3-A Basic of Surface Plasmons
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Transcript of 3-A Basic of Surface Plasmons
Surface plasmons and its dispersion relation
(Third Lecture) Techno Forum on Micro-optics and Nano-optics Technologies
Surface plasmons and its dispersion relation 송 석 호, 한양대학교 물리학과, http://optics.anyang.ac.kr/~shsong
TM pol
1. What is the surface plasmon (polaroton)?2. What is the dispersion relation of SPs?Key
TM pol.
2. What is the dispersion relation of SPs?3. How can the SP modes be excited?4. What can we play with SPPs for nanophotonics?
Key notes
Plasmon = plasma wave (oscillation)Plasmon plasma wave (oscillation)
Plasmons = density fluctuation of free electrons
Bulk plasmons
Plasmons in the bulk oscillate at ωp determined by the free electron density and effective mass
+ + +
2
Bulk plasmons
- - -k
0
2
εω
mNedrude
p =
Surface plasmon polaritonsPlasmons confined to surfaces that can interact with light to
form propagating “surface plasmon polaritons (SPP)”+ - +
ε εω
C fi t ff t lt i t SPP dLocalized plasmons
m dsp
m d
kc
ε εωε ε
=+
Confinement effects result in resonant SPP modes
in nanoparticles21 Nedrude
031
εω
mNedrude
particle =
Surface Surface plasmonsplasmons??
물방울중력
표면장력표면장력
용액/기판
SPP
용액/기판
전자기력
TM pol.
44
금속/주파수
Surface plasmons
Definitions: collective excitation of the free electrons in a metal
p(Gary Wiederrecht, Purdue University)
Definitions: collective excitation of the free electrons in a metalCan be excited by light: photon-electron coupling (polariton)Thin metal films or metal nanoparticlesBound to the interface (exponentially decaying along the normal)Bound to the interface (exponentially decaying along the normal)Longitudinal surface wave in metal filmsPropagates along the interface anywhere from a few microns to
l illi t (l l ) b t lseveral millimeters (long range plasmon) or can be extremely confined in nanostructures (localized plasmon)
Note: SP is a TM wave!
표면표면 플라즈몬플라즈몬 vs. vs. 표면표면 플라즈몬플라즈몬 폴라리톤폴라리톤
•• 표면표면 플라즈몬플라즈몬 (Surface Plasmon, SP)(Surface Plasmon, SP)Dielectric
dε x
z
y⊗
– 금속표면의 전하(자유전자) 진동 → 표면 플라즈마
– 양자화된 표면 플라즈마 진동 → 표면 플라즈몬
ze 1κ−
ze 2κ– – – – + + + + + +
Metal mε
•• 표면표면 플라즈몬플라즈몬 폴라리톤폴라리톤 (Surface Plasmon (Surface Plasmon PolaritonPolariton, SPP), SPP)
– 표면 플라즈몬 (자유전자 진동)과 전자기파가 결합되어 있는 상태 SPP
– 금속과 유전체의 경계면을 따라 진행금속과 유전체의 경계면을 따라 진행
– 금속 표면에 수직한 TM 편광 특성
– 전송거리는 수십~수백 mm로 제한
TM pol.
66
Local field intensity depends on wavelength
Surface plasmons
y p g
(small propagation constant k) (large propagation constant k)(small propagation constant, k) (large propagation constant, k)
Note: Dielectric constants of optical materialsMaterial permittivity
0( ) ( ) ( )rD Eω ε ε ω ω= (spatially local response of media)
( )ε ω : relative dielectric constant( )rε ω= relative permittivity= dielectric function
Insulating media (dielectric) : Lorentz model
Conducting media (Metal, in free-electron region) : Drude model
Conducting media (Metal, in bound-electron region) : Drude-Sommerfeld model
Extended Drude model (Lorentz-Drude model)( )
Dielectric constant (relative permittivity)Material permittivity
2 / Nωj jN α∑C
Lorentz model for dielectric (insulator)
2 2
/,p j
jj
Ni
ωα
ω ω γω=
− −13
( ) 1 ,1
j jj
rj j
jN
ε ωα
= +−
∑∑
2 jj
Cm
ω =
Drude model for metal in free-electron region2 2 2
2 2 2 3 2( ) 1 1p p pr i
ω ω ω γε ω
⎛ ⎞ ⎛ ⎞= − = − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
Modified Drude model for metal in bound-electron region
2 2 2 3 2( )r iω ωγ ω γ ω ωγ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+ + +⎝ ⎠ ⎝ ⎠
Modified Drude model for metal in bound electron region 2 2 2
2 2 2 3 2( ) p p pr i
iω ω ω γ
ε ω ε εω ωγ ω γ ω ωγ∞ ∞
⎛ ⎞ ⎛ ⎞= − = − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+ + +⎝ ⎠ ⎝ ⎠
Extended Drude (Drude-Lorentz) model
2 2⎛ ⎞⎛ ⎞ Δ Ω
γ γ γ⎝ ⎠ ⎝ ⎠
( )2 2
2 2 2( )
p L
rL Li i
ω εε ω εω ωγ ω ω∞
⎛ ⎞⎛ ⎞ Δ Ω⎜ ⎟= − −⎜ ⎟⎜ ⎟ ⎜ ⎟+ − Ω + Γ⎝ ⎠ ⎝ ⎠
2 2 2⎛ ⎞ ⎛ ⎞
Drude model for metals: Dielectric constant of free-electron plasmaMetal permittivity
2 2 2
2 2 2 3 2( ) 1 1p p pr i
iω ω ω γ
ε ωω ωγ ω γ ω ωγ
⎛ ⎞ ⎛ ⎞= − = − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+ + +⎝ ⎠ ⎝ ⎠
2
22 2
0 00
pN ecm
ω γ σ με
= = =
(1) f
2
0 :N e static conductivitym
σγ
=
2 2 2 2
2 3 2 3( ) 1 1p p p pi iω ω ω ω
ε ω⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
≈ − + = − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
(1) For an optical frequency, ωvisible >> γ
2 3 2 3( ) 1 1/r i iε ω
ω ω γ ω ω τ+ +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
(2) Ideal case for metals as an undamped free-electron gas
• no decay (infinite relaxation time)• no interband transitions
2ω0( )r
τγε ω →∞
→⎯⎯⎯→ 2( ) 1 pr
ωε ω
ω= −
Dispersion relation for bulk plasmonsBulk plasmons
( )k
• Dispersion relation:
( )kω ω=
surface plasmon plaritons
Dispersion relation for surface plasmon polaritons
Let’s solve the curl equations for TE & TM modes with boundary conditions
0
0
( , , ) ( ) : ( 0) & ( 0)
( , , ) ( )
xi
xi
jk xi i i i i
jk xi i i i
H i E E x y z E z e i d z i m z
E i H H x y z H z e
ωε ε
ωμ
⎡ ⎤∇× = − = = > = <⎣ ⎦⎡ ⎤∇× = + =⎣ ⎦
TE mode TM mode
0 ( , , ) ( )i i i iyμ ⎣ ⎦
TE mode TM mode
( ) (0, ,0), ( ) ( ,0, )
(0) (0)
(0) (0)
i yi i xi zi
yd ym
E z E H z H H
E E
H H
= =
=
( ) ( ,0, ), ( ) (0, ,0)
(0) (0)
(0) (0)
i xi zi i yi
yd ym
E z E E H z H
H H
E E
= =
=
=(0) (0)xd xmH H= (0) (0)xd xmE E=
surface plasmon plaritons
TE modes : ( ) (0, ,0)
( ) ( 0 )i yiE z E
H H H
=
( ) ( ,0, )i xi ziH z H H=
0 0 0 xi zi xii i i i yi xi zi i yi
H H HH i E i E ik H i Eωε ε ωε ε ωε ε∂ ∂ ∂∇× = − → − = − → − = −
∂ ∂ ∂
0 0 0
y y
yi yizii i xi xi
yi xi
z x zE EEE i H i H i H
y z zE E H k E
ωμ ωμ ωμ
∂ ∂ ∂∂ ∂∂
∇× = + → − = + → = −∂ ∂ ∂
∂ ∂ H
22 202 ( ) 0yi
i xi yi
Ek k E
zε
∂+ − =
∂
0 yi xizi xii H ik E
x yωμ∂
→ − = + →∂ ∂ 0 yi zii Hωμ=
We want wave solutions propagating in x-direction, but confined to the interface with evanescent decay in z-direction.
[ ]( ) : ( ), ( ); Re 0 xi zijk x k zyi i ziE z Ae e i d i m k±= − = + = >
0 ( ) x zyi ik x k zzixi xi i
E ki H H z iA e eωμ ±∂= − → = ±Curl equation 0
0
( )xi xi izμ
ωμ∂
(0) E (0) & (0) (0)yd ym xd xmE H H= =
Curl equation
Boundary cond.
& ( ) 0d d d zd zmA A A k k= + =
0A A= = 0d mA A= =
No surface modes exist for TE polarization !
TM modes : ( ) ( ,0, )i xi ziE z E E=
surface plasmon plaritons
TM modes :( ) (0, ,0)i yiH z H=
),0,( ziixii EiEi ωεωε −−),0,( yixiyizi HikHik−
xmmymzm EHk ωε=xiiyizi EHk ωε=
xm xdE E=
xddydzd EHk ωε=
zm zdk k=
xm xd
ym ydH H=
kkm dε εyd
d
zdym
m
zm HkHkεε
=
surface plasmon plaritons
TM modes :
• For any EM wave:2
2 2 2i x zi x xm xdk k k , where k k kωε ⎛ ⎞= = + ≡ =⎜ ⎟⎝ ⎠
y i x zi x xm xd,c⎜ ⎟
⎝ ⎠
SP Dispersion Relation
m dx
m d
kc
ε εωε ε
=+
TM modes :surface plasmon plaritons
1/ 2
' " m dx x xk k ik
cε εω
ε ε⎛ ⎞
= + = ⎜ ⎟+⎝ ⎠x-direction: ' "
m m miε ε ε= +m dc ε ε+⎝ ⎠
1/ 22
' ii i ik k ik εω ⎛ ⎞
= + = ± ⎜ ⎟z-direction:2
2 2zi i xk kωε ⎛ ⎞= −⎜ ⎟
⎝ ⎠
m m m
For a bound SP mode:
zi zi zim d
k k ikc ε ε
+ ± ⎜ ⎟+⎝ ⎠zi i xc⎜ ⎟
⎝ ⎠
kzi must be imaginary: εm + εd < 0
2 2ω ω ω⎛ ⎞ ⎛ ⎞ ⎛ ⎞2 2 zi i x x i x ik k i k kc c cω ω ωε ε ε⎛ ⎞ ⎛ ⎞ ⎛ ⎞= ± − = ± − ⇒ >⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
+ for z < 0
k’x must be real: εm < 0- for z > 0
So, 'm dε ε< −
1/ 2' " m d
x x xk k ikc
ε εωε ε
⎛ ⎞= + = ⎜ ⎟+⎝ ⎠
' "m m miε ε ε= +
surface plasmon plaritons
( ) 21
2"4221
⎤⎡ ++⎤⎡
m dc ε ε+⎝ ⎠ m m m
( )( )
1
2
2"2'
'
2)(
⎤⎡
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡ ++
⎥⎥⎦
⎤
⎢⎢⎣
⎡
++= dmee
mdm
dx c
kεεεε
εεεεω
( )( )
( )
2
2"42
2"21
2"2'
"
2)(⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎭⎬⎫
⎩⎨⎧ ++⎥
⎥⎦
⎤
⎢⎢⎣
⎡
++=
dmee
dm
mdm
dx c
kεεεε
εεεεε
εω
( ) ( ) '2"2'2 , mdmmewhere εεεεε ++=⎥⎦⎢⎣ ⎭⎩
metals,ofmost in and , ,0 "'''mmdmm εεεεε >>><
,
,,, mmdmm
2/1'' dεεω ⎟
⎞⎜⎛
"2/3'
''
dm
dmx c
k
εεεω
εεεεω
⎟⎞
⎜⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛+
≈
( )2''"
2 m
m
dd
dmx c
kεε
εεεεω
⎟⎟⎠
⎞⎜⎜⎝
⎛+
≈
surface plasmon plaritons
h l h f hi h h i i d /
Propagation length
3/ 2
The length after which the intensity decreases to 1/e :
⎛ ⎞( )3/ 2' "
1" " 1 2 1' ' 21 2 1
2 , where 2( )i x xL k k
cε ε εω
ε ε ε− ⎛ ⎞
= = ⎜ ⎟+⎝ ⎠
Plot of the dispersion relation : For ideal free-electronssurface plasmon plaritons
• Plot of the dielectric constants:
2
2
1)(ωω
ωε pm −=
ε εω
• Plot of the dispersion relation:
22 )( εωωm dx
m d
kc
ε εωε ε
=+ 22)1(
)(
pd
dpspx c
kkωωεεωωω
−+
−==
p
dm
ωωω
εε
=≡∞→⇒
−→•
k
, When
dspωω
ε+=≡∞→⇒
1 ,k x
Surface plasmon dispersion relation:Surface plasmon dispersion relation surface plasmon plaritons
2/1
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=d
dmx c
kεε
εεω1/ 22
izi
m d
kc
εωε ε
⎛ ⎞= ⎜ ⎟+⎝ ⎠⎠⎝ + dmc εε
ωreal kx
xckRadiative modes
2 2 2 2p xc kω ω= +
m d⎝ ⎠
ωp
real kx real kz
dε (ε'm > 0)
ω
imaginary kx real kz
Quasi-bound modes(−εd < ε'm < 0)
d
p
ε
ω
+1
l kBound modesDielectric: real kx imaginary kz
Bound modesDielectric: εd
Metal: ε ε ' + xz
(ε'm < −εd)
Re kx
Metal: εm = εm + εm
"
Dispersion relation for bulk and surface plasmonssurface plasmon plaritons
2/1
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=dm
dmx c
kεε
εεω
2 2 2 2
2 2 3 311
p pm i
ω τ ω τε
ω τ ωτ ω τ⎛ ⎞ ⎛ ⎞
= − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠
Cut-off frequency of SP
d
psp
d
pdpd
pm ωωω
εω
εω
εωεωω
ε+
=⇒+
=⇒−=−−=−=11
, 1When 2
2222
2
Ag/air, Ag/glasssurface plasmon plaritons
2 2 2 2' " p pi i
ω τ ω τε ε ε ε
⎛ ⎞ ⎛ ⎞= + = − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟2 2 3 31m m m Bi iε ε ε ε
ω τ ωτ ω τ= + = − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠
Silver(Ag) dispersion 5
SP Ag/air
For noble metals : J&C measured constants( g) p
3
4
light line air
[eV]
SP Ag/air SP Ag/glass
light line glass 300
[nm
]
0 10 20 30 40 50 601
2
E
1
15001200900
600
0.1 1 10 100
λ
L [ ]
kx [um-1] L [um]Gold(Au) dispersion
4
5
light line air
SP Au/air
300
1
2
3
E [e
V] SP Au/glass
light line glass
15001200900
600
λ [n
m]
0 5 10 15 20 25 30 35 40
kx [um-1]
0.1 1 10 1001500
L [um]
Copper(Cu) dispersion
4
5
SP Cu/air light line air 300300
2
3
4
E [e
V]
SP Cu/glass
light line glass
600600
λ [n
m]
λ [n
m]
0 10 20 30 40 50 601
kx [um-1]
0.1 1 10 10015001200900
0.1 1 10 10015001200900
L [um]
L [um]
X-ray wavelengths at optical frequencies
surface plasmon plaritons
Very small SP wavelength
λvac=360 nm
AgSiO2
Penetration depthsurface plasmon plaritons
' 1'1 2
1At large ( ), z .ixx
kk
ε ε→ − ≈ Strong concentration near the surface in both media.
( : , : - )z xE iE air i metal i= ± +
'1At low ( 1),xk ε >> Larger Ez component'
1 in air :z
x
Ei
Eε= −
1 i t lzEi Smaller Ez component
'1
in metal :z
x
iE ε
=
Gooood waveguide!
Generalization : Surface Electric Polaritons and Surface Magnetic Polaritons
: Energy quanta of surface localized oscillation of electric or magnetic dipoles in coherent manner
Surface Electric Polariton (SEP) Surface Magnetic Polariton (SMP)
+q -q +q -q SNSN
E Hq q q q SNSN
Coupling to TM polarized EM wave Coupling to TE polarized EM wave
Common Features
- Non-radiative modes → scale down of control elements
- Smaller group velocity than light coupling to SP
- Enhancement of field and surface photon DOS
Importance of understanding the dispersion relation :Total external reflection
Slow Propagation, Anomalous Absorption, and Total External Reflection of Surface Plasmon Polaritons in Nanolayer Systemsof Surface Plasmon Polaritons in Nanolayer Systems
A layer of a high-permittivity dielectric on the surface of a metal plays the role of a near-perfect i i th t t l fl ti f SPP f it t t l t l fl timirror causing the total reflection of SPPs from it. total external reflection
Importance of understanding the dispersion relation :Broadband slow and subwavelength light in air
Importance of understanding the dispersion relation : Negative group velocityNegative group velocity
0<dkdω
SiO2
Si N21 SiO
p
εω+
0=dkdω
dk
Re kx
Si3N4
431 NSi
p
εω
+x
SiO2
Si3N4
Al
Excitation of surface plasmonsp
The large k of SP needs specific configurations!
0( )
( )m d
sp spk n k ε ω ε ω⎛ ⎞= = ⎜ ⎟⎝ ⎠
2
1 pk kω ω⎛ ⎞
⎜ ⎟0 ( )sp sp
m d cε ω ε ⎜ ⎟+ ⎝ ⎠0 21 pbp bpk n k
cω⎛ ⎞= = − ⎜ ⎟⎝ ⎠ 2
2( ) pm i
ωε ω ε∞= −
ωreal kx c kω =
Radiative modes( ' > 0)
2 2 2 2p bpc kω ω= +
2( )m iω ωγ∞ +
ωp
real kz
imaginar k
0d
kωε
=
Quasi-bound modes
(ε m > 0)
d
p
ε
ω
+1
imaginary kx real kz
Quasi bound modes(−εd < ε'm < 0)
d
real kx imaginary kz
Bound modesDielectric: εd
M l ' "xz
(ε'm < −εd)
Re k
Metal: εm = εm' + εm
"
hnh
2 2 2⎛ ⎞ ⎛ ⎞( )
2 2 2
2 2 2 3 2( ) p p pr i
iω ω ω γ
γ ω ε ω ε εω ωγ ω γ ω ωγ∞ ∞
⎛ ⎞ ⎛ ⎞← = − = − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+ + +⎝ ⎠ ⎝ ⎠
Note: R does to zero at resonance when i radΓ = Γ
//,d spk k mG= ±
dε// sin sind d dk k ωθ ε θ= =
//,sp dk k mG= ±
dε
metal
//, sin sind d dk kc
θ ε θ
metal
dc kωε
=dε
G+
spk
//, sind dk k θ=dk
Localized surface plasmons (Particle plasmons)
Localized surface plasmons
p ( p )(“Plasmons in metal nanostructures”, Dissertation, University of Munich by Carsten Sonnichsen, 2001)
Lycurgus cup, 4th century(now at the British Museum London)
Focusing and guidance of light at nanometer length scales
(now at the British Museum, London).The colors originates from metal
nanoparticles embedded in the glass.At places, where light is transmittedthrough the glass it appears red, atl h li ht i tt dplaces where light is scattered near
the surface, the scattered light appearsgreenish.
Rayleigh Theory for metal = dipole surface-plasmon resonance of a metal nanoparticle
(“Plasmons in metal nanostructures”, Dissertation, University of Munich by Carsten Sonnichsen, 2001)
Localized surface plasmons
ε ε−
The polarizability α of the metal sphere is
30 0 04
2p
p
p R E Eε ε
πε ε αεε ε
= =+
The scattering and absorption cross-section are then
(particle, )pε ε(surrounding medium, )mε ε
유도-숙제!유도 숙제!
Scattering and absorption exhibit the plasmon resonance where,
For free particles in vacuum, resonance energies of 3.48 eV for silver (near UV) and 2.6 eV for gold (blue) are calculated.
Re ( ) 2 0pε ω ε⎡ ⎤ + =⎣ ⎦→
“Frohlich condition”
p g ( ) g ( )
When embedded in polarizable media, the resonance shifts towards lower energies (the red side of the visible spectrum).
Beyond the quasi-static approximation : Mie scattering TheoryFor particles of larger diameter (> 100 nm in visible), the phase of the driving field significantly changes over the particle volume.
Localized surface plasmons
Mie theory valid for larger particles than wavelength from smaller particles than the mean free-path of its oscillating electrons.Mie calculations for particle shapes other than spheres are not readily performed.
The spherical symmetry suggests the use of a multipole extension of the fields, here numbered by n. The Rayleigh-type plasmon resonance, discussed in the previous sections, corresponds to the dipole mode n = 1.y g yp p p p p
In the Mie theory, the scattering and extinction efficiencies are calculated by:1Re ( )p embedded
nn
ε ω ε +→ ⎡ ⎤⎡ ⎤ = −⎣ ⎦ ⎢ ⎥⎣ ⎦
“Frohlich condition”
(“Plasmons in metal nanostructures”, Dissertation, University of Munich by Carsten Sonnichsen, 2001)
For the first (n=1) TM mode of Mie’s formulation is
For a 60 nm gold nanosphere embedded in a medium with refractive index n = 1.5.(use of bulk dielectric functions (e.g. Johnson and Christy, 1972))
Localized surface plasmons
By the Rayleigh theory for ellipsoidal particles.
By the Mie theoryfor spherical particle
By the Mie theoryfor cross-sections
a/b = 1+3.6 (2.25 − Eres / eV)
The red-shift observed for increasing size is partly due to increased damping and to retardation effects.The broadening of the resonance is due to increasing radiation damping for larger nanospheres.
I fl f th f ti i d f th b ddi di Resonance energy for a 40 nm gold nanosphereInfluence of the refractive index of the embedding mediumon the resonance position and linewidth of the particle plasmon resonance of a 20 nm gold nanosphere. Calculated using the Mie theory.
Resonance energy for a 40 nm gold nanosphere embedded in water (n = 1.33) with increasing thickness d of a layer with refractive index n = 1.5.
Experimental measurement of particle plasmons
Scanning near field microscop (SNOM) SNOM images gold nanodisks
Localized surface plasmons
Scanning near-field microscopy(SNOM) SNOM images gold nanodisks
633 nm
SEM image 550 nm
Total internal reflection microscopy(TIRM)Dark-field microscopy in reflection
Dark-field microscopy in transmissionDark field microscopy in transmission
Interaction between particlesLocalized surface plasmons
an isolated sphere is symmetric, so the polarization direction doesn’t matter.
LONGITUDINAL: restoring force reduced by coupling to neighbor
p
restoring force reduced by coupling to neighbor Resonance shifts to lower frequency
TRANSVERSE: restoring force increased by coupling to neighbor
Resonance shifts to higher frequency
pair of silver nanospheres with 60 nm diameter
using metal nanorods and nanotipsNanofocusing of surface plasmons
Nanofocusing of surface plasmons
using metal nanorods and nanotips
M I Stockman “Nanofocusing of Optical Energy in Tapered Plasmonic Waveguides ” Phys Rev Lett 93 137404 (2004)
D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” PR B 76,035420 (2007)
M. I. Stockman, Nanofocusing of Optical Energy in Tapered Plasmonic Waveguides, Phys. Rev. Lett. 93, 137404 (2004)
Dispersion relation of metal nanorodsD E Chang A S Sørensen P R Hemmer and M D Lukin “Strong coupling of single emitters to surface plasmons ” PR B 76 035420 (2007)
Nanofocusing of surface plasmons
D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” PR B 76,035420 (2007)
For the special case a TM mode ( Hz = 0) with no winding m=0 (fundamental mode).
(TM mode with m = 0) : ai = 0 Eφ = 0, Hz = 0
Continuity of the remaining tangential field components Ez and Hφ at the boundary requires thatε1 (dielectric)
ρ φ
ε2 (metal)R
z
Setting the determinant of the above matrix equal to zero (det M=0) immediately yields the dispersion relation,
In the limit of where Im, Km are modified Bessel functions
When (nanoscale-radius wire)
Dispersion relation of metal nanotipsNanofocusing of surface plasmons
xy
εd
εm
For a thin, nanoscale-radius wire
k k0k nk=
For , the phase velocity / ( ) 0pv c n z= → and the group velocity [ ]/ ( ) / 0gv c d n dω ω= →p g
The time to reach the point R = 0 (or z = 0)
Intensity Energy density
In Summary
Permittivity of a metal2 2
2 2 2 2( ) 1 p pm i
ω ω γε ω ⎛ ⎞= − + ⎜ ⎟+ + ⎝ ⎠
y
2 2 2 2
2 21 /p
ωω γ ω γω ω
⎜ ⎟+ + ⎝ ⎠≈ −
Dispersion relations
1/ 2
d mkε εω ⎛ ⎞
= ⎜ ⎟SPPd m
kc ε ε
= ⎜ ⎟+⎝ ⎠
Type-A : low k
Type-A
Low frequency region (IR)- Low frequency region (IR)
- Weak field-confinement H. Won, APL 88, 011110 (2006).
- Most of energy is guided in clad
- Low propagation lossLow propagation loss
► clad sensitive applicationsclad sensitive applications
► SPP waveguides applications
Type-B : middle k
Type-B
- Visible-light frequency regionNano-hole
- Coupling of localized fieldand propagation field
Nano-hole
- Moderated field enhancement
► Sensors, display applications
► Extraordinary transmission of light
Type-B : SPR sensors
p-GaN (20nm 120nm)
Ag (20nm)Type-C : high k
Type-C n-GaN
p GaN (20nm, 120nm)
ΛQW
yp
- UV frequency region Light emission
- Strong field confinement
V l l itQW
- Very-low group velocity
► Nano-focusing, Nano-lithography
► SP-enhanced LEDs
21 1 ( )( )
R f i ρ ω= = ⋅p E
SE Rate :
0
( )( ) 2
f ρτ ω ε
p
Electric field strengthof half photon (vacuum fluctuation)
Photon DOS(Density of States)
Type-C : SP Nano Lithography
1. What is the surface plasmon (polaroton)?
Final comments
2. What is the dispersion relation of SPs?3. How can the SP modes be excited?4. What can we play with SPPs for nanophotonics?
Key notes
Ekmel Ozbay, Science, vol.311, pp.189-193 (13 Jan. 2006).Challenges of SPs
Some of the challenges that face plasmonics research in the coming years are(i) demonstrate optical frequency subwavelength metallic wired circuits
with a propagation loss that is comparable to conventional optical waveguides; (ii) develop highly efficient plasmonic organic and inorganic LEDs with tunable radiation properties;(iii) achieve active control of plasmonic signals by implementing electro-optic, all-optical,
and piezoelectric modulation and gain mechanisms to plasmonic structures;(iv) demonstrate 2D plasmonic optical components, including lenses and grating couplers,
h l i l d fib di l l i i ithat can couple single mode fiber directly to plasmonic circuits; (v) develop deep subwavelength plasmonic nanolithography over large surfaces.
N t l t t 07/14Next lecture at 07/14(06/23) Introduction: Micro- and nano-optics based on diffraction effect for next generation technologies(06/30) Guided-mode resonance (GMR) effect for filtering devices in LCD display panels(07/07) Surface-plasmons: A basic(07/07) Surface-plasmons: A basic(07/14) Surface-plasmon waveguides for biosensor applications(07/21) Efficient light emission from LED, OLED, and nanolasers by surface-plasmon resonance