mag. field H isparallel to surfaceand perpendicular to propagation

d

m

x

zd iEE

εε−

=

+ + - -xE

zE

zx

⊗yH

El. field

Smmary: SPP lenght scalesSmmary: SPP lenght scales

W.L.Barnes et. al., Nature 424, 825 (2003)

Excitation of SPPsExcitation of SPPs

thin metal filmdielectric

zx

Kretschmann configuration

photon indielectric

k of photon in air is always < k of SPP

photon in air

kx

ω

SPP dispersion

no excitation of SPP is possible

in a dielectric k of the photon is increased

SPP can be excited by p-polarized light (SPP has longitudinal component)

k of photon in dielectric can equal k of SPP

E0θ

R

kx

Methods of SPP excitationMethods of SPP excitation

nprism > nL !!

Excitation by Kretschmann configurationExcitation by Kretschmann configuration

photon indielectric

photonin air

xk

ω

SPP dispersion

ck=ω

z

x

0θ

dε

mε0ε

0εωc

k =

ck ω

=

( )00 sin θεωc

kx =

( )00 sin/ θεω xkc=

0ω

( )0000

sin1

θεεεω cc

k m

m

x

=+

= Resonancecondition

0xk

1+=

m

mx c

kε

εω

Kretschmann configuration – angle scanKretschmann configuration – angle scan

0θ R

0θ

p-polarized

s-polarized-> no excitation of SPPs

illumination freq. ω0= const.

photonin air

kx

ω

0ω

R

0θ

Towards switchable coupling into SPP modes

Note: kSPP > k0 special geometry required to couple free-space

Radiation into SPP modes

Λ

Kretschmann gratingconfiguration „Wood´s anomaly“

Dielectric properties of goldVisible dielectric response strongly affected by d-band

resonance

W

d-band

conduction band

EF

2.38 eV

L Γ

400 450 500 550 600-2

0

2

4

6

Im{ε

}Wavelength (nm)

εIm

ΔεIm(500K) ΔεIm(1000K)

Large changes of the dielectric function by heatingHere: transient optical heating!

Experimental setup: NIR pump, visible probe

810 nm100 fs4 µJ

sapphire

4f pulse-shaper

gold grating:Λ = 830 nm

PCHWP: TM/TE

400 fs

Results: pump-probe of a Wood’s anomaly

TE / TMreflectivity(zero order)

period: 830 nmprofile: Sawtooth, blaze: 29.90

here: Θ = -180, q = 2

peak electronTemperature: 600 K

Discussion: shift of the resonance

measured reflectivity change

wavelength derivative of the linear reflectivity

Indication for a spectralShift of the anomalyby 0.75 nm

Decay time: ~1 ps

Quantitative discussion: Thermomodulationof the dielectric response

Recall: q = 2 in