Surface plasmons and its dispersion relation

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 ω

_p

determined by the free electron density and effective mass

+ + +

2

Bulk plasmons

- - -

k

⁰

2

ω ε

m

drude

Ne

p

=

Surface plasmon polaritons

Plasmons confined to surfaces that can interact with light to form propagating “surface plasmon polaritons (SPP)”

+ - +

ε ε ω

C fi t ff t lt i t SPP d

Localized plasmons

m d sp

m d

k c

ε ε ω

ε ε

= +

Confinement effects result in resonant SPP modes in nanoparticles

1 Ne

2

drude

3

0

1 ω ε

m Ne

drude

particle

=

Surface

Surface plasmons plasmons? ?

물방울 중력

표면장력 표면장력

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 metal

‹ Can be excited by light: photon-electron coupling (polariton)

‹ Thin metal films or metal nanoparticles

‹ Bound to the interface (exponentially decaying along the normal)

‹ Bound to the interface (exponentially decaying along the normal)

‹ Longitudinal surface wave in metal films

‹ Propagates along the interface anywhere from a few microns to

l illi t (l l ) b t l

several millimeters (long range plasmon) or can be extremely confined in nanostructures (localized plasmon)

Note: SP is a TM wave!

표면

표면 플라즈몬 플라즈몬 vs. vs. 표면 표면 플라즈몬 플라즈몬 폴라리톤 폴라리톤

•• 표면 표면 플라즈몬 플라즈몬 (Surface Plasmon, SP) (Surface Plasmon, SP)

Dielectric

εd

x z

y⊗

– 금속표면의 전하(자유전자) 진동 → 표면 플라즈마 – 양자화된 표면 플라즈마 진동 → 표면 플라즈몬

e^−κ1z

e^κ2z

– – – – + +

+ + + +

Metal

^εm

•• 표면 표면 플라즈몬 플라즈몬 폴라리톤 폴라리톤 (Surface Plasmon (Surface Plasmon Polariton Polariton, SPP) , SPP)

– 표면 플라즈몬 (자유전자 진동)과 전자기파가 결합되어 있는 상태 Æ 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 y p g

(small propagation constant k) (large propagation constant k)

(small propagation constant, k) (large propagation constant, k)

Note: Dielectric constants of optical materials

( ) 0 _r ( ) ( )

D ω = ε ε ω E ω (spatially local response of media)

ε ω _r ( ) ( ) : relative dielectric constant

ε 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

Æ Extended Drude model (Lorentz-Drude model) ( )

Dielectric constant (relative permittivity)

2

/ N

j j

ω

N α

∑ C

Lorentz model for dielectric (insulator)

2 2

/ ,

p j

j j

N i α ω

ω ω γω

= − −

1 3

( ) 1 ,

1

j j j

r

j j j

ε ω N

= + α

−

∑

∑ ^ω

^2j

^{= C m}

^j

Drude model for metal in free-electron region

2 2 2

2 2 2 3 2

( ) 1

^p

1

^{p p}

r

ω ω i ω γ

ε ω = − = − ^⎛ ⎜ ⎜ ^⎞ ⎟ ⎟ + ^⎛ ⎜ ⎜ ^⎞ ⎟ ⎟ Modified Drude model for metal in bound-electron region

2 2 2 3 2

r

( )

ω + i ωγ ^{⎜ ⎜} ⎝ ω + γ ^{⎟ ⎟} ⎠ ^{⎜ ⎜} ⎝ ω ωγ + ^{⎟ ⎟} ⎠

Modified Drude model for metal in bound electron region

2 2 2

2 2 2 3 2

( )

^{p p p}

r

i

i

ω ω ω γ

ε ω ε ε

ω ωγ ω γ ω ωγ

∞ ∞

⎛ ⎞ ⎛ ⎞

= − + = ⎜ ⎜ ⎝ − + ⎟ ⎟ ⎠ + ⎜ ⎜ ⎝ + ⎟ ⎟ ⎠

Extended Drude (Drude-Lorentz) model

2

⎛

2

⎞

⎛ ⎞ Δ Ω

γ _⎝ γ _{⎠ ⎝} γ _⎠

( )

2 2

2 2 2

( )

^{p L}

r

L L

i i

ω ε

ε ω ε

ω ωγ ω ω

∞

⎛ ⎞

⎛ ⎞ ⎜ Δ Ω ⎟

= − ⎜ ⎜ ⎝ + ⎟ ⎟ ⎜ ⎠ ⎝ − − Ω + Γ ⎟ ⎠

2

⎛

2

⎞ ⎛

2

⎞

Drude model for metals: Dielectric constant of free-electron plasma

2 2 2

2 2 2 3 2

( ) 1

^p

1

^{p p}

r

i

i

ω ω ω γ

ε ω ω ωγ ω γ ω ωγ

⎛ ⎞ ⎛ ⎞

= − + = − ⎜ ⎜ ⎝ + ⎟ ⎟ ⎠ + ⎜ ⎜ ⎝ + ⎟ ⎟ ⎠

2

2

2 2

0 0

0 p

c N e ω γ σ μ m

= = = ε

(1) f

2

0

N e :

static conductivity σ m

= γ

2 2 2 2

2 3 2 3

( ) 1 ω

^p

i ω

^p

1 ω

^p

i ω

^p

ε ω ≈ − ^⎛ ⎜ ⎜ ^⎞ ⎟ ⎟ + ^⎛ ⎜ ⎜ ^{⎞ ⎛} ⎟ ⎜ ⎟ ⎜ = − ^⎞ ⎟ ⎟ + ^⎛ ⎜ ⎜ ^⎞ ⎟ ⎟ (1) For an optical frequency, ω

_visible

>> γ

2 3 2 3

( ) 1 1

r

i / i

ε 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

ω

2

( )

0 r

τ

ε ω ⎯⎯⎯

γ^→∞_→

→

_r

( ) 1 ω

₂^p

ε ω ^{= −} ω

Dispersion relation for bulk plasmons

( ) k

• Dispersion relation:

( ) k

ω ω =

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 x

i i i i i

jk x

i 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

⎡ ⎤

∇× = − ⎣ = ⎦ = > = <

⎡ ⎤

∇× = + ⎣ = ⎦

G G G G

G G G G

TE mode TM mode

0

( , , ) ( )

i

μ

i

_⎣

i

y

i

_⎦

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

= =

=

G G

( ) ( , 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

= =

=

=

G G

(0) (0)

xd xm

H = H E

_xd

(0) = E

_xm

(0)

TE modes : ^{( ) (0, , 0)}

( ) ( 0 )

i yi

E z E

H H H

G =

G

_i

( ) (

_xi

, 0,

_zi

) H z = H H

0 0 0

xi zi xi

i i i i yi xi zi i yi

H H H

H i

ωε ε

E ^{∂ ∂} i

ωε ε

E ^∂ ik H i

ωε ε

E

∇× = − → − = − → − = −

∂ ∂ ∂

G G

0 0 0

y y

yi yi

zi

i i xi xi

yi xi

z x z

E E

E i H E i H i H

y z z

E E

H k E

ωμ ωμ ωμ

∂ ∂ ∂

∂ ∂

∇× = + → ∂ − = + → = −

∂ ∂ ∂

∂ ∂

G G

H

2

2 2

2 ^yi

(

0 _{i xi}

)

_yi

0

E k k E

z ε

∂ + − =

∂

^{yi xi} i 0H_zi ik E_xi x ^∂ y

ωμ

→ − = + →

∂ ∂ ^yi =i

ωμ

⁰H^zi

We want wave solutions propagating in x-direction, but confined to the interface with evanescent decay in z-direction.

[ ]

( )

^{jk xxi k zzi}

: ( ), ( ); Re 0

yi i zi

E z = A e e

^±

− i d = + i m = k >

0

( )

^{x z}

yi zi ik x k z

xi xi i

E k

i ωμ H H z iA e e

^±

∂ = − → = ±

Curl equation ₀

0

xi xi

( )

i

z μ

∂ ωμ

(0) E (0) & (0) (0)

yd ym xd xm

E = H = H

Curl equation

Boundary cond.

& ( ) 0

d d d zd zm

A = A A k + k = A

_d

= A

_m

= 0 0

A = A =

No surface modes exist for TE polarization !

TM modes : ^{E z}

ⁱ

^{( ) ( = E}

^xi

^{, 0, E}

^zi

⁾

G

TM modes :

( ) (0, , 0)

i yi

H z G = H

) ,

0 ,

( − i ωε

_i

E

_xi

− i ωε

_i

E

_zi

)

, 0 ,

( − ik

_zi

H

_yi

ik

_xi

H

_yi

xm m ym

zm

H E

k = ωε

xi i yi

zi

H E

k = ωε

xm xd

E = E

xd d yd

zd

H E

k = ωε

zm zd

k k

=

xm xd

ym yd

H = H

k k

m d

ε ε

yd d

ym zd m

zm

k H

k H

ε

ε ⁼

TM modes :

• For any EM wave:

2

2 2 2

i x zi x xm xd

k = ε ^{⎛ ⎞} ⎜ ⎟ ⎝ ⎠ ω = k + k , where k ≡ k = k

y

_{i x zi}

,

_{x xm xd}

⎜ ⎟ c

⎝ ⎠

SP Dispersion Relation

m d x

m d

k c

ε ε ω

ε ε

= +

TM modes :

1/ 2

' "

^{m d}

x x x

k k ik

c

ε ε ω

ε ε

⎛ ⎞

= + = ⎜ ⎝ + ⎟ ⎠

x-direction:

^{' "}

m m

i

m

ε = ε + ε

m d

c ⎝ ε + ε ⎠

2 1/ 2

'

ⁱ

i i i

k ₌ k ₊ ik ω ^⎛ ε ^⎞

= ± ⎜ ⎟

z-direction:

2

2 2

zi i x

k ⁼ ε ^{⎛ ⎞} _{⎜ ⎟} ω ⁻ k

⎝ ⎠

m m m

For a bound SP mode:

zi zi zi

m d

k k ik

c ε ε

+ ± ⎜ ⎝ + ⎟ ⎠

zi i x

⎜ ⎟ c

⎝ ⎠

k

_zi

must be imaginary: ε

_m

+ ε

_d

< 0

2 2

ω ω ω

⎛ ⎞

2 2

⎛ ⎞ ⎛ ⎞

zi i x x i x i

k k i k k

c c c

ω ω ω

ε ^{⎛ ⎞} ε ^{⎛ ⎞} ε ^{⎛ ⎞}

= ± ⎜ ⎟ ⎝ ⎠ − = ± − ⎜ ⎟ ⎝ ⎠ ⇒ > ⎜ ⎟ ⎝ ⎠

+ for z < 0

k’

_x

must be real: ε

_m

< 0

- for z > 0

So, '

m d

ε < − ε

1/ 2

' " m d

x x x

k k ik

c

ε ε ω

ε ε

⎛ ⎞

= + = ⎜ ⎝ + ⎟ ⎠

' "

m m

i

m

ε = ε + ε

( )

^{" 2 21}

4 2 2

1

⎡ + + ⎤

⎤

⎡

m d

c ⎝ ε + ε ⎠

^{m m m}

( ) ( )

1 2

" 2 2

' '

) 2 (

⎤

⎡

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ + +

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

+

= +

^{e e m d}

m d

m x d

k c ε ε ε ε

ε ε

ε

ε ω

( ) ( )

( )

2

" 2 4

2

" 2 2

1

" 2 2

'

"

) 2

( ⎥ ⎥ ⎥ ⎥

⎦

⎤

⎢ ⎢

⎢ ⎢

⎣

⎡

⎭ ⎬

⎫

⎩ ⎨

⎧ + +

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

+

= +

d m e

e

d m m

d m x d

k c

ε ε ε

ε

ε ε ε

ε ε

ε

ω where ^, ε

e²

= ( ) ( ) ε

m^{' 2}

+ ε

m^{" 2}

+ ε

d

ε

m^'

⎥⎦

⎢⎣ ⎩ ⎭

metals, of

most in

and ,

,

0

^{' ' "}

'

m m

d m

m

ε ε ε ε

ε < > >>

,

, ,

,

_{m d m m}

m

2 / ' 1

'

ω _⎜ ^⎛ ε ε

d

_⎟ ^⎞

"

2 / ' 3

' '

d m

d x m

k c

ε ε

ε ω

ε ε

ε ε ω

⎟ ⎞

⎜ ⎛

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

≈ +

( )

^{' 2}

'

"

2

_m

m d

d d m

x

c

k ε

ε ε

ε ε ε

ω ⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

≈ +

h l h f hi h h i i d /

Propagation length

3 / 2

The length after which the intensity decreases to 1/e :

⎛ ⎞

( )

^{" 1 "} ' ^{1 2' 3 / 2 1"}' 2

1 2 1

2 , where

i x x

2( )

L k k

c

ε ε ε

ω

ε ε ε

−

⎛ ⎞

= = ⎜ ⎝ + ⎟ ⎠

Plot of the dispersion relation : For ideal free-electrons

• Plot of the dielectric constants:

2 2

1 )

( ω

ω ω

ε

_m

= −

^p

ω ε ε

• Plot of the dispersion relation:

2 2

) ( ω ω ε

m d x

m d

k c

ω ε ε

ε ε

= + _{( 1 )}

^{2 2}

) (

p d

d p sp

x

k c

k ε ω ω

ε ω ω ω

− +

= −

=

p d

m

ω

ω ω

ε ε

=

≡

∞

→

⇒

−

→

•

k

, When

d

ω

sp

ω ≡ = + ε

∞

→

⇒ k , 1

_x

Surface plasmon dispersion relation:

Surface plasmon dispersion relation

2 / 1

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

= +

d d m

x

c

k ε ε

ε ε

ω

i^{2 1/ 2}

zi

m d

k c

ε ω

ε ε

⎛ ⎞

= ⎜ ⎝ + ⎟ ⎠

⎠

⎝

_m

+

_d

c ε ε

ω ^x real k

_x

ck Radiative modes

2 2 2 2

p

c k

x

ω = ω +

m d

⎝ ⎠

ω

_p

real k

_x

real k

_z

ε d _(ε'

_m

_{> 0)}

ω

imaginary k

_x

real k

_z

Quasi-bound modes (−ε

_d

< ε'

_m

< 0)

d p

ε ω

+ 1

Bound modes l k

Dielectric:

real k

_x

imaginary k

_z

Bound modes

Dielectric:

ε_d

Metal: ε ε ^'+

x

z (ε'

_m

< −ε

_d

)

Re k _x

Metal: ε_m = ε_m + ε_m^"

Dispersion relation for bulk and surface plasmons

2 / 1

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

= +

d m

d x m

k c

ε ε

ε ε ω

2 2 2 2

2 2 3 3

1 1

p p

m

ω τ i ω τ

ε ω τ ωτ ω τ

⎛ ⎞ ⎛ ⎞

= − ⎜ ⎜ ⎝ ⎝ + ⎟ ⎟ ⎠ ⎠ + ⎜ ⎜ ⎝ ⎝ + ⎟ ⎟ ⎠ ⎠

Cut-off frequency of SP

d p sp

d p d

p d

p

m

ω ω ω

ε ω ε

ε ω ω

ω ε ε ω

= + + ⇒

=

⇒

−

=

−

−

=

−

= 1 , 1 1

When

2 2

2 2

2 2

D

Ag/air, Ag/glass

2 2 2 2

' " p p

i ω τ i ω τ

ε

^m

= ε

^m

+ i ε

^m

= ^⎛ ⎜ ⎜ ε

^B

− 1

^{2 2}

^⎞ ⎟ ⎟ + i ^⎛ ⎜ ⎜

^{3 3}

^⎞ ⎟ ⎟

ε 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 60

1 2

E

1

15001200 900 600

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 [eV]

SP Au/glass light line glass

15001200 900

600 λ [nm]

0 5 10 15 20 25 30 35 40

kx [um-1]

0.1 1 10 100

1500

L [um]

Copper(Cu) dispersion

4 5

SP Cu/air

light line air 300300

2 3 4

E [eV]

SP Cu/glass light line glass

600 600 λ [nm]λ [nm]

0 10 20 30 40 50 60

1

kx [_um^-1] ^{0.1 1 10 100}

15001200 900

0.1 1 10 100

15001200 900

L [um]

L [um]

X-ray wavelengths at optical frequencies

Very small SP wavelength

λ

_vac

=360 nm

Ag

SiO

₂

Penetration depth

'

 1

'

1 2

At large (

x

), z

i

1 .

x

k ε → − ε ≈ k Strong concentration near the surface in both media.

( : , : - )

z x

E = ± iE air + i metal i

'

At low ( k

_x

ε

1

>> 1),

^'

_{Larger E}

_z

_component

1

in air :

z x

E i

E = − ε

1 i t l

E

z

i _{Smaller E}

z

component

' 1

in metal :

z x

E ⁼ i ε

Gooood waveguide!

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 N S N S

E K K

H

q q q q N S N S

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 Systems

of 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

Æt t l t l fl ti

mirror causing the total reflection of SPPs from it. Æ

total external reflection

Broadband slow and subwavelength light in air

Importance of understanding the dispersion relation : Negative group velocity

Negative group velocity

< 0 dk d ω

SiO

₂

Si N

1

_SiO2 p

ε ω 0 +

dk =

d ω ^dk

Re k _x

Si

₃

N

₄

4

1

_Si3_N p

ε ω +

x

SiO₂

Si₃N₄

Al

Excitation of surface plasmons p

The large k of SP needs specific configurations!

0

( ) ( )

m d

sp sp

k ₌ n k ₌ ε ω ε _{⎛ ⎞} ω

⎜ ⎟ ⎝ ⎠

2

1

^p

k k ω _{⎛ ⎞} ω

⎜ ⎟

^{sp sp 0}

ε ω ε

_m

( ) +

_d

^{⎜ ⎟} ⎝ ⎠ c

0

1

2^p

bp bp

k n k

ω c

= = − ⎛ ⎞ ⎜ ⎟ ⎝ ⎠

₂

( )

2 ^p

m

i

ε ω = ε

_∞

− ω

ω

real k_x

c k

ω = Radiative modes

( ' > 0)

2 2 2 2

p c kbp

ω =ω +

( )

2

m ^∞

ω + i ωγ

ω

_p

real k_z

imaginar k

0 d

ω k

= ε

Quasi-bound modes

(ε_m > 0)

d p

ε ω

+ 1

imaginary k_x real k_z

Quasi bound modes

(−ε_d < ε'_m < 0)

d

real k_x

imaginary k_z

Bound modes

Dielectric: ε_d

Mx l ^{' "}

z (ε'_m < −ε_d)

Re k

Metal: ε_m = ε_m^'+ ε_m^"

( )

_r( ) 2 2 2 i 3 2

γ ω ε ω ε i ε

ω ωγ ω γ ω ωγ

∞ ∞

← = − + =⎜⎜⎝ − + ⎟⎟⎠+ ⎜⎜⎝ + ⎟⎟⎠



Note: R does to zero at resonance when Γ = Γ

_{i rad}

//,d sp

k = k ± mG

ε d

//_{d d}

sin

_d

sin

k = k θ = ε ω θ

sp //, d

k = k ± mG

ε d metal

//,_{d d}

sin

_d

sin

k k

θ ε c θ

metal

d

c k ω ⁼ ε ε

_d

+ G

k

sp

//,_{d d}

sin

k = k θ

k

d

Localized surface plasmons (Particle 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 transmitted through the glass it appears red, at

l h li ht i tt d

places where light is scattered near the surface, the scattered light appears

greenish.

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 G G

The polarizability α of the metal sphere is

3

0 0 0

4 2

p p

p R ε ε E E

πε ε αε

ε ε

= =

+

G G

G

The scattering and absorption cross-section are then

(particle, )

εp ε

(surrounding medium, _m)

ε 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 ⎡ ⎣ ε ω

_p

( ) ⎤ + ⎦ 2 ε = 0

→

“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 Theory

For particles of larger diameter (> 100 nm in visible), the phase of the driving field significantly changes over the particle volume.

Mie theory Î valid for larger particles than wavelength from smaller particles than the mean free-path of its oscillating electrons.

Î 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:

Re _p( ) _embedded n 1

ε ω ε n⁺

→ ⎡⎣ ⎤ = −⎦ ⎡⎢⎣ ⎤⎥⎦

“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 theory for spherical particle By the Mie theory

for cross-sections

a/b = 1+3.6 (2.25 − E_res/ 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 nanosphere Influence of the refractive index of the embedding medium

on 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 dof 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 transmission Dark field microscopy in transmission

Interaction between particles

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 Æ Resonance shifts to lower frequency

TRANSVERSE:

restoring force increased by coupling to neighbor Æ Resonance shifts to higher frequency

pair of silver nanospheres with 60 nm diameter

using metal nanorods and nanotips

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 nanorods

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) 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 ( H

_z

= 0) with no winding m=0 (fundamental mode).

(TM mode with m = 0) : a_i= 0 Î E_φ= 0, H_z= 0

Continuity of the remaining tangential field components E_zand H_φat the boundary requires that

ε₁ (dielectric)

ρ ˆ φ ˆ

ε₂ (metal)

R

z ˆ

Setting the determinant of the above matrix equal to zero (detM=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 nanotips

y x

ε

_d

ε

_m

For a thin, nanoscale-radius wire Æ

k k₀ k_& =nk

For , the phase velocity v_pp =c n z/ ( )→0 and the group velocity vg_g =c d n^/

[

⁽ ω^{) /}dω

]

→⁰ The time to reach the point R = 0 (or z = 0)

Intensity Energy density

In Summary

Permittivity of a metal

2 2

2 2 2 2

( ) 1

^{p p}

m

ω i ω γ

ε ω = − + + + ^{⎛ ⎞} ⎜ ⎟ ⎝ ⎠

y

2 2 2 2

2 2

1

_p

/

ω

ω γ ω γ

ω ω

+ + ⎜ ⎟ ⎝ ⎠

≈ −

Dispersion relations

1/ 2

k ω ^⎛ ε ε d m ^⎞

= ⎜ ⎟

SPP

d m

k ^{= ⎜} c ⎝ 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 loss Low propagation loss

► clad sensitive applications clad sensitive applications

► SPP waveguides applications

Type-B : middle k

Type-B

- Visible-light frequency region

Nano-hole

- Coupling of localized field

and 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)

Λ

QW

yp

- UV frequency region

Light emission

- Strong field confinement

V l l it ^QW

- Very-low group velocity

► Nano-focusing, Nano-lithography

► SP-enhanced LEDs

1 1 2

( ) ( )

R = = f p E ⋅ i ρ ω

SE Rate :

0

( ) 2 f ρ ( )

τ ω ε ^{= p}

Electric field strength

of half photon (vacuum fluctuation)

Photon DOS

(Density of States)

Type-C : SP Nano Lithography

1. What is the surface plasmon (polaroton)?

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 i

that can couple single mode fiber directly to plasmonic circuits;

(v) develop deep subwavelength plasmonic nanolithography over large surfaces.

N t l t t 07/14 Next 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