Surface plasmon-polaritons (SPP)

Dispersion relation of surface plasmon-polaritons

Surface plasmon-polaritons (SPP)

Water drop

gravity

Surface tension

Can light flow down on the surface?

EM force

Frequency of light Viscosity of liquid

metal Light

Dielectric

Surface plasmon-polaritons?

Plasmonics: the next chip-scale technology



limit

e limit

 Micro-optics : Display and Sensors  Holography, LCD display, LEDs

 Nano-optics : Beyond the diffraction-limit  Surface-plasmon photonics

Plasmon = plasma wave

 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)

 Longitudinal surface wave in metal films

 Propagates along the interface anywhere from a few microns to several millimeters (long range plasmon) or can be extremely confined in nanostructures (localized plasmon)

Surface plasmons

(Gary Wiederrecht, Purdue University)

Note: SP is a TM wave!

표면 플라즈몬 vs. 표면 플라즈몬 폴라리톤

• 표면 플라즈몬 (Surface Plasmon, SP)

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

e^1z

e^2z

Metal Dielectric

– – – –

m

d

+ +

+ + + +

x z

y

6

TM pol.

• 표면 플라즈몬 폴라리톤 (Surface Plasmon Polariton, SPP)

– 표면 플라즈몬 (자유전자 진동)과 전자기파가 결합되어 있는 상태  SPP – 금속과 유전체의 경계면을 따라 진행

– 금속 표면에 수직한 TM 편광 특성 – 전송거리는 수십~수백 mm로 제한

Local field intensity depends on wavelength

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

Plasmons in the bulk oscillate at _p determined by the free electron density and effective mass

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

Confinement effects result in resonant SPP modes in nanoparticles

+ + +

- - -

+ - +

k

Plasma oscillation = density fluctuation of free electrons

0 2

 

m

drude Ne

p



0 2

3 1

 

m Ne

drude

particle



Bulk plasmons

Surface plasmon polaritons

Localized plasmons

m d sp

m d

k c

 



 

 

Dispersion relation for bulk plasmons

( ) k

  

• Dispersion relation:

Bulk plasmons

Let’s solve the curl equations for TE & TM modes with boundary conditions

TE mode TM mode

surface plasmon plaritons

Dispersion relation for surface plasmon polaritons

( ) (0, , 0), ( ) ( , 0, ) (0) (0)

(0) (0)

i yi i xi zi

yd ym

xd xm

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

xd xm

E z E E H z H

H H

E E

 





 

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

 



 

         

 

     

   

   

surface plasmon plaritons

TE modes :

^{( ) (0, , 0)}

( ) ( , 0, )

i yi

i xi zi

E z E

H z H H









0 0 0

0 0 0

0

xi zi xi

i i i i yi xi zi i yi

yi yi

zi

i i xi xi

yi xi

zi xi

H H H

H i E i E ik H i E

z x z

E E

E i H E i H i H

y z z

E E

i H ik E

x y

     

  



  

          

  

 

          

  

 

    

 

 

 

yi i0Hzi

2

2 2

2 ^yi ( 0 _{i xi}) _yi 0

E k k E

z 

   



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

0

( ) ^{x z}

yi zi ik x k z

xi xi i

E k

i H H z iA e e

z 



 

    



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

yd ym xd xm

E  H  H

& ( ) 0

d d d zd zm

A  A A k  k 

Curl equation

Boundary cond.

d m

0

A  A 

No surface modes exist for TE polarization !

TM modes :

^{( ) ( , 0, )}

( ) (0, , 0)

i xi zi

i yi

E z E E

H z H









surface plasmon plaritons

zm zd

m d

k k

 ^ 

xm xd

E  E

ym yd

H  H

xm m ym

zm

H E

k  

yd d

ym zd m

zm k H

k H



 ^

) ,

0 ,

(i_iE_xi i_iE_zi )

, 0 ,

(ik_ziH_yi ik_xiH_yi

xi i yi

zi

H E

k  

xd d yd

zd

H E

k  

• For any EM wave:

2

2 2 2

i x zi x xm xd

k k k , where k k k

c

 ^{ } 

        

m d x

m d

k c

 



 

 

SP Dispersion Relation

surface plasmon plaritons

TM modes :

1/ 2

' "

^{m d}

x x x

m d

k k ik

c

 



 

 

       

x-direction:

For a bound SP mode:

k_zi must be imaginary: _m + _d < 0

k’

_x

must be real: 

_m

< 0 So,

2 1/ 2

'

ⁱ

zi zi zi

m d

k k ik

c





 

 

        

z-direction:

2

2 2

zi i x

k k

c

 ^{ } 

   

 

' "

m m

i

m

    

'

m d

   

2 2

2 2

zi i x x i x i

k k i k k

c c c

  



^{ }



^{ }



^{ }

                

+ for z < 0 - for z > 0

TM modes :

surface plasmon plaritons

   

   

 

2 1

" 2 4

2

" 2 2

1

" 2 2

'

"

2 1

" 2 4

2 2 1

" 2 2

' '

) 2 (

) 2 (























  















 











  















 

d m e

e

d m m

d m x d

d m e

e m

d m x d

k c k c









































   

^{' 2 " 2 '}

2

, _{e m m d m}

where       

1/ 2

' " m d

x x x

m d

k k ik

c

 



 

 

       

metals, of

most in

and ,

,

0

^{' ' "}

'

m m

d m

m

   

   

,

 

^{' 2}

"

2 / 3

' '

"

2 / 1

' ' '

2

_m

m d

d d m x

d m

d x m

k c k c

























 

 





 

 

 





 

' "

m m

i

m

    

surface plasmon plaritons

 

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

1 2 1

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

2 , where

i x x

2( )

L k k

c

  



  



 

      

Propagation length

surface plasmon plaritons

Plot of the dispersion relation : For ideal free-electrons

2 2

1 )

( 

 

_m   ^p

m d x

m d

k c

  

 

 

• Plot of the dielectric constants:

• Plot of the dispersion relation:

d p sp

d m

ω

ω 







 















1 , k

, When

x

2 2 2 2

) 1

(

) (

p d

d p sp

x k c

k   





 





 



surface plasmon plaritons

Surface plasmon dispersion relation:

2 / 1

 

 





 

d m

d m

x c

k

 











_p

d p





 1

Re k

_x

real k_x real k_z

imaginary k_x real k_z

real k_x

imaginary k_z

d

ck

x



Bound modes Radiative modes

Quasi-bound modes

Dielectric:

_d

Metal: _m = _m^'+

_m^"

x z

'_m > 0)

_d < '_m < 0)

('_m < _d)

2 2 2 2

p c kx

  

Surface plasmon dispersion relation

2 1/ 2 i zi

m d

k i

c





 

 

    

surface plasmon plaritons

2 / 1

 

 





 

d m

d x m

k c











2 2 2 2

2 2 3 3

1 1

p p

m   i  

     

   

      

Dispersion relation for bulk and surface plasmons

d p sp

d p d

p d

p

m ω ω ω







 



 

 

 

 

















1 , 1 1

When

2 2

2 2

2 2



Cut-off frequency of SP

surface plasmon plaritons

2 2 2 2 ' "

2 2 3 3

1

p p

m m i m B   i  

   

    

   

        

Ag/air, Ag/glass

surface plasmon plaritons

Silver(Ag) dispersion

0 10 20 30 40 50 60

1 2 3 4 5

light line air

E [eV]

kx [um-1]

SP Ag/air

SP Ag/glass light line glass

15001200 900 600 300

0.1 1 10 100

 [nm]

L [um]

Gold(Au) dispersion

0 5 10 15 20 25 30 35 40

1 2 3 4 5

light line air

E [eV]

kx [um-1]

SP Au/air

SP Au/glass

light line glass

0.1 1 10 100

15001200 900 600 300

L [um]

 [nm]

Copper(Cu) dispersion

0 10 20 30 40 50 60

1 2 3 4 5

E [eV]

SP Cu/glass light line glass

kx [_um^-1]

SP Cu/air light line air

0.1 1 10 100

15001200 900 600 300

0.1 1 10 100

15001200 900 600 300

L [um]

[nm]

L [um]

[nm]

For noble metals : J&C measured constants

Very small SP wavelength

_vac=360 nm

X-ray wavelengths at optical frequencies

Ag SiO₂

surface plasmon plaritons

Penetration depth:

' 1

At large _x ( _{m d} ), _{z x} and z_i .

x

k k ik

      k Strong confinement on the interface

At low (k_x _m'  _d ), ^z _m^' in air : _{Larger Ez component}

x

E i

E   

Smaller E_z component

'

1 in metal :

z

x m

E i E ^ 

Goooooooood waveguide!

surface plasmon plaritons

1/ 2

' "

^{m d}

x x x

m d

k k ik

c

 



 

 

       

2 2 2

2 2 i

zi i x

m d

k k

c c

  

  

 

   

               

2 1/ 2 , ,

,

1 1 /

^{d m}

d m

m d

zd km

z k c

 

 

 

        

At  = 633 nm,

for silver (Ag) : z_m = 24 nm in Ag and z_d = 390 nm in air, for gold (Au) : z_m = 27 nm in Au and z_m = 342 nm in air.

( : , : - )

z x

E  iE air i metal i

surface plasmon plaritons

xm xd

E  E

ym yd

H  H

xm m ym

zmH E

k 

xi i yi

ziH E

k 

xd d yd

zdH E

k 

(ik H_{zi yi}, 0,ik H_{xi yi})  ( i_iE_xi, 0,i_iE_zi)

xi yi i zi

k H   E ^{k Hxm ym}  ^mEzm

xd yd d zd

k H   E ^m _zm ^d _zd

xm xd

E E

k k

 _ 

zm zd

m d

k k

 ^ 

x xm xd

k k  k

2

2 2 2

i x zi

k k k

c

 ^{ }

     

mEzm dEzd

  D_zm  D_zd

xi i yi

ziH E

k 

xi yi i zi

k H   E

ym yd

H  H

zi x

x zi

E k

E   k

xm xd

E  E

1/ 2

' " ^{m d}

x x x

m d

k k ik

c

 



 

 

     

2 2

2

zi x i

i

m d

k j k

c j c

 

 

 

      

  

     

2

zi m d

x i

E j

E

 



  

' 1/ 2

' 1/ 2

/ /

zd m d x

zm d m x

E j E

E j E

 

 

 

 

surface plasmon plaritons

Another representation of SP dispersion relation

2 ' "

( )

/

m m m

p p

k c

   



    



surface plasmon plaritons

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 N S N S

E  

H

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

Generalization :

Surface Electric Polaritons and Surface Magnetic Polaritons

1 2 2 1 1 2

2 2 0

2 1

' ' ( ' ' ' ' ) " "

' ' ', '

SEP       k O  

    

 

     

,1 ,2

1 1 2 2

, 2 2 0

2 1 1 2

' ( ' ' ' ' ) " "

, ,

' ' ' ' ' '

SEP SEP

i

SEP i      k O    

      

 

     

Dispersion Relation & Decay Constants

 ^{"/ ' , "/ '}  ^{(1,1) ,}

For

     {

J. Yoon, et al., Opt Exp 2005.

In Summary

1/ 2 d m

SPP

d m

k c

  

 

 

     

Dispersion relations

2 2

2 2 2 2

2 2

( ) 1

1 /

p p

m

p

 i  

      

 

          

 

Permittivity of a metal

Type-A

- Low frequency region (IR) - Weak field-confinement

- Most of energy is guided in clad - Low propagation loss

► clad sensitive applications

► SPP waveguides applications

Type-A : low k

H. Won, APL 88, 011110 (2006).

Type-B

- Visible-light frequency region - Coupling of localized field

and propagation field - Moderated field enhancement

► Sensors, display applications

► Extraordinary transmission of light

Nano-hole

Type-B : middle k

Type-C

- UV frequency region - Strong field confinement - Very-low group velocity

► Nano-focusing, Nano-lithography

► SP-enhanced LEDs

n-GaN

p-GaN (20nm, 120nm) Ag (20nm)



QW QW

2

0

1 1

( ) 2 ( )

R f i  

  

  p E 

 SE Rate :

Electric field strength

of half photon (vacuum fluctuation)

Photon DOS

(Density of States)

Type-C : high k

Light emission

Importance of understanding the dispersion relation : Negative group velocity

Re k

_x

SiO₂

Si₃N₄

1 _SiO2 p







4

1

_Si3_N p







SiO₂

Si₃N₄

Al

 0 dk d

 0 dk d

Importance of understanding the dispersion relation : Total external reflection

Slow Propagation, Anomalous Absorption, and Total External Reflection of Surface Plasmon Polaritons in Nanolayer Systems

Importance of understanding the dispersion relation :

Broadband slow and subwavelength light in air