Post on 14-Apr-2018
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
OverviewOverviewOverview
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,
OverviewOverviewOverview
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
Metals - Drude modelMetals Metals -- Drude modelDrude model
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=
Metals - Drude modelMetals Metals -- Drude modelDrude model
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
SPP at metal/dielectric interfacesSPP at metal/dielectric interfacesSPP at metal/dielectric interfaces
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
SPP at metal/dielectric interfacesSPP at metal/dielectric interfacesSPP at metal/dielectric interfaces
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
SPP at metal/dielectric interfacesSPP at metal/dielectric interfacesSPP at metal/dielectric interfaces
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=
+
SPP at metal/dielectric interfacesSPP at metal/dielectric interfacesSPP at metal/dielectric interfaces
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 +
=
SPP at metal/dielectric interfacesSPP at metal/dielectric interfacesSPP at metal/dielectric interfaces
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
Dispersion and excitation of SPPDispersion and excitation of SPPDispersion and excitation of SPP
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
Dispersion and excitation of SPPDispersion and excitation of SPPDispersion and excitation of SPP
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 !!
Dispersion and excitation of SPPDispersion and excitation of SPPDispersion and excitation of SPP
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)
Dispersion and excitation of SPPDispersion and excitation of SPPDispersion and excitation of SPP
Periodicity a = 760 nm & p-pol beam Periodicity a = 760 nm & s-pol beam
Periodicity a = 700 nm & p-pol beam
Dispersion and excitation of SPPDispersion and excitation of SPPDispersion and excitation of SPP
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⎛ ⎞= >⎜ ⎟⎝ ⎠
Dispersion and excitation of SPPDispersion and excitation of SPPDispersion and excitation of SPP
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).
Localized Surface PlasmonLocalized Surface PlasmonLocalized Surface Plasmon
OverviewOverview
Lycurgus cupLycurgus cup
Illuminated from insideIlluminated from inside Illuminated from outsideIlluminated from outside
Localized Surface PlasmonLocalized Surface PlasmonLocalized Surface Plasmon
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.
Localized Surface PlasmonLocalized Surface PlasmonLocalized Surface Plasmon
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
Localized Surface PlasmonLocalized Surface PlasmonLocalized Surface Plasmon
( ) ( )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
Localized Surface PlasmonLocalized Surface PlasmonLocalized Surface Plasmon
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
Localized Surface PlasmonLocalized Surface PlasmonLocalized Surface Plasmon
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
Localized Surface PlasmonLocalized Surface PlasmonLocalized Surface Plasmon
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
Localized Surface PlasmonLocalized Surface PlasmonLocalized Surface Plasmon
( ) 2 0dε ω ε+ → Complex polarizability enhancementComplex polarizability enhancement
[ ] [ ]Re ( ) 2 Im ( )d if is smallε ω ε ε ω= −
FrFrööhlichlic conditioncondition
Localized Surface PlasmonLocalized Surface PlasmonLocalized Surface Plasmon
( )
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=
Localized Surface PlasmonLocalized Surface PlasmonLocalized Surface Plasmon
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= −
Localized Surface PlasmonLocalized Surface PlasmonLocalized Surface Plasmon
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)
Localized Surface PlasmonLocalized Surface PlasmonLocalized Surface Plasmon
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
Localized Surface PlasmonLocalized Surface PlasmonLocalized Surface Plasmon
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−
= = −
Localized Surface PlasmonLocalized Surface PlasmonLocalized Surface Plasmon
(a) 60 ± 12 nm(b) 31 ± 6 nm(c) 20 ± 4 nm(d) 10.3 ± 1.0 nm(e) 5.4 ± 0.8 nm
Localized Surface PlasmonLocalized Surface PlasmonLocalized Surface Plasmon
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