Surface Plasmon Polaritons (SPPs) Introduction and … · Surface Plasmon Polaritons (SPPs) -Introduction and basic properties ... Overview articles on Plasmonics: ... Surface Plasmon

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- Light-matter interaction- SPP dispersion and properties

OverviewOverviewOverview

Replace Replace ‘‘slowslow’’ electronic devices with electronic devices with ‘‘fastfast’’ photonic ones photonic ones Photonic crystalsPhotonic crystals

SPsSPs is the way to concentrate and channel light using subwavelengthis the way to concentrate and channel light using subwavelength structuresstructures

SPsSPs structures at the interface between a metal and a dielectric mastructures at the interface between a metal and a dielectric materialterialTransverse magnetic in character Transverse magnetic in character electric field normal to the surfaceelectric field normal to the surfacePerpendicular direction Perpendicular direction evanescent fieldevanescent fieldMomentum mismatch in the SP dispersion curveMomentum mismatch in the SP dispersion curve


Gold stripes

surface plasmon polariton optics (SPP) surface plasmon polariton optics (SPP) BandBand--gap effects, SPP waveguiding along straight and bent line, gap effects, SPP waveguiding along straight and bent line,


enhanced optical transmission enhanced optical transmission

Elementary excitations and polaritonsElementary excitations and polaritons

Elementary excitations:• Phonons (lattice vibrations)• Plasmons (collective electron oscillations)• Excitions (bound state between an excited electron and a hole)

Polaritons: Commonly called coupled state between an elementary excitation and a photon= light-matter interaction

Plasmon polariton: coupled state between a plasmon and a photon.

Phonon polariton: coupled state between a phonon and a photon.

PlasmonsPlasmonsPlasmons

A plasmon is the quantum of the collective excitation of freeA plasmon is the quantum of the collective excitation of free electrons in electrons in

solids. solids.

Electron plasma effects are most pronounced in freeElectron plasma effects are most pronounced in free--electronelectron--like metals. like metals.

The dielectric constant of such materials can be expressed as The dielectric constant of such materials can be expressed as

ωω < < ωωpp εεmm < 0< 0 wavevector of light in the medium is imaginary wavevector of light in the medium is imaginary no no

propagating electromagnetic modes propagating electromagnetic modes

ωω > > ωωpp εεmm→→11 altered by intraband transitions in noble metals altered by intraband transitions in noble metals

A combined excitation consisting of a surface plasmon and a pA combined excitation consisting of a surface plasmon and a photon is called hoton is called

a surface plasmon polariton (SSP). a surface plasmon polariton (SSP). different nature, such as phonondifferent nature, such as phonon––polariton, polariton,

excitonexciton––polariton, etc.polariton, etc.

2

( ) 1 pm

ωε ω ω⎛ ⎞= − ⎜ ⎟⎝ ⎠

• free electrons in metal are treated as an electron liquid of high density

• longitudinal density fluctuations (plasma oscillations) at eigenfrequency

• quanta of volume plasmons have energy , in the order 10eV

propagate through the volume for frequencies

0

24mne

pπω 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

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

Surface Plasmon PhotonicsSurface Plasmon PhotonicsOptical technology using- propagating surface plasmon polaritons- localized plasmon polaritons

Topics include:

Localized resonances/ - nanoscopic particleslocal field enhancement - near-field tips

Propagation and guiding - photonic devices- near-field probes

Enhanced transmission - aperture probes- filters

Negative index of refraction - perfect lensand metamaterials

SERS/TERS - surface/tip enhanced Raman scattering

Molecules and - enhanced fluoresencequantum dots

Also called:• Plasmonics• Plasmon photonics• Plasmon optics

Nanophotonics using plasmonic circuitsNanophotonics using plasmonic circuits

Atwater et.al., MRS Bulletin 30, No. 5 (2005)

• Proposal by Takahara et. al. 1997

Metal nanowireDiameter << λ

• Proposal by Quinten et. al. 1998

Chain of metal nanoparticlesDiameter and spacing << λ

First experimental observation byMaier et. al. 2003

ωh

ωh

Nanoscale plasmon waveguides

Subwavelength-scale plasmon waveguidesSubwavelength-scale plasmon waveguides

Krenn, Aussenegg, Physik Journal 1 (2002) Nr. 3

Some applications of plasmon resonant nanoparticlesSome applications of plasmon resonant nanoparticles

• SNOM probes

• Sensors

• Nanoscopic waveguides for light

• Surface enhancedRaman scattering (SERS)

T. Kalkbrenner et.al., J. Microsc. 202, 72 (2001)

S.A. Maier et.al., Nature Materials 2, 229 (2003)

EM waves in matterLorenz oscillator

Isolators, Phonon polaritonsMetals, Plasmon polaritons

Light-matter interactions in solids

Literature:

- C.F.Bohren, D.R.Huffman, Absorption and scattering of light by small particles

- K.Kopitzki, Einführung in die Festkörperphysik

- C.Kittel, Einführung in die Festkörperphysik

EM-waves in matter (linear media) - definitionsEM-waves in matter (linear media) - definitions

χχε lity suszeptibi with 0 EP =

( ) EEPED εεχεε 000 1 =+=+=

+1= χεεε ′′+′= i

εκ =inN +=

κεκε

nn2

22

=′′+=′

Polarization

Electric displacement

Complex dielectric function

Complex refractive index

Relationship between N and ε

k ⋅ k( ) =ω 2

c 2 ε ε(ω) and thus k = k´ + ik´´ are complex numbers!

( ) ( ) rkrkkr EEE ′′−−′− == eee titi ωω00

propagating wave exponential decay of amplitude

E = E0ei kr −ωt( )

B = B0ei kr −ωt( )

wavevector k = 2πλ

frequency ω = 2πf

knckc

==ε

ω

Dispersion in transparent media without absorption:

ε > 0

k and ε are real

Dispersion generally:

EM-waves in matter (linear media) - dispersionEM-waves in matter (linear media) - dispersion

Harmonic oscillator (Lorentz) modelHarmonic oscillator (Lorentz) model

ω0

tieeeKbm ω0EExxx ==++ &&&

0

0

EpPExpχε

αε==

==Ne

γωωωω

χεi

p

−−+=+= 22

0

2

11

meAe

ime i EEx Θ=

−−=

γωωω 220

ω0

+

-

ω02 = K m

γ = b mωp

2 = Ne2 / mε0 plasma frequency

One-oscillator (Lorentz) modelOne-oscillator (Lorentz) model

from Bohren/Huffman

( )( ) 22

22

11

κκ

+++−

=nnR

εκ =inN +=

Weak and strong molecular vibrationsWeak and strong molecular vibrations

weak oscillator: ε‘ > 1

examples: PMMA, PS, proteins

wavenumber / cm -1

-0,8

-0,4

0

0,4

0,8

1,2

1,6

2

-0,4

-0,2

0

0,2

0,4

0,6

0,8

1

860 880 900 920 940

eps' eps''

caused by: molecular vibrations

wavenumber / cm -1

strong oscillator: ε‘ < 0

SiC, Xonotlit, Calcite, Si3N4

-10

-5

0

5

10

15

20

25

-10

-5

0

5

10

15

20

25

860 880 900 920 940

eps' eps''

crystal lattice vibrations

Optical properties of polar crystalsOptical properties of polar crystalsnote:lattice has transversal T and longitudinal L oscillations but only transversal phononscan be excited by light

( ) εω2

2

c=⋅kk

εκ =+= inNεκ == ,0n

( )( ) 22

22

11

κκ

+++−

=nnR

( )tie ω−= krEE 0

A

0 , == κεnB

AB Bεω

ck =

εωc

ik =A

B

A: total reflectionB: transmission and reflection

γ = 0

SiC - single oscillator modelSiC - single oscillator model

γ > 0

permanent dipolese.g. water

phononsmolecular vibrations

ω0 = 0

ω0 > 0

resonancesrestoring forces

General dispersion for nonconductorGeneral dispersion for nonconductor

Dispersion in polar crystals - phonon polaritonsDispersion in polar crystals - phonon polaritons

( ) 22

22

2

2

2

22

ωωωω

ωεωεω−

−==

T

Lscc

k

Dispersion relation forbetween TO and LO there is no solution forreal values ω and k

photon like

γ = 0

strong coupling of mechanical and electromagnetic waves, polariton like

krti eeE −−∝ ω

no propagation

reflection

γ = 0

frequency gap:

Metals - Drude modelMetals Metals -- Drude modelDrude model

( )

220

1

22220

220

tan;)()(

1ωωωω

A

iωωe/mxeKxxbxm

−=

+−=

−−=→=++

− γωθγω

γωEE&&&

2

2 20

22 2

00

;

p

p

e Neω ω i

Ne Kωm m

ω εγω

ωε

= → = =− −

= =

0 Ep x P x

( ) ( ) ( )( ) 1 ( )

( ) ( ) ( ) ( ) ( )ω ε χ ω ω

ε ω χ ωω ε ε ω ω ε ω ω

=→ = +

= = +0

0 0

P ED E E P

2

2 20

( ) 1 p

ω ω iω

ε ωγω

= +− −

Metals - drude modelMetals - drude model- Intraband transitions- longitudinal plasma oscillations- ω0 = 0 i.e. no restoring force- ωP = plasma frequency

γωωωω

εi

p

−−+= 22

0

2

1

ω0 = 0

γωωω

εip

−−= 2

2

1

2

2

22

2

11ωω

γωω

ε pp −≈+

−=′

( ) 3

2

22

2

ωγω

γωωγω

ε pp ≈+

=′′

ω >> γ = 1/τ (1/ collision time) collisions usually by electron-phonon scattering

εκ == ,0n

( )tie ω−= krEE 00 , == κεn

εωc

k =εωc

ik =

total reflection transmission

generally:γ > 0 leads to damping of transmitted wave n > 0, κ > 0

γ = 0

ωT = 0 ωL = ωP

AluminiumAluminium


εεmm ((ωωpp)= 0 )= 0 longitudinal field longitudinal field bulk bulk plasma oscillation plasma oscillation caused by Coulomb caused by Coulomb

forcesforces. .

if if γγ≠≠0,0, εεmm ((ωω)= 0)= 0 ωω = ωωpp+i+iγγ/2 /2 plasmonplasmon is the quantum of plasma is the quantum of plasma

oscillation with energy oscillation with energy ħħ ωωpp and lifetime and lifetime ττ=2/=2/γγ..

plasmonplasmon is not an electron but a collection of electrons.is not an electron but a collection of electrons.

The damping constant The damping constant γγ is related to the average collision time 1/ is related to the average collision time 1/ ττ

interactions with the lattice vibrations: interactions with the lattice vibrations: electronelectron--phonon scatteringphonon scattering..

non conductor = metals at high frequencies: non conductor = metals at high frequencies: intraband transitions intraband transitions acts mainly acts mainly

at low frequencies at low frequencies Drude model as well.Drude model as well.

at frequencies >> at frequencies >> ωωpp metals are transparent: metals are transparent: ultraviolet transparencyultraviolet transparency..

The Drude model needs to consider the effect of bound electroThe Drude model needs to consider the effect of bound electrons ns lower lying shellslower lying shells

Equation of motion has to include also the Equation of motion has to include also the ““restoring forcerestoring force””2

2 20

( ) 1 pInterbandmx bx x e

ω ω iω

α ε ωγω

+ + = → = +− −

E

20ω m

α=


Free and bound electrons in metalsFree and bound electrons in metals

Bound electrons contribute like a Lorenz oscillator

where

bounddrudemetal εεε +=

drudeε

boundε

ωγωω

εd

dpdrude i−

−= 2

2,1

∑ −−=

j j

jpbound i ωγωω

ωε 22

0

2,

Metals - plasmon polaritonsMetals - plasmon polaritons

Plasmon polariton dispersion (γ = 0)

ωp

2222 kcp += ωω

0

ck=ωkrti eeE −−∝ ω

no propagation

reflection

k

ω

Dielectric function of metals and polar crystalsDielectric function of metals and polar crystalsPolar crystal

strong lattice vibrations (phonons)Metal

collective free electron oscillations (plasmons)no restoring force

plasma frequency(longitudinal oscillation)

transversal opticalphonon frequency, TO

longitudinal opticalphonon frequency, LO

Reststrahlenband

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- Light-matter interaction- SPP dispersion and properties

SPP at metal/dielectric interfacesSPP at metal/dielectric interfacesSPP at metal/dielectric interfaces

From MaxwellFrom Maxwell’’s equations, combining the two curl s equations, combining the two curl eqeq. with . with JJextext==ρρextext=0=0

02

22

2

2

0 =∂∂

∇→∂∂

−=×∇×∇ttD

c-EDE 2

εμ

Assuming in general a harmonic time dependenceAssuming in general a harmonic time dependence

( ) ( ) 2 20

0

, - 0i tt e k

kc

ω εω

−= → ∇ =

=

E r E r E E

Helmholtz equationHelmholtz equation


In the defined geometry In the defined geometry εε = = εε(z) (z) the propagating waves can be described asthe propagating waves can be described as

( ) ( ), , i x

x

x y z ek

β

β=

=

E E z

Propagation constantPropagation constant

inserted into the Helmholtz equation gives the inserted into the Helmholtz equation gives the wave equationwave equation

( ) ( )2

2 202 0

zk

zε β

∂+ − =

∂E

E

which is the starting point for any general guided EM modeswhich is the starting point for any general guided EM modes

Two set of solutionsTwo set of solutions

TM or p modesTM or p modes EExx, , EEzz and and HHyy ≠≠ 00

TE or s modesTE or s modes HHxx, H, Hzz and and EEyy ≠≠ 00


where Re[where Re[εε11]<0 and 1/]<0 and 1/|k|kzz| defines the evanescent decay length | defines the evanescent decay length ┴┴ to the interfaceto the interfacewave confinementwave confinement

( )

( )

( )

2,

2,

2,

2

2 20 2

10 2

1

1

z

z

z

ik zi xy

k zi xx

k zi xz

H z A e e

E z iA k e e

E z A e e

β

β

β

ωε ε

ωε ε

−

−

−

=

=

= −

( )

( )

( )

1,

1,

1,

1

1 10 1

20 1

1

1

z

z

z

k zi xy

k zi xx

k zi xz

H z Ae e

E z iA k e e

E z A e e

β

β

β

ωε ε

ωε ε

=

= −

= −

ε1(ω)

ε2(ω)

TM or p modesTM or p modes


Continuity of Continuity of εεiiEEzz

Continuity of Continuity of EEi,xi,x

1 2

2, 2

1, 1

z

z

A A

kk

εε

=

= −

HHi,yi,y has to fulfill the wave has to fulfill the wave eqeq..2 2 2

1, 0 1

2 2 22, 0 2

0

0z

z

k k

k k

ε β

ε β

+ − =

+ − =

1 20

1 2

k ε εβε ε

=+

Dispersion relationsDispersion relations

Valid for both real and complex Valid for both real and complex εε

TE or s modesTE or s modes

Continuity of Continuity of EEyy HHxx ( )1 1, 2, 1 20 0z zA k k A A+ = → = =

SPP only exist for TM (p) polarizationSPP only exist for TM (p) polarization

2

, 01 2

ii zk k ε

ε ε=

+


The interface mode have to fulfill some conditions in order to eThe interface mode have to fulfill some conditions in order to existxist

Im[ ( )] Re[ ( )]i iε ω ε ω<

Propagating interface waves (The Propagating interface waves (The dispersion relation is valid) dispersion relation is valid) 1 2 1 2 1 2 1 2( ) 0 ( ) 0real orβ ε ε ε ε ε ε ε ε⇒ ∧ + < ∧ + >

Bound solution Bound solution vertical vertical components are imaginary components are imaginary

ββ complex complex damped propagation along the interfacedamped propagation along the interface

1 2 0ε ε+ <

( ) ( ) ( ) ( )1 2 1 2 0ε ω ε ω ε ω ε ω∧ + <⎡ ⎤⎣ ⎦

One of the dielectric functions must be negative and > than the One of the dielectric functions must be negative and > than the otherother

dm

dmx c

kεε

εεω+

⎟⎠⎞

⎜⎝⎛=

22

( )dm

dzd c

kεε

εω+

⎟⎠⎞

⎜⎝⎛=

222

( ) xdmm k real and 0Re →>< εεε

kzd and kzm are imaginary

Dispersion relation of SPPsDispersion relation of SPPs

( )dm

mzm c

kεε

εω+

⎟⎠⎞

⎜⎝⎛=

222

( )tzkxki zxe ω−±±= 0SP EE

dpSP ε

ωω+

=1

1

ωp

photonin air

kx

ω

xck=ω

dεε −→′

surface plasmon polariton

xdm

dm ckεεεεω +

=


surface plasmon polaritons are bound waves surface plasmon polaritons are bound waves SPP excitations lie on the right of the SPP excitations lie on the right of the light linelight line

Radiation into metal occurs if Radiation into metal occurs if ωω > > ωωpp

Between the bound and the radiative regime Between the bound and the radiative regime ββ is imaginary is imaginary no propagationno propagation

for small k (<IR),for small k (<IR), ββ is close to kis close to k00 and the light lineand the light line

for large k, for large k, ωωspsp = = ωωpp / (1+/ (1+εε22))1/21/2 ~ ~ ωωpp / (2)/ (2)1/21/2

in the limit in the limit ImIm[[εε11((ωω)] = 0)] = 0, , ββ →→ ∞∞ and and vvgg →→ 0 0 the mode acquire an electrostatic the mode acquire an electrostatic character character Surface PlasmonSurface Plasmon

but real metals suffer also from intraband transitions but real metals suffer also from intraband transitions dampingdamping and and εε11 is complex is complex the the quasibound regimequasibound regime is allowedis allowed

at at ωω ~ ~ ωωspsp better confinement of the SPP but small propagation length ( better confinement of the SPP but small propagation length ( increased increased damping)damping)

photonin air

kx

ω

dpSP ε

ωω+

=1

1

xck=ω

dεε −→′

surface plasmon polariton

2222xp kc+= ωω

ωp

plasmonpolariton

xdm

dm ckεεεεω +

=

Volume vs. surface plasmon polaritonVolume vs. surface plasmon polariton

surface plasmonsnon-propagatingcollective oscillationsof electron plasmanear the surface

with dampingvolume plasmon

ωp

photonin air

xk

ω

1−→′ε

surface plasmonpolariton

ωLO

photonin air

1−→′εsurface phononpolaritonωTO

SP dispersion - plasmon vs. phononSP dispersion - plasmon vs. phonon

ω

xk

Plasmon polaritons:

Light-electron coupling in • metals • semiconductors

Phonon polaritons:

Light-phonon coupling in polar crystals• SiC, SiO2• III-V, II-VI-semiconductors

εεω 1+

= xck

Drude model

1=dε

SP propagation lengthSP 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 1

2−=

=

ε

ωω p

SP

( ) 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 ,, === mzmz LLλ

nm 31 and nm 280 :nm 600 ,, === mzmz 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

ω

mε ′

SPω

1−

The mag. field H isparallel to surfaceand perpendicular to propagation

d

m

x

zd iEE

εε−

=

+ + - -xE

zE

zx

⊗yH

El. field

Dispersion and excitation of SPPDispersion and excitation of SPPDispersion and excitation of SPP

SPP are 2D EM waves propagating at the interface conductorSPP are 2D EM waves propagating at the interface conductor--dielectric (bound waves)dielectric (bound waves)

ββ > > kkdd evanescent decay at both interfaces evanescent decay at both interfaces confinement confinement

SPP dispersion curve lies to the right of the light lineSPP dispersion curve lies to the right of the light line

excitation by 3D light beams is excitation by 3D light beams is not possiblenot possible

phasephase--matching techniques are requiredmatching techniques are required

Excitation by charged particle impactExcitation by charged particle impactR. H. Ritchie, Phys. Rev. 106, 874 (1957) theoretical investigations of plasma losses in thin metallic films

predicted an additional loss at ħωp/(2)1/2

measured by C. J. Powell and J. B. Swan, Phys. Rev. 118, 640 (1960)

( ) ( )2

20 11

p pm d m sp

d

ω ωε ω ε ε ω ω

ω εβ

+ = ∧ = − ⇒ =+

⇒ →∞

Quasi-static electromagnetic surface modes


Bulk plasmonSurface plasmon

Progressive oxidation

Dispersion and excitation of SPPsDispersion and excitation 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

zx

E0θ

R

kxdε

mε

0ε

Excitation by ATRExcitation by ATR

Kretschmann configuration Otto configuration

zx

E0θ

R

kx

dεmε

0ε

total reflection at prism/metal interface-> evanescent field in metal-> excites surface plasmon polariton at

interface metal/dielectric medium

metal thickness < skin depth

total reflection at prism/dielectric medium-> evanescent field excites surface plasmon

at interface dielectric medium/metal

usful for surfaces that should not be damagedor for surface phonon polaritons on thick crystals

distance metal – prism of about λ

ATR: Attenuated Total Reflection

zx

θ

R

kSP,xdε

mε

0ε kphoton,x

< kSP,xkphoton,x

θ

R

kSP,x

kphoton,x

= kSP,xkphoton,x

no SPP excitation SPP excitation

Excitation by ATRExcitation by ATR

SPP excitation requires = kSP,xkphoton,x

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


HeNeHeNe SPP excitation at AuSPP excitation at Au--air interface (ATR)air interface (ATR)

36 40 44 48

0.3

0.6

0.9

43.92º

41.92º

810nm p-polarized, Δ~0.207º 1160nm p-polarized, Δ~0.099º 632nm p-polarized, Δ~0.98º

Ref

lect

ivity

[arb

.]

Incident angle [degrees]

42.57º

64 nm thick Au film

36 38 40 42 44 46 48

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

43.92º

633nm p-polarized, data 633nm p-polarized, theory

Ref

lect

ivity

[arb

.]

Incident angle [degrees]

64 nm thick Au film

Methods of SPP excitationMethods of SPP excitation

nprism > nL !!


Excitation by grating couplingExcitation by grating couplingk//

kθ

kg // sinsin2

g

k kk g

k ga

θβ θ νπ ν ν

=→ = +

= =

E. Devaux, T. W. Ebbesen, J.C. Weeber, A. Dereux, Appl. Phys. Lett. 83, 4936 (2003)


Periodicity a = 760 nm & p-pol beam Periodicity a = 760 nm & s-pol beam

Periodicity a = 700 nm & p-pol beam


Excitation by highly focused optical beamsExcitation by highly focused optical beams

A. Bouhelier and G. P. Wiederrecht, Opt. Lett. 30, 884 (2005)

0arcsinSPP cnk

βθ θ⎛ ⎞= >⎜ ⎟⎝ ⎠


Small aperture Small aperture wavevector wavevector components kcomponents k0 0 < < ββ < k < k phase phase

matched excitationmatched excitation

NearNear--field excitationfield excitation

B. Hecht, H. Bielefeld, L. Novotny, Y. Inouye, D. W. Pohl Phys. Rev. Lett. 77, 1889 (1996)

Localized Surface PlasmonLocalized Surface PlasmonLocalized Surface Plasmon

OverviewOverview

True color image of a sample containing gold and silver nanospheTrue color image of a sample containing gold and silver nanospheres as well as gold nanorods photographed with a dark res as well as gold nanorods photographed with a dark field microscope.field microscope.

Each dot corresponds to light scattered by an individual nanoparEach dot corresponds to light scattered by an individual nanoparticle at the plasmon resonance. The resonance ticle at the plasmon resonance. The resonance wavelength varies from blue (silver nanospheres) via green and ywavelength varies from blue (silver nanospheres) via green and yellow (gold nanospheres) to orange and red (nanorods).ellow (gold nanospheres) to orange and red (nanorods).


OverviewOverview

Lycurgus cupLycurgus cup

Illuminated from insideIlluminated from inside Illuminated from outsideIlluminated from outside


OverviewOverview

Localized Surface PlasmonLocalized Surface Plasmon

SPP are 2D, dispersive EM waves propagating at the interface cSPP are 2D, dispersive EM waves propagating at the interface conductoronductor--dielectric dielectric

SPSP are are nonnon--propagating collective oscillations of electron plasma near the propagating collective oscillations of electron plasma near the surfacesurface

LSP are LSP are nonnon--propagating excitations of the conduction electrons of a metallipropagating excitations of the conduction electrons of a metallic c nanostructure coupled to an EM field.nanostructure coupled to an EM field.


The curved surface of the nanostructure act as a restoring forThe curved surface of the nanostructure act as a restoring force ce immobile positively immobile positively charged core ioncharged core ion

The curved surface of the nanostructure allows the excitation ofThe curved surface of the nanostructure allows the excitation of the LSP by 3D lightthe LSP by 3D light

The resonance falls into the visible region for Au and Ag nanoThe resonance falls into the visible region for Au and Ag nanoparticlesparticles


( ) ( )2 2 2

2 2 2 2 21 1p p pi

iω ω ω Γ

ε ωω Γω ω Γ ω ω Γ

= − = − ++ + +

2

02iwt

e em m e et t

−∂ ∂+ Γ =

∂ ∂r r E Fv

lΓ =

it is not possible to ignore the contribution to the dielectric function of the interband transitions

Drude Free Electron Model

νF Fermi velocity (about 1.4 nm/fs in the case of Au and Ag) l is the electron mean free path (lAu=42 nm and lAg=52 nm at 273K )

Particle in an electrostatic fieldParticle in an electrostatic fieldnanoparticle acts as an nanoparticle acts as an

electric dipoleelectric dipole

Mie TheoryMie Theoryrigorous electrodynamic rigorous electrodynamic

approachapproach


SubSub--wavelength metal particles wavelength metal particles

E0

εd

P

za

electrostatic approach electrostatic approach Laplace equationLaplace equation for the electric for the electric potential + boundary conditionspotential + boundary conditions

02∇ Φ = → ∇ΦE = -

0

30 0 2

3 cos2

coscos2

din

d

dout

d

E r

E r E ar

ε θε ε

ε ε θθε ε

Φ = −+

−Φ = − +

+

Applied fieldApplied field Oscillating dipole fieldOscillating dipole field

ε(ω)

θ


0 30

30 0

cos4

42

outd

dd

d

E rr

a

θπε ε

ε επε εε ε

⋅Φ = − +

−=

+

p r

p E0 0dε ε α=p E

0

0 30

32

3 ( ) 14

d

d

d r

εε ε

πε ε

=+

∇Φ →⋅ −

= +

in

out

E EE = -

n n p pE E


3 ( )4( ) 2

d

d

a ε ω εα πε ω ε

−=

+

Complex polarizabilityComplex polarizability of a subof a sub--wavelength diameter wavelength diameter in the electrostatic approximationin the electrostatic approximation


( ) 2 0dε ω ε+ → Complex polarizability enhancementComplex polarizability enhancement

[ ] [ ]Re ( ) 2 Im ( )d if is smallε ω ε ε ω= −

FrFrööhlichlic conditioncondition


( )

242 4 6

3

( )86 3 ( ) 2

( )Im 4 Im( ) 2

dsca

d

dabs

d

k k a

k ka

ε ω επσ απ ε ω ε

ε ω εσ α πε ω ε

−= =

+

⎡ ⎤−= = ⎢ ⎥+⎣ ⎦

Corresponding Corresponding absorptionabsorption & & scatteringscattering cross sectionscross sections calculated via the Pointing vector calculated via the Pointing vector from the full EM field associated with an oscillating dipolefrom the full EM field associated with an oscillating dipole

ext sca absσ σ σ= +

( ) ( ) ( )0ext ext

II

Sω

ω σ ω=


Corresponding Corresponding absorptionabsorption & & scatteringscattering cross sectionscross sections calculated via Mie Theory in the case of large calculated via Mie Theory in the case of large particles where the electrostatic approx. brakes down and the reparticles where the electrostatic approx. brakes down and the retardation effects are to be consideredtardation effects are to be considered

( )( )( ) ( )

2 22

1

21

2 2 1

2 2 1 Re

sca n nn

ext n nn

n a bk

n a bk

πσ

πσ

∞

=

∞

=

= + +

= + +

∑

∑

abs ext scaσ σ σ= −


Damping mechanismDamping mechanism

2

*2 1 2

2

1 1 12

T

T T T

Γ =

= +

*2 1 2 1

2

25 10T T T T

fs T fs→ =

≤ ≤

In a quasiIn a quasi--particle picture, damping is described as particle picture, damping is described as population decay population decay raradiative (by emission of a photon), or diative (by emission of a photon), or

nonradiativenonradiative

DrudeDrude--SommerfeldSommerfeld model model plasmon is aplasmon is asuperposition of many independent electron oscillationssuperposition of many independent electron oscillations

Nonradiative dampingNonradiative damping due to a dephasing of the due to a dephasing of the oscillation of individual electrons (scattering events with oscillation of individual electrons (scattering events with

phonons, lattice ions, other conduction or core electrons, the phonons, lattice ions, other conduction or core electrons, the metal surface, impurities, etc.)metal surface, impurities, etc.)

PauliPauli--exclusion principle, the electrons can only be excited exclusion principle, the electrons can only be excited into empty states in the conduction band into empty states in the conduction band interinter-- and and

intraband excitationsintraband excitations by electron from either the dby electron from either the d--band or band or the conduction bandthe conduction band

Pure dephasing time Pure dephasing time (elastic collisions)(elastic collisions)

Decay time (radiative & non Decay time (radiative & non energy loss processes)energy loss processes)


Fbulk

vAR

Γ Γ= +

Damping mechanism for very small Damping mechanism for very small nanoparticles (< 10nm)nanoparticles (< 10nm)

Experimental methodsExperimental methods

Total Internal Reflection Microscopy Total Internal Reflection Microscopy (TIRM)(TIRM)

Dark Field Microscopy Dark Field Microscopy in reflectionin reflection

Dark Field Microscopy Dark Field Microscopy in transmissionin transmission


Detection and Spectroscopy of Gold Nanoparticles Using SupercontDetection and Spectroscopy of Gold Nanoparticles Using SupercontinuuminuumWhite Light Confocal MicroscopyWhite Light Confocal Microscopy

K. Lindfors, T. Kalkbrenner, P. Stoller, V. Sandoghdar, PRL 93, 037401-1 (2004)

3 ( )( ) ( )2 ( ) 2

is i i

dd

d

E sE s e E

Ds

ϕ

ε ω επλ ηα λ ηεε ω ε

= =

−→ = =

+

2r iE rE e π−=

{ }2 2 22 2 sinm r s iI E E E r s r s ϕ= + = + −

22

2

( ) ( )( ) ( ) 2 ( ) sin( )

m r

r

I II r rλ λ η ησ λ α λ α λ ϕ

λ−

= = −


(a) 60 ± 12 nm(b) 31 ± 6 nm(c) 20 ± 4 nm(d) 10.3 ± 1.0 nm(e) 5.4 ± 0.8 nm


0 1 2 3 4 5

5,4x105

5,6x105

5,8x105

6,0x105

6,2x105

6,4x105

6,6x105

cps

Position (μm)

20nm Au particles

240 nm

250 nm

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Surface Plasmon Polaritons (SPPs) Introduction and … · Surface Plasmon Polaritons (SPPs) -Introduction and basic properties ... Overview articles on Plasmonics: ... Surface Plasmon