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=


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=


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

=


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= +


( ) 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

⎟⎟⎠

⎞⎜⎜⎝

⎛+

≈


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


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)


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.


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))


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


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

Documents

3-A Basic of Surface Plasmons